2015 6 14homepage.fudan.edu.cn/jgying/files/2011/06...0 w o ~ E X t − X 0 = Z t 0 A(s,X s)ds+ Z t...

$: 2015 6 14homepage.fudan.edu.cn/jgying/files/2011/06...0 w o ~ E X t − X 0 = Z t 0 A(s,X s)ds+ Z t 0 σ(s,X s)dW s. K @. C, 2 J | X c F Z σ(t,X t)dW t. (o ~ , Q v K) d < V X \. K.$
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iC:|oTUn 15 678JSE o!3!X. $!3OVo.xhtTh~a:, ^SuSe;a:, 92a:, &ga:p:"o76XoSW X7S, $.xZ#swL, L$-, ?e1po,$ww Ito t~SQ4oO$J`kOw.oYLOoa.O6LtE.bkLx (Q_P`oo!s0) otE, Q.(dXt

dt= A(t, Xt) + σ(t, Xt)Wt .+o4*|e\) Wt _)Fk!S*, bwwfU, _))eo!w.ko. IoLxQSesf Brown _Tw$-, O~ Brown .>gN^4oQOko?:$-, TwO8, Q.k': R. Brown 6K7%>_o, Brown o?:.(Q*.h A. Einstein OH"o 19056 Annalen der Physik

17, 549-560 ow “On the motion of small particles suspended in liquids at

rest required by the molecular-kinetic theory of heat”o4kMo. ae!, 1900 6, L. Bachelier v$wPow “Theorie de la speculation” o'*, Q4&fw Brown _$3FK2;, Po-T" Ann. Sci.

Ecole Norm. sup., 17 (1900), 21-86, |w.tOH Brown _fo1o.v , Brown oO6?:zC Brown oRmo70.19236hS?: N. WienerQ “Differential space”, J. Math. Phys.

2, 132-1744pCo, kJ Brown o(2o-TS)=<o03.h P. Levy 6+Æo 30 l 40 6 O()o, Q4 Levy " w Brown oeOkRm.QQto, ZWQJo!ok? Wt d<VX)\, G kJoLtE9)'CwdXt = A(t, Xt)dt + σ(t, Xt)dWt

iiQ3.0wo~EXt −X0 =

∫ t

0

A(s,Xs)ds+

∫ t

0

σ(s,Xs)dWs .K.C, 2J|X F∫

σ(t, Xt)dWt.(o~, QvK)vd<VX\. K. Ito 1940 6 6XwBrown kw~e, ZWkwLtEoe, NwQQ?:uo_f, Ito eoQ6oQwtso_f.1eu. 7LIto z%kyn Fields , hPo"Lho4_f Ito t~eo,M;Tyn Nobel o9: (j 1990 6o H. Markowitz, W. Sharpe

and M. Miller, 1997 6o Robert Merton, and M. Scholes) +z;$.:O$,o1w,$L$-Q1LV1puo_f.,moz\=wJoL$$, b,$B1U[?:, t^, X, z, ?:eS PDE po:Kk`Yo3O(. .(ao_rNe:, P 2014 6T1L!Iw:, )Mw`. KJ.(&ga:>oH, e!.(>Wao#F. [5 University of Oxford, UK_0 &ga:

94ux K 1

1.1 Q5- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Lus4p0 . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 ZX? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 a?:Ps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5 s._ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23u 6l 25

2.1 d!Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 s ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 o7!Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 s._ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47uD Brown z 54

3.1 LUEsfxU o+5 . . . . . . . . . . . . . . . 54

3.2 y\us Brown . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Brown o . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Brown o03 . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5 Brown o8 . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.6 s._ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83uU Ito 89

4.1 \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 9xL~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3 ÆX8UE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4 o7YoL~ . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.4.1 Joo7NQ~YoL~ . . . . . . . . . . . . . . 109

4.4.2 Kunita-Watanabe )p( . . . . . . . . . . . . . . . . . . 114

iii

HF iv

4.4.3 ^.o7A,Y . . . . . . . . . . . . . . . . . . . . . . 115

4.4.4 ^.o7Y . . . . . . . . . . . . . . . . . . . . . . . 117

4.5 s._ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119ui Ito M< 122

5.1 Ito >( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2 Ito >(o_f . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2.1 L)? . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2.2 Levy o Brown YRj . . . . . . . . . . . . . . . . . 131

5.2.3 o7A,Y. Brown o!r . . . . . . . . . . . 133

5.2.4 Girsanov |e . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2.5 Y")|e . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.3 Po|+> (*) . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.4 s._ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147u2 Wbj 150

6.1 \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.2 LtEoO$jJ . . . . . . . . . . . . . . . . . . . . . . 152

6.2.1 0 Gauss ^ . . . . . . . . . . . . . . . . . . . . . . . 152

6.2.2 ^ Brown . . . . . . . . . . . . . . . . . . . . . . 154

6.2.3 Cameron-Martin >( . . . . . . . . . . . . . . . . . . . . 156

6.3 LtE||e . . . . . . . . . . . . . . . . . . . . . . . 157

6.4 _.: \vO0 . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.5 Ys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163f fn 168Y 169

vy ^ÆLL=J.)Jo Brown o~e, .h~?: K.ItokU_o, Q%OSZ#, .ZwQ)2fd<o~e_.0. ?:o+/g, LeQ#.`&K o, Q14Kk9J 3o_f, h.JC Oze.Ye, O2JotB.kwNo5,QSC, k6tB, .! OLo.?+Æ%,ho Riemann, Lebesguep:o;T, 4Q9Jo5~eiop, ?:e.w 8O6_oJ%qO6 NQyQ0o5, e.wYIo5M'QSkO6!Fo~e, ZW5s~CwoO6|Poe, 5o-KCw?:e4oO6s ?-nO-ep9Jo-. lÆ+Æ6 , Kolmogorov+TwO6Z8o5, &n+kO6? C;o>e^Æ , Lu.Q5X?, LuoPsYY.~. IQS+o|PO, B*JA+K.,mo. PsR9L+;+#X? (Lebesgue 5Q~e) o|$$. O4, fs (a?k" ) uH5+o|+8,k$8q" . $"ZtBL4.<9Jo. o13O E.B.Dynkin o"> [7].

§1.1 #h:4, _ R ")#?, Q ")ke?, Z ") ?, N ")Kv?, ! + ")'yHp^, R+ ")'#?, QP F. ;_oJÆJ.xz 1.1.1 U_ Ω oO6UJ F Bw. Ω o σ- ? (z σ-u), TQo(JQy%J.TH", σ- ?O|Vk ∅, Ω wyHbok$, k%Qy$pJ.o. _ oO6 σ- ?d<NTw_ oQ5-, Ω

1

:TZ X6[K 2Q oO6 σ- ? F PCo= (Ω,F ) Bw.O6Q5U, Q4o_BwQ5. vJ 2Ω s ∅,Ω . Ω o σ- ?, Q. Ω oNσ- ?.h|X)/H" Ω V6 σ- ?o$K.O6 σ- ?, A .Ω O6J , f C(A ) ") Ω V A wJo σ- ?p^, Zw2Ω ∈ C(A ), G C(A ) .Uo,

σ(A ) :=⋂

F∈C(A )

F .

(:4, Z := T1-.) σ(A ) . Ω o σ- ?, Q.hyaOvOt|o σ- ? F : (i) F ⊃ A ; (ii) F ′ . σ- ?b F ′ ⊃ A , F ′ ⊃ F . ZWB σ(A ) .V A oQ! σ- ?zh A Co σ- ?..Ca?9Jo σ- ?o<f.T Ω .O6nOU, QOkMPCo Co σ- ?Bw.Ω o Borel ?, w B(Ω), ZwMo(., GQK.p^Co σ- ?. o Euclid U, B(Rn) z Bn . n- x Euclid URn o Borel ?. Ω, Ω′ .t6U_, f . Ω l Ω′ oO6e. A′ ⊂ Ω′, |X

f−1(A′) := ω ∈ Ω : f(ω) ∈ A′.$/ 1.1.1 H"y03:

(1) f−1(Ω′) = Ω, f−1(∅) = ∅;

(2) f−1[(A′)c] = [f−1(A′)]c;

(3) ^Jy A′i, f−1 (

⋃

iA′i) =

⋃

i f−1(A′

i). A ′ . Ω′ oO6J , f−1(A ′) := f−1(A′) : A′ ∈ A

′.h >03, T A ′ . Ω′ σ- ?, f−1(A ′) . Ω σ- ?.

:TZ X6[K 3$/ 1.1.2 Ω, Ω′ .t6U_, f . Ω l Ω′ oO6e, A ′ . Ω′oO6J . " : σ[f−1(A ′)] = f−1[σ(A ′)].xz 1.1.2 Q5U (Ω,F ) lQ5U (Ω′,F ′) oe f Bw.Q5o (z ts, F/F ′- Q5o), T f−1(F ′) ⊂ F . YQ5U (Ω,F ) l (R,B)oQ5eBwQ5X? (z F - Q5X?). Y (Rn,Bn) l (Rm,Bm) oQ5eBw Borel Q5e.QfoX?.)0X?. A ⊂ Ω, |X A o)0X?1A(ω) =

1, ω ∈ A,

0, ω 6∈ A.|ok6)0X?o0P_BwfX?. v)0X?Q5ib4i_Q5. Ω ^'X? f , fn(ω) :=

n2n∑

k=1

k − 1

2n1[ k−1

2n, k2n

)(f(ω)) + n1[n,+∞)(f(ω))., fn .)0X?ok0P_, Jo n vÆb. f . rHmC, 'X?QS'Cw'fX?vÆyo.BO6J .π- ', TQk$. BO6J .λ- m, TQVk ∅,Ω bo(Js)$Qy%J. v, ?iv.π- , σ- ?. λ- Æ, %). TNMV6 λ- Æo$. λ- Æ, ZW Ω o^J A , vO\O6V A oQ! λ- Æ, w δ(A ), K FsBwh A Co λ- Æ.x) 1.1.1 F0.O6 π- , δ(F0).O6 σ- ?,ZW σ(F0) = δ(F0).3'. h|X, 43H" δ(F0) k$J. l A ∈ δ(F0), |X

κ[A] := B ∈ δ(F0) : A ∩ B ∈ δ(F0).H" κ[A] .O6 λ- Æ. ,# , +2" κ[A] (1) (J; (2) )$yoQy%J. twoH"~T .

:TZ X6[K 4Z F0 . π- , G A ∈ F0 VG κ[A] ⊃ F0 κ[A] ⊃ δ(F0). V|Gi A ∈ δ(F0) !, κ[A] ⊃ F0. ZW κ[A] ⊃ δ(F0), δ(F0) 4yHk$J.hWkM`kfofz |e. fz |ek8">(, 9Q4o+5&foO6. Ω oO6X?U H Bw.O6fz , TQ'vÆy, tasC.)Q4^Jo n wwvÆb.k/o'X?y fn k limn fn ∈ H ; C H VO6J A.)QVOk A 4Ok_o)0X?.x) 1.1.2 H . Ω oO6fz b 1 ∈ H . H Vk π- F0, H V Ω ok/o σ(F0)- Q5X?p^.3'. F . H 4Ok)0X?_o_PCoJ . F . λ- Æ,ZwQV π- F0, GV σ(F0). 'Q5X?QS")wfvÆX?yo, ZWfz H V'k/o σ(F0)- Q5X?p^, h003, kM-.xz 1.1.3 (Ω,F ) .O6Q5U, B F oO6'#(OX (Qlf(o) X? µ w (Ω,F ) o5, T(1) µ(∅) = 0;

(2) An . F 4oO6f)$oy, µ (⋃

nAn) =∑

n µ(An). 603Bw5oQyQ0.!, B (Ω,F , µ) .5U. T A ∈ F , µ(A) = 0, B ⊂ A VG B ∈F , B (Ω,F , µ) .p5U. i µ(Ω) < ∞ !, B µ .k5; iµ(Ω) = 1 !, B µ w+5; i\y An ⊂ F N ⋃

nAn = Ω sµ(An) <∞ !, B µ . σ- k5.$/ 1.1.3 ^5UoVXQSpl. (Ω,F , µ) .5U.

N := N ⊂ Ω : \ N ′ ∈ F &n N ⊂ N ′, µ(N ′) = 0,

Fµ := σ(F ∪ N ).

:TZ X6[K 5

N 4o_d<Bw µ- z5. , F µ = A ∪ N : A ∈ F , N ∈ N . I µ KsAnl F µ : µ(A ∪ N) := µ(A). 2JH"|XSX. " : (Ω,F µ, µ) .O6p5U, Bw.z5Uopl.ZWkJ, MQS5U.po. O65.5Q|o. O5o.Y Lebesgue5oIM_o. YO6- σ ?f`o ?M', Ω oUJ F0 Bw ?, TQVU, bpJk%J. K.C, σ ?oQy%wk%. KvNU_8),h# ?o-QS<f.$/ 1.1.4 Fn . σ- ?bJo n vÆ, " ⋃

n Fn .O6 ?. ? F0 o'X? µ Bw.5, T µ(∅) = 0 b NQyQ0. hgZw F0 Qy%), OS NQyQ0oVD.^ F04of)$y An, T ⋃

nAn ∈ F0, kµ

(

⋃

n

An

)

=∑

n

µ(An).6QyQ0QS.wkQ0XQyQ0. F0 o5 µ Bw. σ- ko, T\ An ∈ F0 &n µ(An) <∞ b Ω =⋃

nAn.x) 1.1.3 µ . ? F0 oO6 σ- k5. \ (Ω, σ(F0)) vOo5 µ′ &nQs µ F0 OI, µ \vO^.6|eoO69J_f. Lebesgue 5o.$/ 1.1.5 " : R oRMljok%op^ A .O6 ?, QMQS'C)$ok%. |Xm

(

n⋃

i=1

(ai, bi]

)

=n∑

i=1

(bi − ai)." m . A o5.6 m B := σ(A ) o^Bw Lebesgue 5, B Bw. Borel σ- ?, Q4o_Bw Borel . # 5U (R,B, m) pldnl

:TZ X6[K 6o5U(R,L , m)/.!o Lebesgue 5U, L 4o_Bw Lebesgue Q5. QS" , L )2V R oOkJ, QKVw B.ok5, kO6:YfoQyQ0pa, ~8H".$/ 1.1.6 µ. Ωo ? F0 oO6'okQoX?b µ(Ω) <

∞. T^vioUoy An ⊂ F0 k µ(An) ↓ 0, " : µ . F0 o5.05o+8, .5YO6Ul|O6Uo;F, Q+4.`9Jo. µ . (Ω,F ) oO65, Q5e f µ ew (Ω′,F ′) o5 µf−1 (zw f(µ)):

µf−1(A′) := µ(ξ−1(A′)), A′ ∈ F′.Bw µ ξ o5, zC f 5 µ kGl (Ω′,F ′) .$/ 1.1.7 H" µf−1 .O65. (Ω,F , µ) .8|o5U, (R,B) .OX#(Q5U (3O#4o|X). (Ω,F ) oQ5X?.) (Ω,F ) l (R,B) oQ5e. Ω oO6Q5X? f Bw.fo, T f o(u.k, \f)eo<? a1, · · · , an ∈ R, &n

f(ω) =n∑

i=1

ai1f=ai(ω), ω ∈ Ω.)X6, W"^(.vOo. !ilkVX!, |Xµ[f ] :=

n∑

i=1

aiµ(f = ai).AVM.| 0 · ∞ = 0. f S+ ") Ω o'fX?p^. )/H", e µ : S+ → [0,+∞] .fzob0o. Ω oV'Q5X? f|Xµ[f ] := supµ(g) : 0 ≤ g ≤ f, g ∈ S+.Bw. f Jo µ o~.

:TZ X6[K 7xz 1.1.4 f . Ω Q5X?, f+, f− #. f o!,',, iµ[f+], µ[f−] t.ÆkO6.k!, B f Jo µ o~\, b f Joµ o~w µ[f ] := µ[f+]− µ[f−]; i µ[f+], µ[f−] tk!, B f Joµ .Q~o.v, *Q5X? f O6 µ-z5 o()*~o(, |oTf 'bO6 µ- !5 po +∞, µ[f ] = +∞. Jo~, QP<foZqk ∫

Ωf(ω)µ(dω),

∫

Ωfdµ, 〈f, µ〉p.|o f Q5 A ∈ F o~|Xw µ[f1A], <'w ∫

Afdµ. k! , _Kt~ofo")~o.(ot"Z d Kf_")5, QSl5>, µf−1(dx) = µ(f ∈ dx)e.w^Q5o A, µf−1(A) = µ(f ∈ A) Ck.Y|X&)QSkMo03.~ofz0: T f1 ≤ f2 .'Q5X?, , µ[f1] ≤ µ[f2]. O~ofz4p|e.~e4Q|oK.Q9Jo|e.x) 1.1.4 fn .O6vÆ4po f o'Q5X?6y,

µ[f ] =↑ limnµ[fn],g ↑ lim ").O6vÆ.3'. hfz0, µ[fn] .O6fzÆo?y, b

µ[f ] ≥ limnµ[fn].%, lO6 f V1o'fX? g 0 < λ < 1, An := fn ≥ λg.Zw f > 0 , k f > λg, G An ↑ Ω.

µ[fn] ≥ µ[fn1An] ≥ λµ[g1An

].Z g .fo, Glimnµ[fn] ≥ lim

nλµ[g1An

] = λµ[g],QdopZif µ S+ o00o70. Z λ .Vo, kMlimnµ[fn] ≥ µ[g].

:TZ X6[K 8h µ[f ] o|XkM limn µ[fn] ≥ µ[f ]. pCw" .O6z5oCko03BweQQCk. , 5U (Ω,F , µ) Q5X? f1 s f2 BweQQp, .)QO6 µ- z5op, w f1 = f2, µ-a.e. t!, 'w f1 = f2 a.e. z f1 = f2. |eofz0QSfeQQfz _. |o, ^'Q5X? f QS")wO6fz o'fQ5X?6yo, f =↑ lim

n→∞

(

n2n∑

k=1

k − 1

2n1k−1

2n≤f< k

2n + n1f≥n

)

,ZWhfz4p|eQSkMQ~.'fQ5X?6yo~ofz , Z ~o03d<+2'fQ5X?H". ffz4p|eTH"^'Q5X? f, g k µ[f + g] = µ[f ] + µ[g]. vdTf, g Q~, , f + g KQ~bZw (f + g)+ + f− + g− = (f + g)− + f+ + g+,GhQ~0s~|XkM

µ[f + g] = µ[f ] + µ[g].QK/H"Qfz0sQPO$f03. if6DKT" oJo~ou_r>(.x) 1.1.5 f .Q5X?, φ . R Borel Q5X?, φ µf−1 Q~ib4i φf µ Q~, b!kµ[φf ] = µf−1[φ].3'. >(v φ = 1A, A ∈ B Ck, vd_ffz |e. " C , _ffz |e, " Fo>(# +2J)0X?" QSw, 6.+Q|o%O, 3%.0o Fatou\e Lebesgue V14p|e.4Q9Jo;F%O. 0 Fatou \e.x) 1.1.6 (Fatou) fn .'Q5X?6y,

µ[limnfn] ≤ limnµ[fn].

:TZ X6[K 93'. gn := infk≥n fk, gn .O6fzÆo'Q5X?6ybgn ≤ fn, hfz4p|e,

µ[limnfn] = µ[limngn] = lim

nµ[gn] = limnµ[gn] ≤ limnµ[fn].+ 1.1.1 m . [0, 1] Lebesgue 5. |X

f2n−1 := 1[0, 12), f2n := 1( 1

2,1], n ≥ 1. limfn = 0 limm[fn] =

12.. Lebesgue V14p|e.x) 1.1.7 (Lebesgue) fn . Ω oQ5X?6y, T^ ω ∈ Ω,

fn(ω)4pb\O6Jo µQ~oQ5X? g N |fn| ≤ g, µ[lim fn] =

limµ[fn].3'. f := limn fn. Zw g − fn s g + fn .'Q5X?6y, if Fatou \e,

µ[g + f ] = µ[lim(g + fn)] ≤ limµ[g + fn] = µ[g] + limµ[fn].Z g Q~, G µ[f ] ≤ limµ[fn]. |O, g− fn _f Fatou \e, kMlimµ[fn] ≤ µ[f ], hWn µ[f ] = limµ[fn].k5UZ#.+U , k/Q5X?M.Q~o, Z kok/4p|e.`6 1.1.1 (Ω,F , µ) .O6k5U, fn . Ω oQ5X?6y,T^ ω ∈ Ω, fn(ω) 4pb\O6<? M &n |fn| ≤M ,

µ[limnfn] = lim

nµ[fn].05oKo7 Radon-Nikodym k?. µ.Q5U (Ω,F ) 5, f .'Q5X?, |X ν(A) := µ[f ·1A],

A ∈ F . ν K. (Ω,F ) 5, ν w f · µ. A ∈ F , 5 1A · µ # . µ A o1. 8t65 µ, ν, T\O6Q5X? f &n

:TZ X6[K 10

ν = f · µ, ,C ν Jo µ . (Radon-Nikodym) Qko, %B f . ν Jo µ o Radon-Nikodym k?, <'w dν

dµ, vTQk, k? µ- eQQpoVXvO. |oB ν Jo µ Ko7, T A ∈ F b µ(A) = 0VG ν(A) = 0. w ν ≪ µ.$/ 1.1.8 µ, ν . (Ω,F ) t65, ν .ko. " : ν ≪ µ ib4i^ ε > 0, \ δ > 0, &n µ(A) < δ VG ν(A) < ε.vT ν Jo µ . Radon-Nikodym Qko, ν ≪ µ, hOs%), RY5s?5, P.fKo7o, hv?5)Q2JoRY5Qk. h µ .k5!, 5#Ck.x) 1.1.8 (Radon-Nikodym) µ s ν #.Q5U (Ω,F ) O6k5k"Z5. T ν ≪ µ, ν Jo µ Qk.AV |eo" 4# " wTkKo70, , ν =

g · µ + λ, Q4 µ(Dc) = λ(D) = 0. !5 µ s λ BwfRY, >.Bw Lebesgue ..+Qd0D~5Us Fubini |e. (Ω1,F1, µ1), (Ω2,F2, µ2).t6 σ- k5U,

F1 ⊗ F2 := A1 ×A2 : A1 ∈ F1, A2 ∈ F2, F1 ⊗F2 .D~U Ω1 ×Ω2 oO6 π- . F1 ×F2 := σ(F1 ⊗F2),Bw.D~ σ- ?.l Ω1 × Ω2 'Q5X? f , TH"ω1 7→

∫

Ω2

f(ω1, ω2)µ2(dω2).Q5X?, hfz |e, i µ1, µ2 . σ- k!, ^'Q5X? fk∫

Ω2

µ2(dω2)

∫

Ω1

f(ω1, ω2)µ1(dω1) =

∫

Ω1

µ1(dω1)

∫

Ω2

f(ω1, ω2)µ2(dω2). (1.1.1)

:TZ X6[K 11 (# |XwO6D~U oD~5 µ1×µ2, bit. σ- k!, D~QS$r. !ko Fubini o~6$r>(.x) 1.1.9 (Fubini) (Ω1,F1, µ1) (Ω2,F2, µ2) .t6 σ- k5U,

f . (Ω1 × Ω2,F1 × F2) oQ5X?. T f .'ozQ~o, Æ9~pobX~∫

Ω1×Ω2

fdµ1 × µ2 =

∫

Ω1

µ1(dω1)

∫

Ω2

f(ω1, ω2)µ2(dω2)

=

∫

Ω2

µ2(dω2)

∫

Ω1

f(ω1, ω2)µ1(dω1).+ 1.1.2 Fubini |ea4o σ- k0.2o. I = [0, 1], µ1, µ2 #. I o Lebesgue 5s?5. f(x, y) := 1x=y, x, y ∈ I.,TJ

∫

I

dµ1

∫

I

f dµ2 = 1, ∫

I

dµ2

∫

I

f dµ1 = 0.ZW Fubini |e)Ck, zZ.?5). σ- ko.

§1.2 Xb0Q/sxz 1.2.1 O6yP (Ω,F ,P) Bw.O6+U, T Ω .O6U_, F . Ω o σ- ?b P . (Ω,F ) oO6+5. ! , KB Ω.IU, F w,u, F 4oyHw,, P .+.6wO Euclid U, U (Rn,Bn) oO6+5Bw.O6 n- x*. O6 1 x*Bw*.o!0-T# OkuU o+5Ck.x) 1.2.1 µ . Rd oO6*, ^ Borel B, kµ(B) = supµ(F ) : F ⊂ B,F = infµ(G) : G ⊃ B,G M. (1.2.1)

:TZ X6[K 123'. F .&n (Cko Borelp^. Q.O6VMp^o σ ?.,# , Qo(Jo.vo, wH"Qy%J. 6 Fk%.vo. l B =⋃

Bn, Q4 Bn ∈ F )$. ^ ε > 0 n ≥ 1, \Mo Gn so Fn &n, Gn ⊃ Bn ⊃ Fn b µ(Gn \ Fn) < ε/2n+1.|o\ n′ &n µ(

⋃

k>n′ Bk) < ε/2. G =⋃

Gn, F =⋃

n≤n′ Fn. , GM, F , G ⊃ B ⊃ F , bµ(G \ F ) ≤

∑

n≤1

µ(Gn \ Fn) +∑

n>n′

µ(Bn) < ε.ZW B ∈ F ." F VwOkM. ,# , G .M, Fn = x ∈ Rd :

d(x,Gc) > 1/n, n ≥ 1, Q4 d .u. k Fn ↑ G, Y µ(Fn) ↑ µ(G), G ∈ F .Os, +Ud<.Io_ |X, )J. ZW4dZ, \Lu, +gl Euclid U X. 8|+U(Ω,F ,P). O6 n- xLu.) (Ω,F ) l (Rn,Bn) oO6Q5e, O6 1- xLuBwLu. AVsQ5X?)eo., Lu+lk(. ! σ(ξ) := ξ−1(B), Bw.h ξ Co σ- ?, vQ. Ω &n ξ CwLuoQ! σ- ?, FsT ξi : i ∈ I .OOLu,, Ω &nPQ5oQ! σ- ?Bw.hPCo σ- ?, wσ(ξi : i ∈ I). # , σ(ξi : i ∈ I) peoJ ⋃

i∈I σ(ξi) OCo σ ?,TA0 := A1 ∩ · · · ∩An : n ≥ 1, Ak ∈ σ(ξik), ik ∈ I, 1 ≤ k ≤ n,, A0 . π- , b σ(ξi : i ∈ I) = σ(A0). ξ . Ω o n- xLu, µξ (z ξ(P), Pξ−1) .+ P ξ o5, Q. Rn oO6*, Bw ξ o (n_) *, _o*X?Bw. ξ o*X?. 8|* µ, T\+U (Ω,F ,P) Q Lu ξ &n µξ = µ, B ξ . µ (+U (Ω,F ,P) ) oO6#. v^LuQS#, +U (Rn,B(Rn), µ) oapeo*Y

:TZ X6[K 13. µ, 6#Bw.x#, AVQQ5UsLus µ J. t6LuBw.e*o, TQkeo*z*X?. O6*d<QSk)eo#. )4.)#we+U o)eLu, bKQ#pp)eo+U .TO6Lu ξ Jo+5 P Q~, Q~ P[ξ] d<Bw. ξo?:PszL(, <e.w ξ Ω oNL, +4Rfz Ms'w E[ξ]. |o ξ A ∈ F o~K<w E[ξ;A]. hu_r>(nE[ξ] =

∫

R

xµξ(dx).AV"Z P s E k3j#, P Mfo,o+, E foLuoPs, z P(A) = E[1A]. 5O+s, f . R Borel- Q5X?, f ξK.Lu, TQ~, hu_r>(, |e 1.1.5, kE[f(ξ)] =

∫

R

f(x)µξ(dx) =

∫ +∞

−∞f(x)dFξ(x). (l. Lebesgue-Stieltjes VXo~, i f o7!. Riemann-StieltjesVXo.)G0Lu6yo4p0, L4<kf, Qkfo.fo\e.x) 1.2.2 (Borel-Cantelli) An .,y.

(1) ∑∞n=1 P(An) <∞, P(limnAn) = 0;

(2) An .k,yb ∑

n P(An) = ∞, P(limnAn) = 1.3'. (1) 6 P(limnAn) = limn P(⋃

k≥nAk), P(⋃

k≥n

Ak) ≤∑

k≥n

P(Ak) −→ 0,Zw? ∑∞n=1 P(An) 4p.

(2) n < N , ho An k,

P(N⋂

k=n

Ack) =

N∏

k=n

(1− P(Ak)) ≤N∏

k=n

e−P(Ak) = e−∑N

k=n P(Ak).

:TZ X6[K 14n limN P(⋂N

k=nAck) = 0, P(

⋃∞k=nAk) = 1, G P(limnAn) = 1.xz 1.2.2 ξn .O6Lu6y, ξ .O6Lu.

(1) B ξn P+4po ξ, T^ ε > 0

limn

P(ω ∈ Ω : |ξn(ω)− ξ(ω)| ≥ ε) = 0.w ξnp−→ξ.

(2) B ξn eQQ (z+ 1) 4po ξ TP(ω ∈ Ω : lim

nξn(ω) = ξ(ω)) = 1.w ξn

a.s.−→ξ.

(3) B ξn Lr- 4po ξ (r ≥ 1), T ξn, ξ ∈ Lr(Ω) blimn

E[|ξn − ξ|r] = 0,zC, ξn Lr(Ω) 44po ξ, w ξnLr

−→ξ._84pf%oJÆ. 6)/NM84poeQQpoVX%.vOo. h Chebyshev )p(TNM, Tξn

Lr

−→ξ ('6 r > 0), ξnp−→ξ. ww; P+4pseQQ4p%oJÆ, 6TH"

ω : lim ξ(ω) = ξ(ω) =⋂

ε>0

⋃

N≥1

⋂

n≥N

|ξn − ξ| < ε.ZWT ξna.s.−→ξ, V ε > 0, k

P

(

⋃

N≥1

⋂

n≥N

|ξn − ξ| < ε)

= 1.Y h Fatou \elimnP(|ξn − ξ| < ε) ≥ P(limn|ξn − ξ| < ε) = 1.G limn P(|ξn − ξ| < ε) = 1, ξn

p−→ξ, eQQ4pVP+4p.

:TZ X6[K 15x) 1.2.3 ξn .O6Lu6y, ξ .O6Lu.

(1) ξnLr

−→ξ ('6 r > 0) VG ξnp−→ξ;

(2) ξna.s.−→ξ VG ξn

p−→ξ;

(3) ξnp−→ξ, \ ξn oO6J6y ξnk

eQQ4po ξ.two" .if Borel-Cantelli \e. ww5O+:>Ps%o$r, \O/Q~o+8, QK.+4Qw9Jo+8%O.xz 1.2.3 O6Q~LuO ξi : i ∈ I Bw.O/Q~o, lim

N→∞supI

E[|ξi|; |ξi| ≥ N ] = 0.v, ξi : i ∈ I O6Q~LuOV1, ξi .O/Q~o. |e8MO/Q~oO6pa.x) 1.2.4 ξi : i ∈ I .Q~LuO. Q.O/Q~oGJa.(1) O/Ko7: ^ ε > 0, \ δ > 0 &ni A ∈ F , P(A) < δ !,

supi∈I E[|ξi|;A] < ε.

(2) L1- k/: supi∈I E[|ξi|] <∞.3'. J0. V A ∈ F , N > 0, kE[|ξi|;A] = E[|ξi|;A ∩ |ξi| ≥ N] + E[|ξi|;A ∩ |ξi| < N]

≤ E[|ξi|; |ξi| ≥ N] +N · P(A).fO/Q~0, kM ξi .O/Ko7o. ( A = Ω, nE[|ξi|] ≤ E[|ξi|; |ξi| ≥ N] +N,nl ξi o L1- k/0.

:TZ X6[K 16G0. ξi .O/Ko7b L1- k/o. h Chebyshev )p(, iN → ∞ !,

supi

P(|ξi| ≥ N) ≤ 1

Nsupi

E[|ξi|] −→ 0.Y h ξi oO/Ko70nl, ^ ε > 0, \ N > 0, &nE[|ξi|; |ξi| ≥ N] ≤ ε,O/Q~0.$/ 1.2.1 .t6O/Q~oGa (1) O6Q~LuV1oLu ξi : i ∈ I O/Q~. (2) ξn .Luy, \ p > 1 &n

supn E[|ξn|p] <∞, " : ξn .O/Q~o.x) 1.2.5 Q~Lu6y ξn L1- 4po ξ oGJa. ξn .O/Q~ob ξnp−→ξ.3'. J0.6 ξn

p−→ξ.vo, ξno L1-k/0K.vo. ξnoO/Ko70ho)p( ξ .Q~o,#kkM. V A ∈ F ,

E[|ξn|;A] ≤ E[|ξ|;A] + E[|ξn − ξ|].G0. V ε > 0,

E[|ξn − ξ|] ≤ E[|ξn − ξ|; |ξn − ξ| < ε] + E[|ξn − ξ|; |ξn − ξ| ≥ ε]

≤ ε+E[|ξn|; |ξn − ξ| ≥ ε]+E[|ξ|; |ξn − ξ| ≥ ε].Zw limn P(|ξn− ξ| ≥ ε) = 0, Gh ξn oO/Ko70 ξ oKo70kMlQSVs!, ZW ξnL1

−→ξ.

§1.3 [To Rn Vok5 µ, |Xµ(x) :=

∫

Rn

ei(x,y)µ(dy), x ∈ Rn, (1.3.1)

:TZ X6[K 17Q4 (x, y) . Rn o1~, µ Bw µ o Fourier r, +4OBwZX?, Kv"B)e, 3 .OIo. Z#s, T µ . n- xLuξ o*, ,

µ(x) = E[

ei(x,ξ)]

, x ∈ Rn,KBw. ξ oZX?, ivZX?4P`o*.ZX?. Rn O6k/o7o&(X?, QzwoNiE| QSRj µ f~Q5oa!.$/ 1.3.1 n = 1. " : µ zwtXQkib4i∫

R

x2µ(dx) <∞.

Fourier r.4Q9Jo%O, +sLo>4Kk)QÆo9J0.ZX?tO69J03.QSI~lwD~. µ, ν .t6k5, ,|XQoI~µ ∗ ν(A) =

∫

Rn

µ(A− x)ν(dx). (1.3.2)I~.O69Jo+8, Q+4.*#w µ, ν okLuξ, η oo*.x) 1.3.1

µ ∗ ν = µ · ν.ZX?tÆ69J03.QvO0, K.CZX?Twk5lo7X?or.OOo.x) 1.3.2 µ ν . Rn t6k5, T µ(x) = ν(x) Ok x ∈ RnCk, , µ = ν. # , o#p1. µ = ν;

2. ^k/Q5X? f , µ(f) = ν(f);

3. ^k/o7X? f , µ(f) = ν(f);

:TZ X6[K 18

4. µ = ν.

1 V 2 .fz |e, 4 V 1 .vO0|e, QQ.No. 603.4Q9Jo;F, K.C, ww" t65p, +2J" t65oZX?p, `dZvT4. |oit|O6Luo*!, K4QSt|QoZX?, ZX?vOs$#M*.$/ 1.3.2 ξ, η .Lu, dQ~. T^ x ∈ Rn kE[

η · ei(x,ξ)]

= 0," : E[η|ξ] = 0.ZX?QdO69J03.o70, 7LQ!X4%k!&f. CO6k5y µn 4po µ, T^k/o7X? f kµn(f) −→ µ(f).x) 1.3.3 (Levy) µn 4po µ ib4i µn ww4po µ.6|e." >"o4*|eoiV, O&l6, Qe.vOo.$/ 1.3.3 ξn .NQ~oke*L6y, Eξn = 0, Eξ2n = 1. " :

∑ni=1 ξi/

√n o*4po!F!S*.Q#ZX?:?:oz_C)&, ?:4o Fourier ?.

Fourier roO8, o [0, 2π) oQ~X? f , |X f o Fourier Æ?cn(f) :=

∫ 2π

0

f(x)einxdx, n ∈ Z.Q. ?u oO6X?. FoZX?oqk(X?s Laplace r,Q3 .OIo, /fou)e, ZX?QS/foOkLu, (X?/fo?y, O6?y an : n ≥ 0 o(X?|XwO6?z 7→

∞∑

n=0

anzn, (1.3.3)

:TZ X6[K 19i?o?y;w!!, QOIQSvOt|QoÆ?. Laplace r/fo#A!< [0,∞) o5, IO6k5 µ o Laplace r|Xwt 7→

∫ ∞

0

e−txµ(dx), t > 0. (1.3.4) Fs, Laplace rOIQSvOt|5 µ.

§1.4 ^Tv;aa?:Ps.Le4O6Q9Jo+8, Markov UEYo>4.)Qzro. Qs+4a+o+8k Fos nk3oj#. 8MQ|X.xz 1.4.1 (Ω,F ,P).O6+U, A .F oJ σ- ?, ξ. (Ω,F ,P) Q~Lu, ξ Jo A oa?:Ps, w E[ξ|A ], .) NSaoLu η: (1) η . A Q5o; (2) ^o B ∈ A , E[ξ;B] = E[η;B].Z#s, P(B|A ) := E[1B|A ], Bw B Jo A oa+. |o, Tξi : i ∈ I .OOLu. E[ξ|ξi : i ∈ I] := E[ξ|σ(ξi : i ∈ I)].62J" a?:Pso\0svO0. ,# , A ∈ A , µ(A) := E[ξ;A], µ . (Ω,A ) ok"Z5, PA . P A o1, ,TH" µ≪ PA bQ Radon-Nikodym k? N|X4oa(1) s (2), ZW.O6a?:Ps. |oT$ma?:PseQQpoVX%.vOo. SdZ#C , kJa?:Psop(z)p(.eQQpoVX%. |X4, ξ oQ~0aQS, k4:sBXW . h:44XQ~odZ.$/ 1.4.1 g8M\0o|oO6" .

1. T ξ ∈ L2(Ω,F ,P), M := L2(Ω,A ,P), " : E[ξ|A ] . ξ JU M o!$g .

2. if L2(Ω,F ,P) L1(Ω,F ,P) 4oK0, " a?:Pso\0.

:TZ X6[K 20+ 1.4.1 Ω1, · · · ,Ωn . Ω oQ5kb P(Ωi) > 0, 1 ≤ i ≤ n. A := σ(Ω1, · · · ,Ωn).lQ~Lu ξ,_J E[ξ|A ]. 6Zw E[ξ|A ].A -Q5o,GQ.O6fLu, .(w ∑n

i=1 ai1Ωi. if|Xoa (ii) n aiP(Ωi) =

E[ξ; Ωi], ZWE[ξ|A ] =

n∑

i=1

E[ξ; Ωi]

P(Ωi)· 1Ωi

.

E[ξ;Ωi]P(Ωi)

Bw. ξ Ωi oNL. ojJ.fdZa?:Pso&K.0. σ ?e.w+, F ")p,o+. a?:Ps E[ξ|A ] ")R$+ A ξoA,NL, z ξ o'8VXoQYC. 8MkJa?:PsoO$03.x) 1.4.1 ξ, η, ξn .Q~Lu.

(1) E[ξ|F ] = ξ;T ξsA k, E[ξ|A ] = Eξ,Z#s E[ξ|Ω,∅] = E[ξ].

(2) T ξ = a, E[ξ|A ] = a a.s.

(3) a, b .<?, E[aξ + bη|A ] = aE[ξ|A ] + bE[η|A ].

(4) T ξ ≤ η, E[ξ|A ] ≤ E[η|A ].

(5) |E[ξ|A ]| ≤ E[|ξ||A ].

(6) E[E[ξ|A ]] = E[ξ].

(7) T limn ξn = ξ a.s. b |ξn| ≤ η, Q4limn

E[ξn|A ] = E[ξ|A ].3'. (1), (2), (3) .vo, w" (4) ^ A ∈ A ,

E[E[η|A ]− E[ξ|A ];A] = E[E[η − ξ|A ];A] = E[η − ξ;A] ≥ 0.

:TZ X6[K 21 E[η|A ]− E[ξ|A ] . A Q5o, G.'o. (5) h (4) kM. (6) .|e 1.4.2(2) ok. w" (7), Zn := supk≥n |ξk − ξ|, Zn ↓ 0 b |Zn| ≤ 2η,hV14p|e EZn ↓ 0. |E[ξn|A ]− E[ξ|A ]| ≤ E[Zn|A ]b E[Zn|A ] .fzo, Q. Z, Z .'o,

E[Z] = E[E[Z|A ]] ≤ E[E[Zn|A ]] = E[Zn],ZW E[Z] = 0, Z = 0 a.s.x) 1.4.2 ξ, η .Lu.

(1) T ξ . A Q5o, b η ξη .Q~o, E[ξη|A ] = ξE[η|A ];

(2) T A ⊂ B . F oJ σ- ?, b ξ .Q~o, E[E[ξ|A ]|B] = E[E[ξ|B]|A ] = E[ξ|A ].3'. (1) +2'o ξ, η Bw. Z ξE[η|A ] . A Q5o, G+2H"V A ∈ A , k

E[ξE[η|A ];A] = E[ξη;A].i ξ .)0X?!, ξ = 1G, G ∈ A , (vCk, ZW ( A Q5ofX?Ck, Y 'Q5X?Ck.

(2) 6 E[ξ|A ] . B Q5o, ZWk E[E[ξ|A ]|B] = E[ξ|A ]. |O, A ∈ A , A ∈ B, GE[E[ξ|A ];A] = E[ξ;A] = E[E[ξ|B];A].ZW E[E[ξ|B]|A ] = E[ξ|A ].6|ek&Ko.0, # .p+>(okO, QK3Gne.. Qd, qk9Jo Jensen )p(. R oj (a, b) oiX? φ .)^ x, y ∈ (a, b) p, q ≥ 0, p+ q = 1, k

φ(px+ qy) ≤ pφ(x) + qφ(y).

:TZ X6[K 22x) 1.4.3 (Jensen) ξ .Q~Lu, φ . R oiX?b φ(ξ) Q~.φ(E[ξ|A ]) ≤ E[φ(ξ)|A ].3'. i0" φ oRlk?\, A .Qlk?, A vÆb^

x0 ∈ R, A(x0)(x−x0)+φ(x0) ≤ φ(x), x ∈ R. x, x0 #f ξ, E[ξ|A ] :

A(E[ξ|A ])(ξ − E[ξ|A ]) + φ(E[ξ|A ]) ≤ φ(ξ).T E[ξ|A ] k/, (Rtk/, lOQ~. Z A . Borel Q5o, G A(E[ξ|A ]) Jo A Q5. t A la?:Psn" Jensen >(.Os, Gn := E[|ξ||A ] ≤ n. Gn ∈ A b Gn ↑ Ω. ZWφ(E[ξ1Gn

|A ]) ≤ E[φ(ξ1Gn)|A ] = E[1Gn

φ(ξ) + 1Gcnφ(0)|A ].h φ oo70V14p|en >>(.+ 1.4.2 X, Y ko7on_w f(x, y), w_J E[X|Y ]. Zw

E[X|Y ] . Y Q5o, G\ g &n g(Y ) = E[X|Y ]. vdV y ∈ R kE[X ; Y ≤ y] = E[g(Y ); Y ≤ y]._fn_n

∫

R

∫ y

−∞xf(x, t)dxdt =

∫ y

−∞g(t)fY (t)dt,t y hkn ∫

Rxf(x, y)dx = g(y)fY (y), ZW

g(y) =

∫

R xf(x, y)dx

fY (y),Q4 fY . Y oX?. X, Y .!S*o, *.!F!So, JÆ?. ρ. >(k E[X|Y ] = g(Y ), Q4

g(y) =

(

1√2πe−

y2

2

)−1 ∫

R

x1

2π√

1− ρ2e−x2−2ρxy+y2

2(1−ρ2) dx

:TZ X6[K 23

=1

√

2π(1− ρ2)

∫

R

exp

(

−(x− ρy)2

2(1− ρ2)

)

dx = ρy,ZW E[X|Y ] = ρY . :Os, X, Y .!S*, Ps#w a, b, 8#wσ2, τ 2, JÆ?q. ρ. ,

E[(X − a)/σ|Y ] = E[(x− a)/σ|(Y − b)/τ ] = ρ(Y − b)/τ,ZW E[X|Y ] = ρσ(Y − b)/τ + a.|oO8`<o.o\e.) 1.4.1 T A .J σ- ?, X, Y .t6Lu, X ko A , Y .A Q5o, V'zk/o f k

E[f(X, Y )|A ] = E[f(X, y)]|y=Y .v\eo f(x, y) = 1A(x)1B(y) Ck, vd_f Dynkin \eD~Q5 F o)0X? 1F Ck, _ffz4p|eQ. 8+4.<fo, Bwfz .

§1.5 lp1. (Kolmogorov 0-1 ) ξ1, · · · , ξn, · · · .kLu6y,

F := σ(ξ1, ξ2, · · · ), A :=⋂

n

σ(ξn, ξn+1, · · · )." : F s A k, b^ A ∈ A , P(A) = 0 z 1.

2. ξn, ξ .'Q~Lu, ξnp−→ξ, E[ξn] −→ E[ξ]. " : ξn

L1

−→ξ.

3. ξ .Lu, " : ξ sJ σ- ? A kib4i^k/ BorelQ5X? g k E[g(ξ)|A ] = E[g(ξ)], Kpo^ x ∈ R kE[eixξ|A ] = E[eixξ].

4. Fα : α ∈ Σ .O6J σ- ?O, " : ξ .Q~Lu, E[ξ|Fα] : α ∈ Σ O/Q~.

v~ 7mYU|o'j, Q.)O6Io'P. Y. L4o9J;F%O, QÆfo%w+:NlQ9J0o>JQ<oJ.L. Doob 20 +Æ4Mo;T. 0Yo|+8, d!Yso7!Y, Doob ok/ *|e, Y)p(, Yo|03SYo!l|e.d!YeD'&K, Kf, hd!Ye.o7!YL9J.

§2.1 (GJ+4, G90Yo|XO$<fojJ. fsC, Y.>Nz. x4k4x-To,, , -ZJ, O6NoX?.fZq.Zp. zQS^Io, 5/A', +J5/'o7|wP.>!o. >!o|D.: sQ2oyiC!. 2K, 4 o+t, hOg4 , a, zgo.W, ).+. n[\[/, ivO)v;Q, hOyiK4.i R. .Yo|D. 4, Yd!YM', d!Y'&K, Qo|Y. Doob ko.8|+U (Ω,F ,P), Fn : n ≥ 0 . F oJ σ- ?yb2on vÆ, d<Bw.O6+, ")+LG!oÆ Æ. CO6LUE X = (Xn) .Jo (Fn) /_o, T^ n, Xn Jo Fn Q5, !KC (Fn) . X o/_. +%), V8|O6LUEX = (Xn), QKvs8MO6 (+)

F0n = σ(Xk : k ≤ n),Q. X /_oQ!+. +6+8oe.LZ#9J, do7!odZ5O+:>.xz 2.1.1 O6Q~o#(UE X = (Xn) Bw.Jo (Fn) oY, T X

25

:<Z \G9 26. (Fn) /_ob^ n, kE(Xn|Fn−1) = Xn−1. (2.1.1)T^ n, kE(Xn|Fn−1) ≥ Xn−1, (2.1.2)B X .Y.YskJ, Joao/_.YVGJo!o/_K.Y. OSO6YJoQKvM.Y. wwfU, iO68|d, OCo/_SYp+8.o8|+ Bo. Tx)|O6, Y.)JoWUEoKvoY. B X . Y, T −X .Y. d<42JAYsY, Yo03QSYYkM.. &Ks, oO6Y_C, Slw*o+_xP_'!Ro9b.)Q2o, zC, .2$m_o9bJooaPs.z. h|X, kRnlyf03:

(1) Yop^.0U.

(2) YoPs EXn Jo n ). YoPs EXn Jo n vÆ.

(3) h Jensen )p(, T X .Y, φ .iX?, , φ(X) Q~, .Y. ZW |X|, X2 ( X NQ~) .Y. |o, T X .Y, φ .ivÆX?, , φ(X) K.Y. ZW X+ .Y.$/ 2.1.1 T X = Xn Jo (Fn) .Y, bs σ- u G k, F ′t .h Ft G Co σ- u, " : X Jo (F ′

t ) K.Y.+ 2.1.1 ξn : n ≥ 1 .O6 Bernoulli L6y. X0 = 0, Xn :=∑n

i=1 ξi, n ≥ 1, b Fn . X oKv, o n ≥ 1,

E(Xn+1|Fn) = E(Xn+1 −Xn|Fn) +Xn

= Eξi +Xn = Xn + (p− q),ZWi p = q !, X . Z fLj, .6Y, p ≥ q !, X .Y,

p ≤ q !, X . Y. QSNMY_oO6>No'WA, Y

:<Z \G9 27s Y#_sO6kisPkio'WA. (W-, K4 YsYo"BrO:/_#oVX.)Y.Y Doobo||eM'o, (Ω,F ,P).+U, (Fn : n ≥ 0).. O6L6y Hn : n ≥ 1 Bw.Qxxo, T^ n ≥ 1, Hn .Fn−1 Q5o. X ./_UE, Hn .QxxUE, |XO6L'(w Y0 oLUE Y = (Yn) N

Yn := Yn−1 +Hn(Xn −Xn−1), n ≥ 1.Bw.UE H Jo X oL~, Q.OL~od.(. (ww"Z j#D~sL~, N2, 'D~!O)fw.)+ 2.1.2 L~k<&Ko.0. O2; kO65w S = (Sn :

n ≥ 0) o H9iw r oqO6FkL'H9 X0 ogHzo0(UE. O6gH4.)!R n − 1 J|t n !Fk Hn H9,oH1Aq, n− 1 !RoH9MwXn−1 = HnSn−1 + (Xn−1 −HnSn−1).,QgHP_!R n o(wXn = HnSn + (1 + r)(Xn−1 −HnSn−1).hWkM

Xn − (1 + r)Xn−1 = Hn(Sn − (1 + r)Sn−1),teDS (1 + r)−n n(1 + r)−nXn − (1 + r)−(n−1)Xn−1 = Hn[(1 + r)−nSn − (1 + r)−(n−1)Sn−1],K.C, do0(UE (1 + r)−nXn .gH4 Hn JodoH95UE (1 + r)−nSn oL~.f H.X ") H Jo X oL~. 6|eBw Doob oY||e, Q. 6LotOX| , 9J0s.

:<Z \G9 28x) 2.1.1 X .O6/_UE, H .QxxUE&n H.X .Q~o. TX .Y, ,UE H.X .Y. T X .Yb H ', , H.X .Y.3'. v H.X ./_o, b n ≥ 1,

E[(H.X)n − (H.X)n−1|Fn−1] = E(Hn(Xn −Xn−1)|Fn−1)

= HnE(Xn −Xn−1|Fn−1),ZW X .Y (_s, X .Y H o'0) VG H.X .Y (_s,Y).B H .A,k/, T^ n, Hn .k/o. )/" , T X Q~,b H A,k/, , H.X .Q~o. Doob oY||ek&Ko.0, |e ojJR94IL~o&KVX, ,6|e.CTdoH95UE.O6Y, ,)L"IogH4nlo0(UEq.Y, Y+o%N)v:YK)v:n. # .xo<$, a?zK4~<x4v._l8me, OSiNl6|e!, 1*v.V Doob ox<$o,7mI2m.2J\ !o+8, !.O6(uwLUE! (QSl ∞) oLu τ , N^ n k τ ≤ n ∈ Fn. Io !KBw(Fn)- !. TL!e.w',do!, , !oVD.,d.! n !RQSh n !Ro+_. Z#s, t|0o!. !. Qx-o !.64!, X = (Xn) .Jo (Fn) /_oL6y, |X

τ(ω) := infn : Xn(ω) ∈ A,Bw._ A ⊂ R o64!. i A . Borel !, kτ ≤ n =

⋃

k≤n

Xk ∈ A ∈ Fn,OS τ .64!. !.LUEeQ&KKvo+8, Qo\oLUE>oVX.e=<o, _LUEz+qd)M5oY, h !o\&nLUEkwKBAo. Ct6 !ojJ.

:<Z \G9 29+ 2.1.3 X , A ⊂ E, ω ∈ Ω, |XLA(ω) := supn > 0 : Xn(ω) ∈ A.

LA .RmQdOX A 4o!, Bw. A o%d!. Os LA ). !,ZwRm n !Rd)5 A Io, LA ≤ n )2+fRm n !Ro+_o.|oO6!F2 9<vvl, zPsFK5Qq!Q2!BM. w N > 0 I|,

T = infn ≤ N : Xn = max0≤k≤N

Xk,6!). !, Zw^! o'6!RLUE.!^lw 6!oQa(. 8#.T4kFK5^l'62!BM, ,6k.Q/o; hT4kFK^lQ2w!BM, ,6k.)Q/o. . !s !o9Jj#.r 4o03`9J, hTH".$/ 2.1.2 T τ , σ . !, ,l! τ ∧ σ K. !.k !, k *0, Xτ . τ !R X OQo0, K.C,

Xτ (ω) := Xτ(ω)(ω),zCi τ = n !, Xτ = Xn, # Xτ +2|X Ω oJ τ < ∞ .TwO6Zj, τ . !, TH"oap(Xτ∧n −Xτ∧(n−1) = 1τ≥n(Xn −Xn−1), (2.1.3)Q4o Hn := 1τ≥n = 1 − 1τ≤n−1 . Fn−1 Q5obk/o. uW |Xo *UE Xτ

n := Xτ∧n, nlo Doob k/ *|e.x) 2.1.2 (Doob) X .O6Y, τ .O6 !, τ *UE Xτ K.O6Y. ZWT τ .k/ !, ,EXτ = EX0. (2.1.4)

:<Z \G9 30x) 2.1.3 (Doob) X .O6Y, σ, τ .k/ !b σ ≤ τ , Xσ, Xτ.Q~o, bkEXσ ≤ EXτ .i X .Y!, pZCk.$/ 2.1.3 (1) " |e 2.1.3. (2) X .Q~/_UE, T^k/ ! σ ≤ τ kEXσ ≤ EXτ ," X .O6Yb

E[Xτ |Fσ] ≥ Xτ .

Doob *|e.>s !Jo9J;F, wYLjQ64!M'. j. Doob *|eoO69xo_f.+ 2.1.4 ξn .O6Ox5w oLj, Q.kL6ybP(ξn = 1) = p, P(ξ = −1) = 1− p = q. X0 = 0,

Xn = X0 +

n∑

k=1

ξk.Q.Y 0 MoLj. a > 0, τa . a o64!τa := infn > 0 : Xn = a.tO6Kvo. τa .ko? tÆ6.Tk, QPs.Æ?z"*? YQSu_$, wwu_$, Q9Jo.l_/oY. l z > 0, , zXn .kLuoD~, b

E[zXn |Fn−1] = zXn−1E[zξn ] = zXn−1(zp + z−1q),Q4 (Fn) )NCw X oKv. ZWYn := zXn(zp + z−1q)−n

:<Z \G9 31.O6Y. 6YBw)?Y, .6<kfoY. fQ_J P(τ <∞)S τa o(X? E[zτa ]. w"J(X?? ZwPso|XE[zτa ] =

∑

n

znP(τa = n)(X?o Taylor M(oÆ?. τa o*. iv (o4p;.Æ.1. h Doob o|e, _)k

E[

zXτa (zp + z−1q)−τa]

= 1. (2.1.5)O6vo,#., i τa < ∞ !, k Xτa = a, K.C X Q6Xl^ ao! T|YY a Q. ZWkE[(zp + z−1q)−τa ] = z−a,Ld+2w (zp+ z−1q)−1 = x .M z nl τa o(X?. h.gk62J.Jo, 6 Doob o|e+/fok/ !, k^ehC τa .k/ !, OS)2&)f Doob o|e. hQS τa ∧ n f Doob|e, ZwQ.k/ !. ZW %S (2.1.5) k

E[

zXτa∧n(zp + z−1q)−τa∧n] = 1. (2.1.6)w n iof, , τa ∧ n io τa, h.Ps.!2B$r?i z > 1 !, Zw τa ∧ n ≤ τa, h τa o|X$ Xτa∧n ≤ a. |oTkzp + z−1q > 1, ,

zXτa∧n(zp + z−1q)−τa∧n ≤ za,_fk/4p|e, PsQS$r, ZWE[

(zp + z−1q)−τa ; τa <∞]

= z−a. /2e! N z > 1 zp + z−1q > 1 t6a0? i p ≥ q !, V|G z > 1, i p < q !, V|G z > q/p. O8dZ, w z ↓ 1, nP(τa <∞) = 1; dO8dZ, w z ↓ q/p, n

P(τz <∞) = (q/p)−a < 1.

:<Z \G9 32K.C, i+Ilo! , Ljvk!1l^low, hk!1l^Rwo+!o 1. O8dZ, JM τa o(X?wE[zτa ] =

(

1 +√

1− 4pqz2

2pz

)−a

, z ∈ [0, 1]. (2.1.7)!E[τa] = lim

z↑1

d

dzE[zτa ] =

1

|p− q|

(

1 + |p− q|2p

)−a

,i p = q = 1/2 !, E[τa] = +∞. b < 0 < a, τa, τb #.Lj6Xl^ a s b o!. τ = τa ∧ τb .Lj6Xl^Q4Owo!, ,h nlo-, +J 0 < p < 1, k

P(τ < +∞) = 1. (2.1.8)K.CLj|vk!1dM6j. $mlrLjY b w (zY a w) dMo+ka? .9xoj9N.q.2J_/oY. i p = q = 12!, X .Y. ^ n ≥ 0,

b ≤ Xτ∧n ≤ a. oh Doob |ek/4p|ekM,

EXτ = EX0 = 0.ZEXτ = bP(τb < τa) + aP(τb > τa),G

P(τb < τa) =a

a− b.i p > q !, X ).Yw, !n2JO6)?Y. Z E(q/p)ξn =

p+ q = 1, G)/H" (q/p)Xn .Y. eekE[

(q/p)Xτ]

= 1.hoE (q/p)Xτ = (q/p)bP(τb < τa) + (q/p)a P(τa < τb).

:<Z \G9 33ZWpb =

1− (q/p)a

(q/p)b − (q/p)a._KQSfp+>(yM8E.M.$/ 2.1.4 /ioYJ jJ4 τ o(X?." Doob ot6|)p(, a)p( Z)p(. QQ#. Doob ||eo_f.) 2.1.1 X .O6Y. ,^ λ > 0 ! ? N , k

λP( max0≤n≤N

Xn ≥ λ) ≤ E(XN ; max0≤n≤N

Xn ≥ λ). (2.1.9)3'. τ := min0 ≤ n ≤ N : Xn ≥ λ, τ .O6 !b τ ≤ N , GEXN ≥ EXτ

= E(Xτ ; max0≤n≤N

Xn ≥ λ) + E(Xτ ; max0≤n≤N

Xn < λ)

≥ λP( max0≤n≤N

Xn ≥ λ) + E(XN ; max0≤n≤N

Xn < λ),ltÆQlQR, kMJo)p(.x) 2.1.4 (Doob) X .O6'Y.

(1) ^ λ > 0 ! ? N ,

λP( max0≤n≤N

Xn ≥ λ) ≤ EXN .

(2) ^ p > 1 ! ? N ,

E[ max0≤n≤N

Xpn] ≤

(

p

p− 1

)p

EXpN .3'. (1).\e 2.1.1o&)k. (2) ξ := XN , η := maxn≤N Xn, q :=

pp−1

.h\e 2.1.1 $mtP(η ≥ t) ≤ E(ξ; η ≥ t),

:<Z \G9 34-_ Fubini |e Holder )p(E[ηp] = E

∫ η

0

ptp−1dt =

∫ ∞

0

ptp−1P(η ≥ t)dt

≤∫ ∞

0

ptp−2E(ξ; η ≥ t)dt

≤ pE

(

ξ

∫ η

0

tp−2dt

)

=p

p− 1E[ξηp−1]

≤ q(E[ξp])1p (E[η(p−1)q])

1q

= q(E[ξp])1p (E[ηp])

1q .teN (E[ηp])

1q n.i X .NQ~Y!, X2

n .'Y, ZW^ λ > 0 ! ? N ,

P( max0≤n≤N

|Xn| ≥ λ) ≤ 1

λ2EX2

N .6)p(kOwkL6y;_>"o Kolmogorov )p(. Zw|Xn| .'Y, fotÆ6)p( p = 2 odZ, n

E[ max0≤n≤N

X2n] ≤ 4EX2

N .6)p(QSkM o)p( (86<?).X Z)p(. X .#(/_L6y, −∞ < a < b <

∞, |Xτ0 = 0;

τ1 := infn > 0 : Xn ≤ a;

τ2 := infn > τ1 : Xn ≥ b;

· · · · · ·

τ2k+1 := infn > τ2k : Xn ≤ a;

τ2k+2 := infn > τ2k+1 : Xn ≥ b;

· · · · · ·

:<Z \G9 35

(| inf ∅ = +∞) τn : n ≥ 1.O6?5fz o !6y, N ≥ 1,UXN [a, b] := maxk : τ2k ≤ N,Lu UX

N [a, b] wL6y X !R 0 s N %Y a b. b o ZX?. .>"o Doob Z)p(.x) 2.1.5 (Doob) X .O6Y, ^! ? N , <? a < b,

EUXN [a, b] ≤ 1

b− a[E(XN − a)+ − E(X0 − a)+].3'. Yn := (Xn − a)+, v Y = (Yn) K.O6Y. w τ1, τ2, · · · .

0, b− a, Y #l a, b,X d |Xo !y, Kv UXN [a, b] = UY

N [0, b− a].ZWYN − Y0 =

∑

n≥1

(Yτn∧N − Yτn−1∧N),

=∑

n≥1

(Yτ2n∧N − Yτ2n−1∧N) +∑

n≥0

(Yτ2n+1∧N − Yτ2n∧N)

≥ (b− a)UYN [0, b− a] +

k−1∑

n=0

(Yτ2n+1∧N − Yτ2n∧N),AV τn Jo n .?5vÆo, G o# .k, QCw ZosZot6,, t6oQepp)e, Zo.ifIRm03, Zo.ifYL(ovÆ03., h|e 2.1.3 kMlZ4oOoPs.'o, ZWEYN − EY0 ≥ (b− a)EUX

N [a, b] +k−1∑

n=0

(EYτ2n+1∧N − EYτ2n∧N)

≥ (b− a)EUXN [a, b].nlOJho-.6" R9pC, hIOom, )p(o" D.<Jvo. w"C0*oQ2R9AVl" 4# kO6&K oÆ. v Zo,.!o, ,Z,_).'o. o

:<Z \G9 37

= E(Xn : Xn > λ) + E(Xn : Xn ≥ −λ)− EXn

≤ E(Xk : Xn > λ) + E(Xk : Xn ≥ −λ)− EXk + ε

≤ E(Xk : Xn > λ) + E(−Xk : Xn < −λ) + ε

≤ E(|Xk| : |Xn| > λ) + ε,|oP(|Xn| > λ) ≤ 1

λE|Xn| =

1

λE(2X+

n −Xn)

=1

λ(2EX+

n − EXn) ≤1

λ(2EX+

0 − x),hWkM X .O/Q~o.

(2) Fo|e 2.1.6 o" . X∗, X∗ #.i n → −∞ ! Xn o s. vX∗ > X∗ =

⋃

a,b∈QX∗ < a < b < X∗.! ? N , Z)p(foY X−N , X−N+1, · · · , X0, f UX

N [a, b] ") [a, b] o ZX?, EUX

N [a, b] ≤ 1

b− a(E|X0|+ a).ZWeIk limN U

XN [a, b] < +∞ a.s. h.

X∗ < a < b < X∗ ⊂ limNUXN [a, b] = +∞,Gk P(X∗ < a < b < X∗) = 0, kM X∗ = X∗ a.s. Xn eQQ4p, w X−∞. h Xn : n ≥ 0 oO/Q~0, Xn K. L1 4po X−∞, ZW X−∞ .Q~o.|o A ∈ F−∞ m < n ≤ 0, k E(Xn;A) ≥ E(Xm;A), w m→ −∞,

E(Xn;A) ≥ limm

E(Xm, A) = E(X−∞;A),ZW E(Xn|F−∞) ≥ X−∞.Tw_f, _" Kolmogorov _a?|. QkOw2JE*Bko Borel _a?.

:<Z \G9 38x) 2.1.8 (Kolmogorov) ξn .ke*Q~L6y. 1

n(ξ1 + ξ2 + · · ·+ ξn)eQQb L1 4po Eξ1.3'. X−n := 1

n(ξ1 + ξ2 + · · · + ξn), F−n := σ(X−k : k ≥ n). ,

(X−n,F−n : n ≥ 1) .Y ( ). h o|e 2.1.7 kM X−n a.s. b L14poO6Q~Lu X .) Eξ1 = 0, k EX = 0. w φ ") ξ1 oZX?. , φ 0 wQtb φ′(0) = 0. ZWEeitX = lim

nEeitX−n = lim

n

(

φ(t

n)

)n

= 1. X = 0 a.s.

§2.2 1_JdOo7!Y, Kv3 o7!sd!ekRaoj#, h.=+ o7!eJ&O$, o7!Y.SS P.A.Meyer ko Strassburg :?DwU_o, P Doob oY Ito oL~e-_U_MwO~oLe.TCOXo+sLUE,qkppld5oY, \o !o+8Rpp>w5oJ. !.9xLUEe4Q2^+&K:o+8%O. QSC, k !oX, )2BwL. k+U (Ω,F ,P) T ⊂ R. uW F oJ σ- ?O (Ft : t ∈ T) Bw, T^ s < t, k Fs ⊂ Ft. ZwOkz+QJl6 Ft 4)v aPs, OSM.6 Ft4Vkz+QJ. ^ t ∈ T, |X Ft+ :=⋂

s>t Fs, , (Ft+) K.. O6 (Ft) Bw.lo7o, T^ t, Ft = Ft+. o7dZ !o|X Fod!.xz 2.2.1 8|+U (Ω,F ,P) (Ft), e τ : Ω −→ T ∪ ∞ Bw.O6 (Ft) !, T^ t ∈ T, ω ∈ Ω : τ(ω) ≤ t ∈ Ft.

:<Z \G9 39 !.oO6 Bo, hi t!, fsCO6 !.

(Ft) !. (Ft+) !.$/ 2.2.1 " : τ . (Ft+) !ib4i^ t, k τ < t ∈ Ft. τ . (Ft) !, |XFτ = A ∈ F : A ∩ τ ≤ t ∈ Ft, Ok t ∈ T Ck,&K ") τ %o+.$/ 2.2.2 " :

1. Fτ .O6 σ ?;

2. τ . Fτ Q5o;

3. T τ ≡ t !, Fτ = Ft;

4. O6Lu ξ . Fτ Q5ib4i^ t ∈ T, ξ · 1τ≤t . Ft Q5o.) 2.2.1 τ, σ, τn . (Ft) !.

(1) τ ∨ σ, τ ∧ σ . !.

(2) i τn fz !, lim τn . !.

(3) i τn fz"b (Ft) lo7!, lim τn . !.

(4) T σ ≤ τ , Fσ ⊂ Fτ .

(5) T (Ft) lo7b τn ↓ τ , ⋂

n Fτn = Fτ ." f, ~T .\o7!LUE. &KsC, P!oLuO.LUE. T = [0,∞), ivQS.Voj.xz 2.2.2 (Ω,F ,P) .O6+U, (E, E ) .O6Q5U, O6l( E oQ5eO X = (Xt : t ∈ T) Bw. (Ω,F ,P) S (E, E ) wESUoLUE.

:<Z \G9 40:4, O~oESU E d<lw Euclid U Rd, i d = 1 !_oUEBw#(UE. ww, &f6eK o+8: Q~UE, O/Q~UESNQ~UE. LUEo|X, LUEojJL5Qn, ojJ40O$9JoJAoLUE. oLUE_C, k69Jo+82Jw., # K.+otB. 6.IRm, $m Ω BwIU, w ω ∈ Ω BwIw, iIw ω I|, Xt(ω)Tw t oX?. T l E oe, Bw.w_q. j, O69&-d3H4!o-T.O6IRm; L!o'6FKo5QSNCw.OaIRm; L!o=AK.OaIRm; K7gN^"o8MOaRm, )eogQSNCw)eoIRm; 'zO6A4HolK.IRm. LUEo+K.IRmo*dZ.xz 2.2.3 BLUE X .o7o, TQeOkIRm.o7o, \O6z+ N &ni ω 6∈ N !, t 7→ Xt(ω) .o7o. lo7sRo7o+8QS F|X.64!o|Xsd! F. 8|ESUw E oLUE X = (Xt), A ⊂ E, ω ∈ Ω, |X A o5!64!DA(ω) := inft ≥ 0 : Xt(ω) ∈ A;

TA(ω) := inft > 0 : Xt(ω) ∈ A.Uot/M.|Xwz, ZW TA < ∞ ib4i'6 t k Xt ∈ A.5!s64!oj#oUEoL'0, RmoU'w) A 4, DA = TA; RmY A 4owM, DA = 0 TA )O|. LUEe4, 64!fn:O$. Nl, d!dZ" 64!. !e.No, o7!odZJ&n, g+OMzo64!.+ 2.2.1 X .Jo (Ft) /_oS E wESUoLUE. " A .b X .o7UE, DA . (Ft) !. ,# , !hRmo

:<Z \G9 41o70 DA ≤ t ib4iinf

s∈[0,t]∩Qd(Xs, A) = d(Xs : s ∈ [0, t] ∩Q, A) = 0.h. X oRm+.eQQo7o, ZW DA ≤ t s Ft Q5

infs∈[0,t]∩Q

d(Xs, A) = 0+8O6z+, K.CT Ft VwOkz+, , DA ≤t ∈ Ft, DA . (Ft) !. ww" o64! TA K. !, ^ n,

DnA = inft ≥ n−1 : Xt ∈ A., Dn

A K. !bfzvio TA, ZW, T (Ft) .lo7o, ,TA K. !. K.C, iLUEo7, /_Vz+blo7!,o64!. !. v, TLUE.lo7 ).o7!, o)2fw.ModZtTO$, A .Mb X .lo7o, ,lo7oRm , TA < t ib4i\ke? s < t &n Xs ∈ A, ZW Ft VOkz+, TA < t ∈ Ft, TA . (Ft+)- !, hO). (Ft)- !. &Ks, X oOaRm t !R ( t) o+)24IQ.!kR5O6M. K.C (Ft) VOkz+QJblo7odZ, QS" Mo5!Mo64!. !.Y6jJQSNM, J&nO6zMo64!.Jo (Ft). !, 2Joa. (Ft) klo70bPVkOkz+, !.O~od<a.) 2.2.2 Nd<a. ,O6o7/_UEJoMzo64!. !.gQSNMo7!dZ" 64!. !).dddZ,fN.QdXLUE !o0 Xτ oQ50, 2J\Jo(t, ω) n_Q5o+8, Q#K.s Borel o64!.!. !aJ

:<Z \G9 42o+8, h)6 U>. L! τ : Ω → T ∪ +∞,Kvs|XXτ (ω) := Xτ(ω)(ω),AV, Weo|Xu. τ < ∞, N X+∞ QQk|X. i τ . A o64!!, Xτ .64w. |X X o τ *UE (zA,lUE) w

Xτt := Xτ∧t.

t > 0, *UE.+14O&fo=J`%O.xz 2.2.4 T = [0,∞), X = (Xt)t∈T .SnOU E wESUo(Ft)t∈T /_UE, BUE X .Q5o, Te (s, ω) 7→ Xs(ω) .Y (T ×Ω,B(T) × F ) l (E, E ) oQ5e; B X . (Jo (Ft)) <6Q5o, T^ t ∈ T, e (s, ω) 7→ Xs(ω) .Y ([0, t]×Ω,B([0, t])×Ft) l (E, E )oQ5e.<6Q5UE./o/_UElo7/_UE%oO6+8.x) 2.2.1 T (Ft) VOkz+, ,O6lo7 (zRo7) o(Ft) /_#(UE.<6Q5o.3'. t ∈ T, n ≥ 1,

X(n)s :=

2n∑

k=1

X k2n

t1(k−12n

t, k2n

t] +X010. (s, ω) 7→ X(n)s (ω). ([0, t]×Ω,B([0, t])×Ft)l (E, E )oQ5e. Z X .lo7o, G Xs = limnX

(n)s a.s. ZW (s, ω) 7→ Xs(ω). ([0, t]×Ω,B([0, t])×

Ft) l (E, E ) oQ5e." Xτ # .Jo Fτ Q5o, 6-<9J.x) 2.2.2 T X .S E wESUo (Ft) <6Q5UE, τ . (Ft) !, Xτ 1 Ωτ := τ <∞ . (Ωτ ,Fτ ∩ Ωτ ) l (E, E ) oQ5e.3'. Z X .<6Q5o, G^ t ∈ T, (s, ω) 7→ Xs(ω) .Y ([0, t] ×Ω,B([0, t]) × Ft) l (E, E ) oQ5e. TH" ω 7→ (τ(ω) ∧ t, ω) .(Ω,Ft) l ([0, t] × Ω,B([0, t]) × Ft) oQ5e, ZWt6eo&_ ω 7→

:<Z \G9 43

Xτ(ω)∧t(ω) . (Ω,Ft) l (E, E ) oQ5e. ,^ B ∈ E , Xτ ∈B ∩ Ωτ ∩ τ ≤ t = Xτ ∈ B, τ ≤ t = Xτ∧t ∈ B ∩ τ ≤ t ∈ Ft, Xτ ∈ B ∩ Ωτ ∈ Fτ .# , <6Q50o9J0~WQ)oJa, jT Nd<a, ,O6<6Q5UEJo^O6 Borel o64!. !, )4oM. :d%kfl6-T

§2.3 .uJ+4, wLSwF;T, 6" oO6lo7Y, QS Nd<a, ZWMso64!. !, " Doob ok/ *|eCk, Y Yo *UEv.Y, Qd Doob ot6YY)p(kOlo7!;_. .C+Jklo7, d!Yo-QSkOlo7!Y , z4vAVl'!Y4p|egÆG:9Jo%. T = [0,∞), (Ω,F ,P) .p+U, (Ft)t∈T .+U o.xz 2.3.1 X = (Xt) ./_o#(Q~UE. T^ t > s kE(Xt|Fs) = Xs,,B X .Y. Y Y Fs|X. TO6 () Ye!.lo7zo7LUE, ,B%wlo7zo7 () Y.YzY.lo7o, ,Kvo.Q.).klo71!,oO6Y X .!\O6lo7Y Y &n^ t, k Xt = Yt a.s.?6`9J, h.!X%kfl, OSQ gX.o7!Yo03d!Yk3o)e. lw*, qk!kVXoo7!LUEojJ, OSK8MNoo7!YojJ, OS!o$eq+., T$, O&Jpl\ Brown %d.

:<Z \G9 44+ 2.3.1 Doob Y.YoO6NojJ. ξ .Q~Lu, |XXt := E(ξ|Ft), t ≥ 0., X = (Xt) .O6O/Q~Y, Bw Doob Y.9D6|X, )/NM, QS F0 4VkOk+zo_, !Q$_. 6) aPs, ZwOkJoaPso-T.eQQoVX5>o. 5>lo7Yo6-.x) 2.3.1 T X .O6lo7o (Ft) Y, X K.O6 (Ft+) Y.3'. .')!Y4p|eoO6k. J" t > s k,

E(Xt|Fs+) ≥ Xs.l t0 = t, tn ?5vio s. |X Y−n := Xtn , G−n := Ftn . , (Y−n,G−n :

n ≥ 0) .')!Y. Zw EY−n ≥ EXs > −∞, G_f|e 2.1.7 o-kME(Xt|Fs+) = E(Xt|

⋂

n

G−n) ≥ limnY−n = Xs.Q4 Y−n → Xs .Zwlo7.h 6-, SdCllo7Y!, M.QS_o. Nd<ao, K.C (Ft) Nd<a, ,Mzo64!M. !. o|e. Doob k/ *|eoo7! , .olo7Y5>o.x) 2.3.2 X .lo7Y, τ, σ .k/ !b σ ≤ τ , Xσ, Xτ .Q~ob

Xσ ≤ E(Xτ |Fσ), a.s. (2.3.1)3'. " Xσ .Q~o. n ≥ 0, Dn := k2n

: k = 0, 1, 2, · · · σn(ω) := inft ∈ Dn : t ≥ σ(ω), ω ∈ Ω.Z Dn ⊂ Dn+1, G σn s σn+1 .(uw Dn+1 oJo (Ft : t ∈ Dn+1)ok/ !, _f Doob d!k/ !|eo (Ft : t ∈ Dn+1) Y

:<Z \G9 45

(Xt : t ∈ Dn+1), n$ Xσns Xσn+1 .Q~obXσn+1 ≤ E(Xσn

|Fσn+1). n ≤ 0, Yn := Xσ−n

, Gn := Fσ−n, (Yn : n = 0,−1,−2, · · · ) .Jo (Gn : n = 0,−1,−2, · · · ) oY, b^

n ≤ 0, EYn = EXσ−n≥ EX0, h|e 2.1.4, (Xσn

: n = 0, 1, 2, · · · ) .O/Q~o, bZw X .lo7o, Gi n → ∞ !, XσneQQ4po Xσ, Z K

L1- 4po Xσ, ZW Xσ .Q~o, ee Xτ K.Q~o.)G" (2.3.1). eI|X τn := inft ∈ Dn : t ≥ τ, τn ≥ σn b.k/ !, ^ A ∈ Fσ =⋂

n Fσn, _f Doob *|e, ^ n,

E(Xτn ;A) ≥ E(Xσn;A),h L1 4p0n E(Xτ ;A) ≥ E(Xσ;A).hW|e, T X = (Xt) .lo7 (Ft) Y, τ . !, , *UE

Xτ .Jo *o (Fτ∧t) oY. h.# q)B, " Xτ .Joz_ (Ft) oY. d!odZ, 6-T.hJod!oL~o|e 2.1.1 kMo. h.o7!od., L~o|X~Wf, G2Jf Doob *|e_" . FO6\e.) 2.3.1 σ . !, t ≥ 0, X . Fσ Q5oQ~Lu. ,

E[X|Ft] = E[X|Fσ∧t]. (2.3.2)3'.6TH" Fσ∧t = Fσ ∩ Ft, W, 4k:Oo-T.haPso03, +2" E[X|Ft] Jo Fσ Q5QSw. uW O+oO6r 4OBo, Lu Y . Fσ Q5oib4i^s ≥ 0, k Y · 1σ≤s . Fs Q5o. ZW

E[X|Ft] = E[X · 1σ≤t +X · 1σ>t|Ft]

= X · 1σ≤t + 1σ>tE[X|Ft],

:<Z \G9 46ltOv. Fσ Q5o, ww" tÆ. Fσ Q5o, lV s ≥ 0,

1σ>tE[X|Ft] · 1σ≤s = 1t<σ≤sE[X|Ft],i t < s, l. Fs Q5oX?s Ft Q5oX?D~, ZW. Fs Q5o; it ≥ s !, lpo 0. Y tÆK. Fσ Q5o, E[X|Ft] . Fσ Q5o.x) 2.3.3 τ . !, X .lo7Y (zY), X o τ *UE Xτ =

(Xt∧τ : t ∈ T) K.Jo (Ft) oY (Y).3'.Z t ∧ τ .k/ !, h|e 2.3.2, Xτ . (Ft) /_oQ~UE, %bt > s, _f|e 2.3.2 h\e 2.3.1, Zw Xt∧τ . Ft∧τ Q5o, G

E(Xt∧τ |Fs) = E(Xt∧τ |Ft∧τ∧s)

= E(Xt∧τ |Fs∧τ )

≥ Xs∧τ .G Xτ K. (Ft) Y.|ed.kfo, " Fd!, ~T .x) 2.3.4 (Ft) /_olo7#(Q~UE X .Yib4i^k/ ! τ k EXτ = EX0.|e 2.1.4 4ot6Y)p(1QyKJ Ck, hYolo70 Doob o)p(QkOlo7!Y, Q4tÆ6# .+Q9Jo-T%O, Bw Doob Ya)p(, dLfl.x) 2.3.5 X .O6lo7'Y, T > 0. (1) ^ λ > 0,

λP( max0≤t≤T

Xt ≥ λ) ≤ E[XT ]. (2.3.3)

(2) ^ p > 1,

E[ max0≤t≤T

Xpt ] ≤

(

p

p− 1

)p

E[XpT ]. (2.3.4)

:<Z \G9 47Z#s, T X .Y, ,E[ max

0≤t≤TX2

t ] ≤ 4E[X2T ]. (2.3.5)3'.(1) l [0, T ] oQyKJ D = xn, ^ n, lQ n 6?!la>yw 0 = t0 < tn1 < · · · < tnn = T , Zw X .lo7o, G

max0≤t≤T

Xt = maxXt : t ∈ [0, T ] ∩D =↑ limn

max0≤i≤n

Xtni,vd_f|e 2.1.4 (1) fz4p|en". (2) o" F.o7!dZ, Kolmogorov )p(vCk, T X = (Xt) .lo7Y, ,V T > 0 s λ > 0 k

P(supt≤T

|Xt| ≥ λ) ≤ 1

λ2E[X2

T ]. (2.3.6)h.Sdo;_, d<QS&f Kolmogorov )p(:_o (2.3.5).

§2.4 lp1. (Yn : n ≥ 1) .O6FkkESU E o Markov q, P = (p(x, y)) :

x, y ∈ E .CQB, ^ n ≥ 1 y ∈ E kP(Yn = y|Yn−1, . . . , Y1, Y0) = p(Yn−1, y).

α : E → R . P oY<oZ( λ oZu: Pα = λα. Xn :=

λ−nα(Yn), " : (Xn : n ≥ 1) .O6Y.

2. X .zL(NQ~okÆuUE, " : \vOo T L(wzovÆX? F &n (X2t − F (t)) .O6Y.

3. (Wald Y) Yn : n ≥ 1 .ke*L6y&n φ(t) := E[etYn ] '6 t 6= 0 k. Xn := φ(t)−n exp[t(Y1 + · · ·+ Yn)]." : Xn : n ≥ 1 .Y.

:<Z \G9 48

4. O6dJ4!R 0 kO6bgsO6g. LsYdJ4lO6g,vdQu%O6eDog, s9&WUE. Xn w nXdd4g?sMg?%. " : Xn .Y.

5. Yn : n ≥ 1 .ke*L6y, f0, f1 .t6+X?, f0 >

0. Xn :=

f1(Y1)f1(Y2) · · ·f1(Yn)f0(Y1)f0(Y2) · · ·f0(Yn)

." : T f0 . Yn oX?, , (Xn) .Y.

6. Xn : n ≥ 0 . (Fn) /_oQ~L6y, NE(Xn+1|Fn) = αXn + βXn−1, n ≥ 1,Q4 α > 0, β > 0, α + β = 1. a w^(!, 6y Y0 := X0, Yn :=

aXn +Xn−1 . (Fn) Y.

7. Markov q Xn : n ≥ 0 oESUw 0, 1, · · · , N, CQ+wpij =

(

N

j

)

πji (1− πi)

N−j, 0 ≤ i, j ≤ N,Q4 πi =1−e−2a i

N

1−e−2a . H": Zn := e−2aXn .Y.

8. Yn.ke*!Lu6y&n EYn = 1. Xn := Y1Y2 · · ·Yn.

(a) " : (Xn) .YbeQQ4poO6Lu X ;

(b) Yn S+ 12#l( 1

2s 3

2. H" X = 0 a.s. ZW E

∏

n≥1 Yn 6=∏

n≥1 EYn.

9. (Doob .) " : Y (Xn) QvO.w Xn = Yn + Zn, Q4 (Yn) .Y, (Zn) .Y 0 Mo'QxxoÆUE: 0 = Z1 ≤ Z2 ≤ · · · .

10. (Riesz .) " : Y (Xn) Q.w Xn = Yn + Zn Q4 Y .Y, Z .- ( Z . Yb limn EZn = 0) ib4i EXn k/. !W.vO.

:<Z \G9 49

11. Xn .Yb |X0(ω)|, |Xn(ω)−Xn−1(ω)| O6s ω n Jo<?V1. T τ .O6FkL(o !, " : Xτ Q~b EXτ = EX0.

12. Xn : n ≥ 0 .j 2.1.4 4|XoY a MoLj, τ .6Xl^0, b o!. " : i p = q !, X2

n − n .Y. %hWh Eτ .

13. Xn : n ≥ 0 .j 2.1.1 CoLj, p > q. l ? b > 0, σ := minn : Xn = b. h ! σ o(X?%hWJ σ oL(s8.

14. ξ1, ξ2, · · · , ξn, · · · .ke*Q~Lu. k ≥ 1, Sk :=

ξ1 + · · ·+ ξk, " : · · · , Sk/k, Sk−1/(k − 1), · · · , S2/2, S1 .Y.+(.+2" E(S1|Sn, Sn+1, · · · ) = E(S1|Sn) =

Sn

nNBw.

15. T X .' Y, ,P(XN > 0, inf

n≤NXn = 0) = 0.

16. Xn : n ≥ 0 .Y, τ . !. T(1) P(τ <∞) = 1;

(2) E|Xτ | <∞;

(3) limn E[Xn; τ > n] = 0; , E[Xτ ] = E[X0].

17. (Kolmogorov ?|e) Xn .kL6y. , ∑

Xn eQQ4pib4i'6 A > 0 6aCk(a) P(|Xn| > A) 4p;

(b) Yn := Xn1|Xn|≤A, , ∑

E(Yn) 4p;

(c)∑

D(Yn) 4p.

:<Z \G9 50

18. X .O6 (Ft) /_o, lo7%FkRoUE. A .&n X [0, t) o7oIp^. " : A ∈ Ft.3'. Yt = (Xt−, Xt), iv Y = (Yt) K. (Ft) /_o. |X

τ = inft ≥ 0 : Yt 6∈ d,Q4 d .ESU E×E o%, , τ .Mo64!, G. (Ft+)- !, ZW A = τ ≥ t = τ < tc ∈ Ft.

19. τ .6 !, σ .L!b σ ≥ τ . " : T σ . Fτ Q5o, σ.6 !.

20. τ . (Ft) !, " : Fτ+ =⋂

σ>τ Fσ, Q4 σ . !.

21. τ, σ . (Ft) !, Fτ∧σ = Fτ ∩ Fσ, b, τ < σ, τ ≤ σ ∈Fτ ∩ Fσ.3'.vh\e 2.2.1 (4), Fτ∧σ ⊂ Fτ ∩ Fσ. w"|OoVJÆ,l A ∈ Fτ ∩ Fσ, ^ t ∈ T,

A ∩ τ ∧ σ ≤ t = A ∩ (τ ≤ t ∪ σ ≤ t)

= (A ∩ τ ≤ t) ∪ (A ∩ σ ≤ t) ∈ Ft.ZW A ∈ Fτ∧σ.^ t ∈ T,

τ < σ ∩ σ ≤ t = (⋃

s∈Qτ < s ∩ σ > s) ∩ σ ≤ t

=⋃

s∈Q(τ < s ∩ σ > s ∩ σ ≤ t)

=⋃

s∈Q,s<t

(τ < s ∩ σ > s ∩ σ ≤ t) ∈ Ft.ZW τ < σ ∈ Fσ. |oτ < σ ∩ τ ≤ t = τ < σ, τ ≤ t, σ ≤ t ∪ τ < σ, τ ≤ t, σ > t

:<Z \G9 51

= τ < σ, σ ≤ t ∪ τ ≤ t, σ > t ∈ Ft.ZW τ < σ ∈ Fσ∧τ . B0kM τ > σ ∈ Fσ∧τ , 5 τ ≤ σ, σ ≤τ ∈ Fσ∧τ .

22. (Follmer \e) X = (Xt : t ∈ T) .O6Y, D . T oO6QyKJ, ! ? K > 0, DK := [0, K] ∩ (D ∪ 0, K), (1) eOko ω ∈ Ω, Y D l R oe t 7→ Xt(ω) ^ K > 0

DK .k/ob6w t ∈ T klXD

t+(ω) := lims∈D,s↓↓t

Xs(ω)RXD

t−(ω) := lims∈D,s↑↑t

Xs(ω).

(6y sn ↑↑ t ")^ n ≥ 1, sn < t b sn ↑ t. ↓↓ Fe..)

(2) Ok t ∈ T, XDt+ .Q~ob Xt ≤ E(XD

t+|Ft). T t 7→ EXt lo7, Xt = E(XDt+|Ft).

(3) UE XD+ = (XD

t+ : t ∈ T) .Jol (Ft+) olo7Y, i X.Y!, XD+ K.O6Y.3'. w F := 0 = t0 < t1 < · · · < tN = K . DK oO6Vk 0, KokJ, 0 ≤ n ≤ N , Gn := Ftn , Yn := Xtn , Y = (Yn : 0 ≤

n ≤ N) .O6Y, h Doob o)p(,

EUYN [a, b] ≤ 1

b− a(E(YN − a)+ − E(Y0 − a)+) ≤ 1

b− aE(XK − a)+,

λP( max0≤n≤N

|Yn| > λ) ≤ 2EY +N − EY0 = 2EX+

K − EX0,Q4 a < b, λ > 0. # UYN [a, b] . X 1 F [a, b] o ZX?,Qw UX

F [a, b], UXDK

[a, b] := supF⊂DK

UXF [a, b],

:<Z \G9 52Q. X 1 DK j [a, b] o ZX?, Zw DK Qy, GQkJp^.Qyo, ZW UXDK

[a, b] .O6'Qlf(oLu. h o)p(kMEUX

DK[a, b] ≤ 1

b− aE(XK − a)+, (2.4.1)

λP( supt∈DK

|Xt| > λ) ≤ 2EX+K − EX0. (2.4.2)

(a) N0 w Ω 4&ne t 7→ Xt(ω) '6 DK /z'6w t Q >lzR)\o ω p^. &)H" N0 oQ50, h.TH"N0 ⊂

⋃

K≥1

[

supt∈DK

|Xt| = ∞⋃

NK

]

,Q4NK :=

⋃

a,b∈Q,a<b

UXDK

[a, b] = ∞. ^I|o! ? K a < b, h Doob o)p(P( sup

t∈DK

|Xt| = ∞) = limλ→∞

P( supt∈DK

|Xt| > λ)

≤ limλ→∞

1

λ(2EX+

K − EX0) = 0,

P(UXDK

[a, b] = ∞) = limN→∞

P(UXDK

[a, b] ≥ N)

≤ limN→∞

1

NEUX

DK[a, b]

≤ limN→∞

1

N(b− a)E(XK − a)+ = 0.Z+U.po, G N0 ∈ F b P(N0) = 0.

(b) ^ t ∈ T, l sn : n ≥ 0 ⊂ D N sn ↓↓ t. n ≤ 0, Gn := Fs−n

, Yn := Xs−n, Y = (Yn : n ≤ 0) .Jo (Gn : n ≤ 0) oY, b

EYn = EXs−n≥ EXt.

:<Z \G9 53h|e 2.1.4, Y .O/Q~o, bi n −→ +∞ !, Xsn . L1 4poXD

t+, ZW XDt+ .Q~o, nZw Xt ≤ E(Xsn |Ft), h L1- 4p0kM

Xt ≤ E(XDt+|Ft). T t 7→ EXt lo7, EXt = limn EXsn = EXD

t+,,'Lu E(XDt+|Ft)−Xt oPspo EXD

t+ − EXt = 0, Y Xt = E(XDt+|Ft).

(c) 6ho (Ft+) .po, TNM XD+ . (Ft+) /_oQ~UE,|o t > s, l sn ∈ D, t > sn ↓↓ s, q.h|e 2.1.4,

E(Xt|Fs+) ≥ XDs+,l tn ∈ D, tn ↓↓ t, k E(Xtn |Fs+) ≥ XD

s+, ZWE(XD

t+|Fs+) = limn

E(Xtn |Fs+) ≥ XDs+, XD

+ .Jo (Ft+) oY. X .Y, EXt .<?, , EXDt+K.<?, ZW XD

+ .Y.

23. (Xt) .O6Jo (Ft) olo7' Y. " : i t→ ∞ !, Xt eQQ4poO6Q~Lu, (w X∞) b (Xt : 0 ≤ t ≤ +∞) . (Ft) Y.

24. " : O6Y.O/Q~Yib4iQ.lY (Doob Y).3'.T M = (Mt) .O/Q~Y, ,i t → ∞ !, Mt )4eQQ4pK L1- 4po'6Q~o ξ, bk Mt = E[ξ|Ft]. %Uv.

vE Brown 4, \+4QkfQ9JoLUE, Brown .Q.o7o, Gauss UE, .YK. Markov UE. 7LQoIRm).k/8o, hQkÆX8, &nQS|X~, S Ito t~.

§3.1 Xkh?e%Hs5h|Brown o).ONo,d, 6J$m nO6o7!oLUE, . Kolmogorov Pw+k>eoB>48MoO6, .dUkx*O_xU 5. ww! O|eo:, JC Y*oVX_C, LUE.fxU o+5. (Ω,F ,P) .+U, T .VO6)!, X = (Xt : t ∈ T) .Q oS (E, E ) wESU () E . Euclid U) oLUE (tasC_)BwQ5eO). ww5>: O, 2J\O$"Z"V.f IT ") T okk6Jp^,

IT := (t1, · · · , tn) : n ≥ 1, t1, · · · , tn ∈ T. I = (t1, · · · , tn) ∈ IT, En w EI , f XI z X(t1,··· ,tn) ")eω 7→ (Xt1(ω), · · · , Xtn(ω)). XI . Ω l EI oQ5e. 5

µI := PX−1I.U (EI , E I) h XI mko5, K. X !w I okx*.tas, ^ A1, · · · , An ∈ E ,

µ(t1,··· ,tn)(A1 × · · · × An) = P(Xt1 ∈ A1, · · · , Xtn ∈ An).kx*Ow LX := µI : I ∈ IT.

54

:JZ Brown Y; 55+ 3.1.1 Q9JoO6LUE. Gauss UE. O6LUEBw. GaussUE, TQo^kx*.!S (Gauss) *. wf Hilbert U_O6 Gauss UE. )! T .O61~w 〈·, ·〉o Hilbert U H , lO6!F!$| en : n ≥ 1. lO6+UQ O6kb#Y!F!S*oL6y ξn : n ≥ 1, |XL)!w H oLUEX(h) :=

∑

n≥1

〈en, h〉ξn, h ∈ H., X = (X(h) : h ∈ H) .O6 Gauss LUE (;) bE[X(g)X(h)] = 〈g, h〉, X .O6pGe.xz 3.1.1 #+U (Ω,F ,P) (Ω′,F ′,P′) |XobkeoESU (E, E ) seo)! T oLeO X , X ′ Bw.po (e*o), TQkeokx*O, ^ I = (t1, · · · , tn) ∈ ITT ,

PX−1I = P′X ′−1

I ._C YpoVX C, LUE (LeO) QS")wESUofD~U o+5. I| ω ∈ Ω, t 7→ Xt(ω) . T l E oe. G2JO T l E oePCoU. f ET ")Y T l E 4oe x = (x(t) : t ∈ T) p^PCo_, ET 4oyHk!KBwRm, ETd<BwRmU, Q# . E o T XKDoD~U. o x ∈ ET, Zt(x) := x(t), Q. ET l E og , KBwU!JJ, Zw x(t) KBw x t QoU!. w E T . ET wOkg Zt : t ∈ T CwQ5eoQ!σ- ?,

ET := σ(Zt : t ∈ T),QKBw.?Co σ- ?. U (ET, E T) h E T J|, BwxU, OS Z = (Zt : t ∈ T) . (ET, E T) oQ5eO, BwxUE. b68|+U (Ω,F ,P) S E wESUS T w)!oLUE

X = (Xt) xU 8MO6+5. if5o, +J

:JZ Brown Y; 56O6nÆt6UoO6Q5eBw, O6Kvoe,g, QI ω ∈ Ω ewIRm t 7→ Xt(ω), w Φ.$/ 3.1.1 H" Φ .Q5e.I+ P Φ kGlxU .CO6+µ = PΦ−1.x) 3.1.1 µ %, RmUE Z = (Zt) s X = (Xt) p.$/ 3.1.2 H"Qkeokx*O.S |e4I, LUEpoxU O6+5. T LX = µI : I ∈ IT .LUE X okx*O, ,Qv No0

1. T I = (t1, · · · , tn), A1, · · · , An . E oQ5J, k1, · · · , kn . 1, · · · , noO69>, ,µI(A1 × · · · × An) = µ(tk1 ,··· ,tkn)(Ak1 × · · · ×Akn);

2. I = (t1, · · · , tn) ∈ IT, A1, · · · , An ∈ E , T'6 1 ≤ k ≤ n kAk = E,

µI(A1 × · · · × Ak × · · · ×An)

= µIk(A1 × · · · × Ak−1 ×Ak+1 × · · · × An),Q4 Ik := (t1, · · · , tk−1, tk+1 · · · , tn). K.Ci I ⊂ J . T okJ!, µI . µJ o_u o*.xz 3.1.2 5o_L = µI : I ∈ ITBw. E oO6kx*O, T6 I ∈ IT, µI .D~U (EI , E T) o+5. E okx*O L = µI : I ∈ IT Bw.okx*O, TQ N o0a.

:JZ Brown Y; 57O6LUE9okx*OM.o. ,8| E oO6okx*O L , .!\O6+U (Ω,F ,P) Q oO6LUE X & X okx*OY. L ? T\, Ckx*O L QS#, +U (Ω,F ,P) UE X . L oO6#, fs9> o: O6okx*O.!O|QS#?uUhNNLu, T ξ . (Ω,F ,P) n- xLu, *wµ, Q# . P ξ o, +U (Rn,B(Rn), µ) |XLuI(x) = x, , I o*K. µ. C "0? C O6*QShESU Rn O68|oLu I #. LUEK.W.0LUEo|e, K. Kolmogorovo0|e,QBO6YoESU ookx*O.QS#o, . +o| . 6w0xUK.fxD~Uo+8, QLUEoe4ÆG9Jo%. I = (t1, · · · , tn) ∈ IT, x ∈ ET,

x(I) := (x(t1), · · · , x(tn)) ∈ EI , φI(x) := x(I), φI . ET l EI og . T ot6kJ I ⊂ J , φJI (x(J)) := x(I),Q. EJ l EI og . ^ H ∈ E I ,

φ−1I (H) = x ∈ ET : (x(t1), · · · , x(tn)) ∈ H.

ET o.( oJBw. ET oO6?. ET oOk?wE

T

0 := φ−1I (H) : I ∈ IT, H ∈ E

I,Os E T

0 ). σ- ? (N T .ko).) 3.1.1 ?o_ E T

0 (Jk%b E T = σ(E T

0 ).3'. dO6-.vo. " E T

0 .O6 ?. h|X&)kM ∅,Ω ∈E T

0 , b E T

0 (J. |oTNM, ^ I = (t1, · · · , tn) ∈ IT,

H ∈ E I , t ∈ T, kφ−1I (H) = φ−1

I′ (H′),

:JZ Brown Y; 58Q4 I ′ := (t1, · · · , ti−1, t, ti, · · · , tn) ∈ IT, H ′ :=(x1,· · ·,xn+1)∈EI′ : (x1,· · ·, xi−1,xi+1, · · ·,xn+1) ∈ H.,? φ−1

I1(H1), φ

−1I2(H2), \ I ∈ IT, H

′1, H

′2 ∈ E I , &n

φ−1I1(H1) = φ−1

I (H ′1), φ

−1I2(H2) = φ−1

I (H ′2). φ−1

I1(H1)

⋃

φ−1I2(H2) = φ−1

I (H ′1

⋃

H ′2) ∈ E T

0 . E T

0 k%.CO6kx*OQSxU #, TxU kO6+P &nRmUEokx*OY.8|okx*O.) 3.1.2 O6kx*OkO6#ib4iQQSxU #,Bwx#." Kolmogorov|e. f\o"Z, 0 (3.1.1)poCT I ⊂ J ⊂ T, µI . µJ g %o,

µI = µJ(φJI )

−1. (3.1.1)x) 3.1.2 (Kolmogorov) E .pQuU, E ._o Borel σ- ?, (E, E ) o^okx*OO|kO6#.3'. L = µI : I ∈ IT . (E, E ) oO6okx*O. w? E T

0 O6X? P :

P(φ−1I (H)) := µI(H), I ∈ IT, H ∈ E

I .O6?QSk)eo"), hh*Oo0, >|X)v9SX. j φ−1(t1,t2)

(A1×E) = φ−1(t1)

(A1), h L o0kM µ(t1,t2)(A1×E) = µ(t1)(A1).Zw E T

0 .O6 ?, h5^|e, +J2" P . E T

0 oO6x+5, QQS^l E T CwO6+5, bh o|XTNM P xUEokx*OY. L .TH" P (ET, E T

0 ) ky03:

(1) P(ET) = 1, P(∅) = 0;

:JZ Brown Y; 59

(2) T A ∈ E T

0 , P(Ac) = 1− P(A);

(3) (kQ0) T A,B ∈ E T

0 b A ∩ B = ∅, P(A ∪ B) = P(A) + P(B).$/ 3.1.3 H"03 (3).ww" PQS^w E T o5,q2H" P.. ? E 0

T oo7,poH"^Oyfz"b$wUo? Ank P(An) ↓ 0.)v, \ ε > 0 & P(An) > ε. o?y, MQSlO6!y

tn ⊂ T Hn ∈ E kn , &n An = φ−1(t1,t2,··· ,tkn )

(Hn). ww5>f, kn = n. , µ(t1,··· ,tn)(Hn) = P(An) > ε.$/ 3.1.4 k? A ∈ E 0

TQS'Cw A = φ−1

I (H), Q4 I . T okJ, H ∈ E I . " : o A oJ? B, \kJ T oV I okJJ S K ∈ E J &n B = φ−1

J (K).Z E Y En .pQuU, ^Q ok5.!o, k3 Kn ⊂ Hn &µ(t1,··· ,tn)(Hn \Kn) <

ε

2n, Bn := φ−1

(t1,··· ,tn)(Kn), P(An \Bn) = µ(t1,··· ,tn)(Hn −Kn) <

ε

2n. Cn :=

⋂

k≤nBk, ,P(An \ Cn) = P(

⋃

k≤n

(An \Bk))

≤ P(⋃

k≤n

(Ak \Bk))

≤∑

k≤n

P(Ak \Bk) < ε.G P(Cn) > P(An)− ε > 0, Cn Kv.Uo, l x(n) ∈ Cn. Z Cn .fz"o, G l ≤ n, x(n) ∈ Cl ⊂ Bl, (x(n)(t1), x

(n)(t2), · · · , x(n)(tl)) ∈ Kl,

:JZ Brown Y; 60ho Kl 3, VI| l, wy x(n)(tl) : n ≥ 1 k4pJy, h%, \O6Kv?Jy ni, &^ l, x(ni)(tl)i≥1 4p, xl := limi x(ni)(tl).l x ∈ ET & x(tl) = xl. (x(t1), · · · , x(tl)) ∈ Kl, ZW^ l ≥ 1,

x ∈ Bl ⊂ Al, G ⋂

l≥1Al U, k/Æ.AVl |e4 E Jh.O6pQouU, Zw!Q o^+5.!o.x) 3.1.3 (Ulam) pQuU E o+5 µ .!o, ^ B ∈ B(E), kµ(B) = supµ(K) : K ⊂ B,K 3. (3.1.2)3'.h|e 1.2.1, +2" B = E o! QSw. hQ0, ^ n, \;w 1/n oQy6g An,k : k ≥ 1 $, E. ,^ n k

limi→∞

µ

(

⋃

i≥k≥1

An,k

)

= 1,G\ in &nµ

(

in⋃

k=1

An,k

)

> 1− ε/2n. A =⋂

n≥1

⋃ink=1An,k, , A .ppk/b

µ(Ac) ≤∑

n≥1

µ

(

in⋃

k=1

An,k

)

< ε,f K ") A o, , K .3b µ(K) ≥ µ(A) > 1− ε.$/ 3.1.5 o Borel B " (3.1.2).+ 3.1.2 QfoLUE. Gauss UE, Gauss UEokx*.!S*, !S*.hQPs%8BJ|o. f . T× T o'|X?, N^ n ≥ 1, t1t2, · · · , tn ∈ T, B (f(ti, tj) : 1 ≤ i, j ≤ j) .B'|o. !w µ(t1,··· ,tn) .PswzbSWw%8Bo!S*, TH"QPCO6 N0aokx*O.o7!;_, Q9JoLUE.kÆuUE.

:JZ Brown Y; 61+ 3.1.3 _od!okLuyo, o7!!BwkÆuUE. T = [0,∞), X = (Xt : t ≥ 0) .O6#(LUE, T^n ≥ 1, 0 ≤ t1 < t2 < · · · < tn, LUEoÆu

Xt1 , Xt2 −Xt1 , · · · , Xtn −Xtn−1.ko, , X Bw.kÆuUE; T^ t > s > 0, Xt −Xs sXt−s −X0 e*, ,C X.NÆuUE; T X .NÆuUEn.kÆuUE, ,C X .NkÆuUE.$/ 3.1.6 " : T X .kÆuUEbÆu Xt − Xs o*. νs,t, , (Xt1 , · · · , Xtn) on_*.

P((Xt1 , · · · , Xtn) ∈ A)

=

∫

E

µ(dx)

∫

A

ν0,t1(dx1 − x)νt1,t2(dx2 − x1) · · ·νtn−1,tn(dxn − xn−1),Q4 µ . X0 o*, "Z ν(dy− x) .")JoNQdo5 A 7→ ν(A− x)o~.$/ 3.1.7 " : O6YT. Gauss UE, ,QO|.kÆuo.$/ 3.1.8 νt : t ≥ 0 . R OO+5, N νt ∗ νs = νt+s, " :

(1) \ Levy UE X &n Xt −X0 o*. νt. (2) UE X Lo7ib4i^k/o7X? f , klimt↓0

νt(f) = f(0). Nt6ao+5O (νt) Bw.I~u.Lo7oNkÆuUEBw. Levy UE, SÆ8S+:Paul Levy. d Brown R9_6w, Zw Brown Y.yEo|. – y\u – O_o Levy UE.$/ 3.1.9 µ . Rd O6+5, |X

νt = e−t

∞∑

n=0

tnµ∗n

n!." : (νt) .O6I~u.

:JZ Brown Y; 62$/ 3.1.10 (E, E ) .O6Q5U, p(x, dy) Bw E CQX?, T^ x ∈ E, p(x, ·) . (E, E ) o+5, ^ A ∈ E , p(·, A) . E Q5X?. kCQX?Op(t, x, dy) : t > 0 N^ t, s > 0, A ∈ E , k

p(t + s, x, A) =

∫

E

p(s, x, dy)p(t, y, A).I|+5 µ, ^ I = 0 < t1 < t2 < · · · < tn |XµI(dx1dx2 · · · dxn) (3.1.3)

=

∫

x∈Eµ(dx)p(t1, x, dx1)p(t2 − t1, x1, dx2) · · ·p(tn − tn−1, xn−1, dxn)." : µI : I ∈ IT . E okx*O.

§3.2 B\A Brown O~ Brown , Q.+Æ': Robert Brown OK7lolJN^"oPo?:$-, 6$-koUE4, `/oe: A. Einstein V1eoS'z N. Wiener o;T.30o. +o=J1." Brown o\0. Brown .O6Rmo7oÆu*w!S*o Levy UE.|Xp(t, x) :=

1

(2πt)d/2e−

|x|2

2t ; x ∈ Rd t > 0 .Q.OO+X?, b_o+5O p(t, x)dx .O6I~u, Bwy\u, Zw p(t, x) .yRkE(

1

2∆− ∂

∂t

)

u = 0 (3.2.1)o|. (|..)L(w 0 wofw5o.), Q4∆ =

∑

i

∂2

∂x2i

:JZ Brown Y; 63. Laplace JJ. p(t, x, y) = p(t, x− y), Q. x wMoLUE!R t o0o*X?, Bw.CQ+. ^ t > 0 |XPtf(x) =

∫

Rd

f(y)p(t, x, y)dy, ∀f ∈ Cb(Rd) .Zw

p(t + s, x, y) =

∫

Rd

p(t, x, z)p(s, z, y)dz,G (Pt)t≥0 . Cb(Rd) ou. (Pt)t≥0 Bw Rd oyu, .Zw^f ∈ C2

b (Rd), u(t, x) = (Ptf)(x) .SL(aoyRkE

(

1

2∆− ∂

∂t

)

u(t, x) = 0 ; u(0, ·) = f , (3.2.2)o..$/ 3.2.1 H" u(t, x) = Ptf(x) .E (3.2.2) o..xz 3.2.1 +U (Ω,F ,P) ol(o Rd oLUE B = (Bt)t≥0 Bw. Rd o Brown , T1. (Bt)t≥0 FkkÆu: ^ 0 ≤ t1 < · · · < tn, Lu

Bt1 , Bt2 − Bt1 , · · · , Btn − Btn−1.ko.

2. ^ t > s ≥ 0, Lu Bt − Bs #Y!S* N(0, t− s), PBt −Bs ∈ dx = p(t− s, x)dx .

3. (Bt)t≥0 oeOkIRmo7.$/ 3.2.2 |X4 1,2 twpo (Bt1 , Bt2 , · · · , Btn) on_.p(t1, x1)p(t2 − t1, x2 − x1) · · · p(tn − tn−1, xn − xn−1). (3.2.3)

:JZ Brown Y; 64|o, T PB0 = x = 1, x ∈ Rd, ,C (Bt)t≥0 .Y x MoBrown . Z#s, T PB0 = 0 = 1, Q4 0 . Rd ozw, ,C(Bt)t≥0 .!F Brown . ZwR9CUy\u.O6I~u, OSa 1,2 QSY Kolmogorov |ekM, Q/o. "a 3 Ck.!F Brown B = (Bt) s Laplace JJ ∆ (ZWz) onÆhyap(^M_:

(Ptf) (x) = E (f(Bt + x))

=1

(2πt)d/2

∫

Rd

f(y)e−|y−x|2

2t dy.+ 3.2.1 T B = (Bt)t≥0 . R Brown , , p ≥ 0,

E|Bt − Bs|p = cp|t− s|p/2 for all s, t ≥ 0 (3.2.4)Q4 cp .P` p o<?. ,# E|Bt − Bs|p =

1√

2π|t− s|

∫

R

|x|p exp(

− |x|22|t− s|

)

dx .Su_rnx

√

|t− s|= y ; dx =

√

|t− s|dyZWkE|Bt −Bs|p =

(√

|t− s|)p√2π

∫

R

|x|p exp(

−|x|22

)

dx

= cp|t− s|p/2Q4cp =

1√2π

∫

R

|x|p exp(

−|x|22

)

dx =

(

2p−1

π

)12

Γ(p+ 1

2),Q4 Γ(·) . Γ X?. (3.2.4) Rd o Brown K.o, +.<? cp P`o p s d.$/ 3.2.3 ^ ξ ∈ R, " :

E(

e−ξ(Bt−Bs))

= exp

(

1

2|ξ|2(t− s)

)

. (3.2.5)

:JZ Brown Y; 65+4, f (F 0t )t≥0 ")!F Brown (Bt)t≥0 Co, b

F0∞ = σ

(

⋃

t≥0

F0t

)

.Zw^ 0 ≤ t1 < · · · < tn, Bt1 , Bt2 − Bt1 , · · · , Btn − Btn−1 Co σ- ?peo Bt1 , Bt2 , · · · , Btn Co σ- ?, Ghfz |ekMo\e, Q.kÆu03op">.) 3.2.1 ^ t > s ≥ 0, Æu Bt −Bs ko F 0s .. Brown o Markov 0.x) 3.2.1 t, s > 0, b f .k/ Borel Q5X?. ,

E

f(Bt+s)|F 0s

= Ptf(Bs) a.s. (3.2.6)Q4 (Pt)t>0 .y\u. Z#sE

f(Bt+s)|F 0s

= E f(Bt+s)|BsCk.3'. Zw Bt+s −Bs ko F 0s , Q. p(t, x), Bs . F 0

s Q5o, GkE(f(Bt+s)|F 0

s ) = E(f(Bt+s − Bs +Bs)|F 0s )

= E(f(Bt+s − Bs + x))|x=Bs

=

∫

f(y + x)p(t, y)dy|x=Bs

= Ptf(Bs). (3.2.6) t Bs laPsnlO6p(.$/ 3.2.4 Bt = (B1t , · · · , Bd

t ) . d- x!F Brown . ,6 j,

Bjt .!F Brown , b (Bj

t )t≥0 (j = 1, · · · , d) fk.ZW d- x!F. 1- x Brown o d 6k&1.

:JZ Brown Y; 66

§3.3 Brown s\0.|XO6+8!3J6C o. 601!o+8, t6eI+U, eIESUeI! oLUE Xt Yt Bwfw1!, T^ t k Xt = Yt a.s. vt6fw1!oLUEkeokx*O.x) 3.3.1 (A. Einstein, N. Wiener) Rd \k!F Brown .3'. d = 1, 2xo" . Fo. 6_f Kolmogorov 0|eO6+U (Ω,F ,P) Q oLUEUE X = (Xt) &nQokx* () .h (3.1.3) 8M. TH" (Xt)t≥0 N Brown |X4oa 1,2. Q9Jo., koBp(E|Xt −Xs|2n = (2n− 1)!!|t− s|n. (3.3.1)ZW2J1!LUE Xt &nQIRmo7. D = j

2n: j ∈

Z+, n ∈ N 'Æwp^. v D . R+ oQ?KJ. J" NyO6z+o, OkRm D .A,O/o7o, QSo7^.I|! ? N , _fBp( (3.3.1) 4 n = 2 o;_,

P

(

N2n⋃

j=1

∣

∣

∣X j

2n−X j−1

2n

∣

∣

∣≥ 1

2n/8

)

≤N2n∑

j=1

P

(

∣

∣

∣X j

2n−X j−1

2n

∣

∣

∣≥ 1

2n/8

)

= N2nP

(

∣

∣

∣X 1

2n

∣

∣

∣≥ 1

2n/8

)

≤ N2n(

2n/8)4

E∣

∣

∣X 1

2n

∣

∣

∣

4

=(

2n/8)4N2n3

(

1

2n

)2

=3N

2n/2,

:JZ Brown Y; 67Gh Borel-Cantelli \ekMP

(

limn

N2n⋃

j=1

∣

∣

∣X j

2nX j−1

2n

∣

∣

∣≥ 1

2n/8

)

= 0,ZW, TΩ0 :=

∞⋂

N=1

limn

N2n⋂

j=1

ω :∣

∣

∣X j

2n(ω) X j−1

2n(ω)∣

∣

∣<

1

2n/8

,,Q.Q5bkP(Ωc

0) = P

∞⋃

N=1

limn

N2n⋃

j=1

(

∣

∣

∣X j

2nX j−1

2n

∣

∣

∣≥ 1

2n/8

)

= 0.ZW, ω ∈ Ω0, ^ N , \ l &n^ n > l s j = 1, · · · , N2n k∣

∣

∣X j

2n(ω)−X j−1

2n(ω)∣

∣

∣<

1

2n/8.Zw.I|o ω 5>o, GQSClCO6, QS" (Tw ) ^ ω ∈ Ω0, Xt(ω) : t ∈ D ^k/j O/o7, Q [0,∞) kvOoo7^, w Bt(ω) : t ≥ 0. ω 6∈ Ω0, |X Bt(ω) = 0. h|X, Zw P(Ω0) = 1, G (Bt)t≥0 .O6o7oLUE, b^ t ≥ 0, Xt eQQ4po Bt. o,d." (Bt)t≥0 . X o1!, .Zw

E[(Xt −Xs)2] = t− s,G Xs N4po Xt, ZW Xt s Bt eQQp.Q7~8O6?:o n" (Xt(ω) : t ∈ D∩ [0, N ]) ^ N O/o7.$/ 3.3.1 α > 0 b f . D oX? N^ N , \ l &n^

n ≥ l s j = 1, · · · , N2n k∣

∣

∣

∣

f(j

2n)− f(

j − 1

2n)

∣

∣

∣

∣

≤(

1

2n

)α

.,^ N > 0, \<? CN &n^ s, t ∈ D ∩ [0, N ],

|f(s)− f(t)| ≤ CN |s− t|α,

:JZ Brown Y; 68 f . α-*A, Holdero7o. \): ^ s, t ∈ D∩ [0, N ]b |t−s| < 2−l,,\ m ≥ l &n2−m−1 ≤ |t− s| < 2−m. t Ro 2−m ww i2−m, , t QS")w

t = i2−m + 2−m(1) + · · ·+ 2−m(k),Q4 m < m(1) < · · · < m(k). haH"\<? c1(α) &n |f(t) −f(i2−m)| ≤ c(α)2−αm.hWkM\<? c2(α) &n

|f(t)− f(s)| ≤ c2(α)2−α(m+1) ≤ c2(α)|t− s|α.|o)/" f D ∩ [0, N ] .k/o, ZWi |t− s| ≥ 2−l !,

|f(t)− f(s)| ≤ 2max |f | · 2αl2−αl ≤ 2max |f | · 2αl|t− s|α,AV<? max |f | s l s N kJ.

Brown ok4)eo. Kv. Wiener 8MtO6p o" , hPo" ).8Mo,6. Wiener z_o" ( [15]).U C([0, 1]) O6'0X, vd" Q.65, d_BwWiener5. 8M Wieners PaleyOU\=o|oO6_f Fourier?o, QN nK`kVD. 3O 4o (3.6.2).$/ 3.3.2 B = (Bt) .p+U (Ω,F ,P) o Brown . µ . PdUxe Φ kGlxU (RT,BT) o5, W . RT 4oo7ep^. " : (1) W ).Q5, b µ∗(W ) = 0, µ∗(W ) = 1, Q4 µ∗ s µ∗#.15so5. (2) Φ−1(W ) Tw Ω oJ (OKQ5) oo5s15po 1, ZW Φ−1(W ) ∈ F .& W )Q5KJÆ, ZwQo5w 1, QS+5lW . |X W ,u

B(W ) := W ∩ A : A ∈ BT,S5

µ(W ∩A) := µ(A),

:JZ Brown Y; 69, (W,B(W ), µ) .+UbQ oRmUEs RT oRmUEp.$/ 3.3.3 " : µ .s|Xo, K.CT W ∩A =W ∩B, , µ(A) =

µ(B). " (W,B(W ), µ) oRmUEs RT oRmUEp.

§3.4 Brown ss B = (Bt)t≥0 .p+U Jo (Ft) o d- x!F Brown ,QKvOkz+dow (Ft). . Brown o.03.) 3.4.1 ^#? λ 6= 0

Mt ≡ λBt/λ2. Rd !F Brown .W-Qh Brown o|X&)kM. Z#s, (−Bt)t≥0 K.!FBrown . . Brown o8C)0.) 3.4.2 U . d×d!$B, , UB = (UBt)t≥0 . Rd !F Brown. K.C, Brown !$r.)o.ww" O603, 0 Brown o|oO6Rj. B = (Bt)t≥0. R !F Brown . , B .4*l Gauss UE, Q%8X?wC(s, t) = s∧t. ,# ,6 B o^kx*. Gauss*,b EBt = 0,G B .4*l Gauss UE, JQ%8X?, s < t,

E[BtBs] = E[(Bt − Bs)Bs +B2s ]

= E[(Bt − Bs)Bs] + E[B2s ]

= E[Bt − Bs]E[Bs] + E[B2s ] = s .U_K)/H", ~T . . Brown oO6kfoRj.x) 3.4.1 O6o7o4*l Gauss UE (Xt : t ≥ 0), Fk%8X?

E[XsXt] = t ∧ s, s, t ≥ 0,.!F Brown .

:JZ Brown Y; 70. Brown o!5C)0.) 3.4.3 |XoUEM0 = 0 , Mt = tB1/t, t > 0. R !F Brown .3'. v Mt .4*l Gauss UEbQ%8X?w

E (MtMs) = tsE(

B1/tB1/s

)

= ts

(

1

t∧ 1

s

)

= s ∧ t.s Brown O/, OS+2" (Mt) o7QSw, goJ." M t = 0 !o7, limt↓0 tB1/t = 0 a.s.,# , M s B keokx*, 9D Brown o|e, \ M oO6o71!, w M , Zw M (0,∞) .o7o, G M s M (0,∞) .apo, ZW limt→0Mt = limt→0 Mt = 0.|O6" limt→0Mt = 0 of._f Doob oY)p(. hY)p(P(max|Mt| : 2−(n+1) ≤ t ≤ 2−n > 1

n) ≤ 4n2E[M2

2−n ] =4n2

2n.h Borel-Cantelli \ekMi n→ ∞ !k

max|Mt| : 2−(n+1) ≤ t ≤ 2−n → 0, a.s.Vk Mt → 0 a.s.# R9" wO~o Brown o_a?|.x) 3.4.2 B = (Bt) .!F Brown . limt→∞

Bt

t= 0 a.s." Brown k Markov 0.x) 3.4.3 (_10) ^ (F 0

t+)- ! T , Sk/o7X? f , kE[f(Bt+T − BT )|F 0

T+]1T<∞ = E[f(Bt)]1T<∞. (3.4.1)

:JZ Brown Y; 713'. ,# , ^ t, s > 0, Bt+s − Bs s Fs k, GE[f(Bt+s − Bs)|Fs] = E[f(Bt)]. T (n) . T odlT (n) =

∑

k≥1

k

2n1k−1

2n≤T< k

2n.,^ H ∈ F 0

T+, k H ∩ T < t ∈ Ft ^ t > 0 Ck, ZW^n ≥ 1, h|e 3.2.1

E[f(Bt+T (n) − BT (n));H, T <∞]

=∑

k≥1

E[f(Bt+k/2n −Bk/2n);H ∩ T (n) = k/2n]

=∑

k≥1

E[f(Bt)]P(H ∩ T (n) = k/2n)

= E[f(Bt)]P(H ∩ T <∞) .w n iof, h Brown o70kM (??).|e# .C (Bt+T − BT : t ≥ 0) .O6s FT+ ko!F Brown, hWkME[f(Bt+T )|FT+] = Ptf(BT ) = E[f(Bt+T )|BT ],Q4 Ptf(x) = E[f(Bt + x)]. .9Jo_ Markov 0: 8|, _sUnk.$/ 3.4.1 if o_ Markov 0" : oI|! t k Ft = Ft+. K.C, +J BrownoKv4z+, QKv Nd<a.ze.~ Markov 03:wEaoO6+8. _f6ze_J Brown oajEo*. 4_f#j4, Z#f4, 2JCLUEoajEo*. o Brown B = (Bt)t≥0 _C,ajE sups∈[0,t]Bs o*QShzeo_kM. B = (Bt) .

:JZ Brown Y; 72!F Brown . I|O6! t > 0, w Brown ! s %dP Bs , B′

t =

Bt, t ≤ s;

2Bs − Bt, t > s,, B′ = (B′t) v.o7o, v.4*l Gauss UEb E[B′

uB′v] = u ∧ v.ZW B′ K.!F Brown . 603Bwze, Qo !K.Cko.$/ 3.4.2 H" E[B′

uB′v] = u ∧ v.zei s . !!KCk. b > 0 s b > a, %Tb = inft > 0 : Bt = b ., Tb . !, |X Tb QoB′

t =

Bt, t ≤ Tb;

2b−Bt, t > Tb,, B′ = (B′t) K.!F Brown . ZW

P

sups∈[0,t]

Bs ≥ b, Bt ≤ a

= P

sups∈[0,t]

B′s ≥ b, B′

t ≤ a

= P( sups∈[0,t]

Bs ≥ b, Bt ≥ 2b− a

= P Bt ≥ 2b− aQ4tO6p(.h Brown Tb %d (Y0 b M': BTb= b) CnF Brown , OSQJo y = b .Bo, e!F Brown JozwB. tÆ6p(.B0ok. K.O~oze. oEQS'Cw

P Tb ≤ t, Bt ≤ a = P Tb ≤ t, Bt ≥ 2b− a (3.4.2)

= P Bt ≥ 2b− a ,

:JZ Brown Y; 73Jozeo?5" Jfl Brown o_ Markov 0.hze (3.4.2) QnP

sups∈[0,t]

Bs ≥ b, Bt ≤ a

=1√2πt

∫ +∞

2b−a

e−x2

2t dt ,Q8Mw Brown sQajEon_*.Z#s, i a = b > 0 !, kP(Tb ≤ t) = 2P(Bt > b) = 2P(B1 >

b√t)

=2√2π

∫ +∞

b/√t

exp

(

−x2

2

)

dx,QoX?wp(t) =

b√2πt3

exp

(

−b2

2t

)

.$/ 3.4.3 " : i b 6= 0 !E(

e−sTb)

= e−√2s|b|.>( P(Tb <∞) = 1 V|G Brown T|QSk!1#4^Ow

b, &K .vo, ZwQoRmo7 bv −∞ l +∞ %j.h.603sx?kJ, 2- xS o Brown )aQ2qQS#4O68|ow. tasC, T B . d- x Brown , d ≥ 2, x ∈ Rd, ,P(Tx <∞) = 0, (3.4.3)Q4 Tx . x o64!. !C, ox Brown , fw.. . 1- xsx%oEaj#. oJP`o b > 0. zwo64! T = T0 nv I0?$/ 3.4.4 o6o7RmoI ω, " : b 7→ Tb(ω) .Ro7ohlo7.6r C )2fw b iozo_.J. 2J_f|oO6`, t > 0,

T t := infs > t : Bs = 0 = t + infs > 0 : Bs+t = 0

:JZ Brown Y; 74! t %d6X#4 0 o!R. v T t ↓ T . Zw X = (Bs+t − Bt : s > 0).ko Bt o!F Brown , G T t po t X 64 −Bt o!, K.C,

E[e−sT t

] = e−stE(

e−√2s|Bt|

)

.w t iozkM E[e−sT ] = 1, P(T = 0) = 1.$/ 3.4.5 (*)Brown eaIRmozw_. Lebesgue 5wzoDkwo.OY0.x) 3.4.4 1. Ox Brown (Bt)t≥0 .o7NQ~Y.

2. o d- x Brown (B(i)t : 1 ≤ i ≤ d), Mt = B

(i)t B

(j)t − δijt .o7Y.3'. tO,S" U. i t > s !, Zw Bt − Bs ko Fs. ZWk

E(Bt − Bs|Fs) = E(Bt − Bs) = 0.,E(Bt|Fs) = E(Bs|Fs) = BsK.C, (Bt)t≥0 .o7Y.v+2Ox Brown " 3). 8d.,

E(B2t −B2

s |Fs) = E((Bt −Bs)2 |Fs)

+ E(2Bs (Bt − Bs) |Fs)

= E((Bt −Bs)2) + 2BsE((Bt −Bs) |Fs)

= E (Bt − Bs)2

= t− sG E(B2

t − t|Fs) = E(B2s − s|Fs)

= B2s − sR9" w B2

t − t .O6Y.

:JZ Brown Y; 75|efZX?vO0o%_N.vo.x) 3.4.5 Euclid U Rd Y 0 wM/_o'6 (Ft) oo7LUE B = (Bt) .!F Brown ib4i^ ξ ∈ R s t > s

E exp (i〈ξ, Bt −Bs〉) |Fs = exp

(

−(t− s)|ξ|22

)

. (3.4.4)`6 3.4.1 (Bt) . Rd !F Brown . T ξ ∈ Rd, ,Mt ≡ exp

(

i〈ξ, Bt〉+|ξ|22t

).Y.4) 3.4.1. AVl (3.4.4) ot. ξ o.X?, Gp(^&( ξ KCk. Z#s, f −iξ _ ξ, nE exp (〈ξ, Bt − Bs〉) |Fs = exp

(

(t− s)|ξ|22

)

.ZW^u ξ

exp

(

〈ξ, Bt〉 −|ξ|22t

).O6o7Y. 6-KQSkOl Rd ou; ξ = (ξ(t)), hWnlop(Bw Cameron-Martin >(.wif Brown oY03S Doob |e_.JO$.+ 3.4.1 B = (Bt) . 1 x!F Brown . a > 0, |X Ta . B lwa o64!, ,Q. !, fze" w Brown O|vl^ a,

P(Ta <∞) = 1.KQSfY_._6. :Os, Brown vT|FlOa&o& x = kt− a ? T . Brown B 6XFla&o!, T = inft > 0 : Bt = kt− a.

:JZ Brown Y; 76KQSC.JQ Brown (Bt − kt) 6XFl −a o!, h P(T <∞).) a > 0, i k = 0 !, T = T−a. &KN, i k > 0 !, P(T < ∞) = 1, i k < 0 !.h)?Y03, ^#? z,

exp

(

zBt −z2

2t

)

, t ≥ 0.Y. ,h Doob |e,

E

[

exp

(

zBt∧T − z2

2(t ∧ T )

)]

= 1. (3.4.5)w t iof, oJ.sPs.!QS$r. i z < 0 !,

zBt∧T − z2

2(t ∧ T ) ≤ z(k(t ∧ T )− a)− z2

2(t ∧ T )

= (zk − z2

2)(t ∧ T )− za.ZWJ" (3.4.5) 4o)?Jo t, ω k/, 3 zk − z2/2 ≤ 0. 2t8dZ, O8. k ≥ 0, !+J z < 0; |oO8dZ. k < 0, !+J z < 2k.+8dZ, QS_fk/4p|e,

E

[

exp

(

zBT − z2

2T

)

;T <∞]

= 1.! BT = kT − a, ZWE[

e(zk−z2

2)T ;T <∞

]

= eza.tO8dZ, QSw z ↑ 0, n P(T < ∞) = 1; tÆ8dZ, wz ↑ 2k, n

P(T <∞) = e2ka < 1.tO8 k ≥ 0 dZ, QSJM T o Laplace r, ZwE[

e(zk−z2

2)T]

= eza, −s = zk − z2/2, s > 0, n z = k −√k2 + 2s < 0, ZW

E[e−sT ] = ea(k−√k2+2s). (3.4.6)

:JZ Brown Y; 77Tf Ta ") o T , , (Ta : a ≥ 0) .O6NkÆuUE, Bw|UE.+ 3.4.2 a < 0 < b, Ta, Tb #.zwMo Brown 6XFl a, b o!. fY_h P(Ta < Tb), E[T ]S T o Laplacer,Q4 T = Ta∧Tb.hYoPs)0, ^ t > 0,

EBT∧t = 0.i t→ ∞ !, a ≤ BT∧t ≤ b, ZWhV14p|en EBT = 0. BT = a1Ta<Tb + b1Tb<Ta,ZW aP(Ta < Tb) + bP(Tb < Ta) = 0,

P(Ta < Tb) =b

b− a.JJ ET , 2JfY (B2

t − t), q.hPs)0,

E[B2T∧t] = E[T ∧ t].vdw t→ ∞, R_fV14p|el_ffz4p|en

E[T ] = E[B2T ] = a2P(Ta < Tb) + b2P(Ta > Tb) = −ab. Fs, JJ T o Laplace r, _)f)?Y (exp(λBt − λ2t/2)). Ps)0V14p|enE(

exp(λBT − λ2T/2)

= 1, (3.4.7)ZWeλaE

[

e−λ2T/2;Ta < Tb

]

+ eλbE[

e−λ2T/2;Ta > Tb

]

= 1. (3.4.8)q)NSJM E[

e−λ2T/2]

. O)?Y (exp(−λBt − λ2t/2)), FsnE(

exp(−λBT − λ2T/2)

= 1 (3.4.9)

:JZ Brown Y; 78e−λaE

[

e−λ2T/2;Ta < Tb

]

+ e−λbE[

e−λ2T/2;Ta > Tb

]

= 1. (3.4.10)Y t6EkME[

e−λ2T/2]

=sinh(λa)− sinh(λb)

sinh(λ(a− b))(3.4.11)Q4 sinhx = (ex − e−x)/2. hWkRnl T o Laplace r.

Brown . Levy UEo|jJ, Levy UE. Euclid U olo7NkÆuUE, (3.4.4) .Jo Brown o Levy-Khinchin >(. Os, T (Xt) .O6 d- x Levy UE, , t > s s ξ ∈ Rd

E exp (i〈ξ,Xt −Xs〉) |Fs = exp (ψ(ξ)(t− s))Q4 ψ .o7&(X?, Bw Levy )?, QvOsJ| Levy UE. Levy )?ko"^(, Bw Levy-Khinchin >(,

ψ(ξ) = −1

2〈Sξ, ξ〉+ i〈b, ξ〉

+

∫

Rd\0

(

ei〈ξ,x〉 − 1− i〈ξ, x〉1|x|<1)

ν(dx), (3.4.12)g S .O6 d × d B'|B, b ∈ Rd, ν . Rd '[zwooσ- k5, Bw X o Levy 5, Q NyQ~a

∫

Rd\0(|x|2 ∧ 1)ν(dx) < +∞ . (3.4.13)

§3.5 Brown sbieNlo,I, Ox Brown Bt s Mt ≡ B2t − t .Y, ,

B2t =Mt + AtQ4 At = t. ZW, o7Y B2

t .O6YsO6Æ/_UEo. vNl6.. Ito L~enSkoJ.

:JZ Brown Y; 81ZWlim

m(D)→0E

[

∑

l

|Btl −Btl−1|2 − t

]2

= 0 . o4p.P+4p. h|e4okln:Yom, o4pQSC.eQQo.8\ 3.5.1 (Bt)t≥0 .Ox!F Brown . ,^ t > 0, i n iof!, k2n∑

j=1

[

B j2n

t − B j−12n

t

]2

→ t a.s. (3.5.1)3'. Dn . [0, t] oÆwkDn = 0 =

0

2nt <

1

2nt < · · · < 2n

2nt = t .bf Vn ") VDn

. ,, \e 3.5.1, EVn = t and

E |Vn − EVn|2 = 22n∑

l=1

(

l

2nt− l − 1

2nt

)2

= 2n+1

(

1

2nt

)2

=1

2n−1t2 .ZWh Markov )p(,

P

|Vn − EVn| ≥1

n

≤ n2E |Vn − EVn|2

=n2

2n−1t2G

∞∑

n=1

P

|Vn − EVn| ≥1

n

= t2∞∑

n=1

n2

2n−1< +∞ .Ih Borel-Cantelli \en Vn → t eQQ4p.

:JZ Brown Y; 82,# , eQQ4pn-ofzkyCk. :tasC, ^ n Dn = 0 = t0,n < t1,n < · · · < tkn,n = t. [0, t] okk. T Dn+1 ⊃ Dn b

limn→∞

m(Dn) = limn→∞

max |tni,n − tni−1,n| = 0,,i n iof!n∑

i=1

∣

∣

∣Btki,n

− Btki−1,n

∣

∣

∣

2

→ t a.s. (3.5.2)W-ho#s'Yy4p|e&)kM, " 3 .8\ 3.5.2 f Mn ") (3.5.2) oR, !Co6y· · · · · · ,Mn, · · · ,M2,M1.O6'Y.2B" ()T) p > 2,

supD

∑

l

|Btl − Btl−1|p <∞ a.s.Q4 sup .Ok [0, 1] okklo,

supD

∑

l

|Btl − Btl−1|2 = ∞ a.s.,.C, Brown kk p- 8, p > 2. ,# , α < 1/2 !, BrownoeOkIRm. α-Holder o7o, hQ). α = 1/2-Holder o7o. # qQS" QeOkoIRm^kj QQ)Qk.)`JgM Brown Rmo:>, Qo Brown oL~e).2o.xz 3.5.1 p > 0 .<?. j [0, T ] O6l( Rd oX? f Bwj

[0, T ] Fkk p- 8, supD

∑

l

|f(ti)− f(ti−1)|p < +∞Q4 D C [0, T ] oOkkk. k8.)k 1- 8.

:JZ Brown Y; 83k8X?d<KBwk/8X?. w"0? k8.)QM8k, k/8.)Q8Twk oX?k/, t.eIoVX. k (M) 8oX?.t6vÆX?o8, Z#s, Qo)o7wp^..Qyo.O6LUE V = (Vt)t≥0 Bwk8UE, TQeOkIRmt→ Vt(ω) ^kj kk8. Brown ).k8UE.

§3.6 lp1. TUE X,X ′ .lo7o (zRo7), ,Qfw1!VGQ.)Qjo.

2. " : A ∈ E T\ T oQyJ K &nA ∈ σ(Zt : t ∈ K).

3. C ET oO6J A .QyJ|o, T\O6Qy S ⊂ T, &nx s y S p, x ∈ A ib4i y ∈ A. " : (1) E T 4o_.QyJ|o. (2) T E .oO6wouU, , T l E oo7Rmp^) E T 4.

4. X = (Xt : t ∈ T) .#(lo7LUE. " supt∈[a,b]Xt . (OX#()Lu. x = (x(t)) ∈ R[0,1], |Xf(x) = sup

t∈[0,1]x(t)." : f Tw R[0,1] oX?Jo B(R)[0,1] )Q5.

5. (Kolmogorov) T > 0. T\!<? α, β, C &n#(UE X N^ t, h > 0, t, t + h ∈ [0, T ], kE|Xt+h −Xt|α ≤ C · h1+β . Brown \o" " : X ko71!.

:JZ Brown Y; 84

6. ^ α ∈ (0, 1], " : (1) \ Gauss UE ξt : t ∈ R &nE[|ξt − ξs|2] = |t− s|α. (3.6.1)

(2) UEko71!, Bw? Brown .

7. " : +U ([0, 1],B([0, 1]),P) )Q2k)Q?6<?okLu, Q4 P . Lebesgue 5.

8. d64, B = (Bt) .!F Brown . r < s < t. hE0(Bs|Br, Bt).

9. (*) O6 F oJ σ- ?Bwk 0-1 , TQ4Vk+w 0 z 1 o_. z σ u ⋂

t>0 σ(Bs : s ≥ t) ssL> σ u ⋂

t>0 σ(Bs : s ≤ t) .!k 0-1 ?

10. " :

P(supt>0

Bt = +∞, inft>0

Bt = −∞) = 1.

11. " : (*) ^ tn ↓ 0, kP(limkBtk > 0, limkBtk < 0) = 1.

12. B = (B(t) : t ∈ T) . 1- x!F Brown , Xt := e−tB(e2t), t ∈ R." : X .Fk Markov 0o4*l Gauss UE, J X o%8X?.hW" X .NUE (AV).NÆuUE), Qkx*NQ): ^ t1, · · · , tn, t > 0, (Xt+t1 , · · · , Xt+tn) s (Xt1 , · · · , Xtn) e*. UE X Bw. Ornstein-Ulenbeck UE.

13. a > 0, b .#?, τ := inft > 0 : Bt > a+ bt 6XW&o!,h P(τ < +∞), iW+po 1 !, h τ o Laplace r.

:JZ Brown Y; 85

14. B . d- x!F Brown , (1) " : ^ x ∈ Rd, ||x|| = 1, 〈x,Bt〉. 1- x!F Brown . (2) r > 0, Tr := inft : |Bt − B0| ≥ r. x ∈ Rd, " : (1) Px(Tr < ∞) = 1. (3) " * Px(BTr∈ ·) .g

y : |y − x| = r L*.

15. Brown qQSf Fourier ?o9 (Wiener). ξn : n ≥ 0.+U (Ω,F ,P) o#Y!F!S*okLu6y. (C Q\0.) X?1/√π,

√

2/π cosx, · · · ,√

2/π cos nx, · · · . L2([0, π] o!F!$|, PXw en : n ≥ 0. ^ f ∈ L2([0, π]),H(f) := a0ξ0 + a1ξ1 + a2ξ2 + · · · ,Q4 an . f o Fourier Æ?: an := 〈f, en〉. vEH(f)2 =

∑

n≥0

a2n =

∫

f 2(x)dx, H . L2([0, π]) l L2(Ω,F ,P) oO6pG^. t ∈ [0, π], v 1[0,t]o Fourier ?w1[0,t](x) =

t√π+

√

2

π

∑

n≥1

sin nt

n· cosnx." : /iZd

Xt := H(1[0,t]) = tξ0 +∑

n≥0

2n−1∑

m=2n−1

sinmt

mξm. (3.6.2) t ∈ [0, π] S+ 1 O/4p. ZW (Xt) .o7UE, " Q [0, π] . Brown .+(:

sm,n(t) :=n−1∑

m

sin kt

kξk, tm,n =: max

0≤t≤π|sm,n(t)|.

:JZ Brown Y; 86f Cauchy )p(C E(

t2m,n

)

, Q4q.kO$`(n: , )K33, o)o." ∑

E (t2n−1,2n) <∞.3O [11],

E(

t2m,n

)

≤ E

maxt

∣

∣

∣

∣

∣

n−1∑

m

eikt

kξk

∣

∣

∣

∣

∣

2

≤n−1∑

m

1

k2+ 2E

(

n−m−1∑

l=1

∣

∣

∣

∣

∣

n−l−1∑

j=m

ξjξj+l

j(j + l)

∣

∣

∣

∣

∣

)

≤n−1∑

m

1

k2+ 2

n−m−1∑

l=1

E

∣

∣

∣

∣

∣

n−l−1∑

j=m

ξjξj+l

j(j + l)

∣

∣

∣

∣

∣

2

12

≤n−1∑

m

1

k2+ 2

n−m−1∑

l=1

(

n−−l−1∑

j=m

1

j2(j + l)2

)12

≤ n−m

m2+ 2

(n−m)3/2

m2,l n = 2m n

E

(

∑

n≥1

t2n−1,2n

)

≤∑

n≥1

(

E[t22n−1,2n ])1/2

< +∞.

16. (Wald p() B .!F Brown , σ, τ .t6Q~ !b σ ≤ τ . " : (1) Bτ NQ~b E[B2τ ] = E[τ ]. (2) σb . b ∈ R (b 6= 0) o64!,, E[σb] = ∞. (3)

E[(Bτ −Bσ)2] = E[B2

τ − B2σ] = E[τ − σ].3'.(1) Zw B2

t − t.Y, h Doob *|e, E[B2τ∧n] = E[τ ∧n] ≤ E[τ ] <

∞. h Fatou, E[B2τ ] <∞. h Doob )p(, E[supnB

2τ∧n] ≤ 4E[τ ], ZW

B2τ∧n : n ≥ 1 O/Q~, Y E[B2

τ ] = E[τ ]. (2) ".

:JZ Brown Y; 87

17. (I|!oze) B = (Bt) .!F Brown , V s > 0, |XB′

t =

Bt, t < s;

2Bs − Bt, t ≥ s." B′ = (B′t) K.!F Brown . # B′ . B ! s Q%doRmEG& y = Bs Onlo.

18. ( !Qoze) -i s ! _!KCk.

19. " # 3.5.2.3'.(2) 2J" E[M1|M2,M3, · · · ] =M2. (3.6.3)wfU, D1 = 0, t, D2 = 0, s, t,

E[M1|M2,M3, · · · ] =M2 + 2E[(Bt −Bs)(Bs − B0)|M2,M3, · · · ],) Brown .Rm-o, V ω ∈ Ω s s > 0, θω(u) =

ω(u), u < s;

2ω(s)− ω(u), u ≥ s.,h Brown o\e, bi s . Mn 4oO6w!, Mnθ =

Mn, G E[(Bs −Bt)(Bs − B0)|M2,M3, · · · ] = E[(Bt − Bs)(Bs −B0)|M2,M3, · · · ],ZW (wz, (3.6.3) Ck.

20. (Xt, Yt) .Æx!F Brown , s > 0, Ts . X o s o64!,Zs := YTs

," : Z = (Zs) . Levy UE, JQ Levy )?.

:JZ Brown Y; 88

21. (DO) W = C[0, 1], (W,B(W ), µ) . Wiener U. o h ∈ W , |X W oNQrThx(t) = x(t) + h(t).: µT−1

h .!s µ pz"ap?

vV Ito 4, S9xoM Ito ~e, K.C, fUE (Ft)t≥0 |XL~ ∫ t

0FsdBs, vdif Ito ~oY0|X^.`Oo~UE. BCOH, Ito >L~o.>P.,mo, hLtE

dXt = σ(Xt)dBt + b(Xt)dtOt|o1UE X . P|wIoE.(&K >wlJoO8L$(.

§4.1 |7 B = (Bt)t≥0 .+U (Ω,F ,P) Ox!F Brown , (F 0t )t≥0. (Bt)t≥0 CoKv, ^ t ≥ 0

F0t = σBs for s ≤ tQ") Brown B = (Bt)t≥0 &.! t oh%. Okz+%dw (Ft).|Xy.(o Ito ~

∫ t

0

FsdBs for t ≥ 0BwO6o7LUE, Q4~UE F = (Ft)t≥0 . N'$a (d.) oO6LUE. j, QS|X~∫ t

0

f(Bs)dBso Borel Q5X? f .O [0, 1] ot6X? f, g, Dn .O6_iozoky, |XSn(f) :=

∑

tk ,tk−1∈Dn

f(tk−1)(g(tk)− g(tk−1)).

89

:MZ Ito B> 90O6:O;$o-TC, T^o7X? f , Sn(f) 4p, ,X? g Dn o8k/.$/ 4.1.1 " 6-. \): _f?!|e.U_C, T g ).k/8o, ,S ()Q2Oko7X?4p. ZweOko ω ∈ Ω, Brown oIRm t → Bt(ω) ^j kk/8, GIRmo Riemann .(∑

i

Ft∗i(Bti −Bti−1

)_|X~.kVXo. L~.C, _/s9l t∗i ∈ [ti−1, ti]bUE (Ft)t≥0 /_o (F 0t )t≥0, ,6TwLuP+4poVXos.\o. 6k#.Zw (Bt)t≥0 s (B2

t − t)t≥0 .o7Y.fsC, /_Ro7UE F = (Ft)t≥0 Jo Brown B = (Bt)t≥0 oIto ~

(F.B)t =

∫ t

0

FsdBsv.O6LUE, ikkioz!, o^I|o t, .ZZ8o Riemann o∫ t

0

FsdBs = limm(D)→0

∑

i

Fti−1(Bti −Bti−1

)Q4o\Q2 L2- VXh:o.P+4poVX, kioz.)D = 0 = t0 < t1 < · · · < tn = t N m(D) = maxi(ti − ti−1) → 0. 9Rwoeh.: +kI9, /k

E(

Fti−1(Bti − Bti−1

))

= 0 (4.1.1)sE(

F 2ti−1

(Bti −Bti−1)2 − F 2

ti−1(ti − ti−1)

)

= 0 , (4.1.2)

:MZ Ito B> 91Y &nL~ "oLUE F.B .O6Y, )4W, :9Js, FBrown , QNdnO6ÆUEnloUE

(

(F.B)2t −∫ t

0

F 2s ds : t ≥ 0

)K.O6Y.hWnlO~o Ito pGeE

(∫ t

0

FsdBs

)2

= E

∫ t

0

F 2s ds, (4.1.3)I|! T , [0, T ] oY (Mt) .hQ7( MT OvOJ|oQV|Ge

F 7→∫ T

0

FsdBskw?w ||F || =√

E∫ T

0F 2s ds < ∞ oo7/_UE F oUl?w

||M || =√

E(M2T ) o7NQ~YUoO6pGe. ho6pGe,ifNQ~YUop0, QSfUE|X Ito ~vd^.Q. eNlo, Ito pGeoJ. (B2

t − t) .O6Y, zC (Bt) kÆX8UE.L~ofLUE|X, vdif Ito pG5/^, :OoLUE|X, %difA,l=^lA,k/UEJoo7A,Yo~, Qd^lJoo7Yo~.

§4.2 wX8|+U (Ω,F ,Ft,P) o!F Brown B = (Bt : t ≥ 0). ) (Ft) Nd<a.xz 4.2.1 O6/_LUE F = (Ft)t≥0 BwfUE (z*[UE),TQ.O6QS2B")w.(ok//_UEFt(ω) = f(ω)10(t) +

∞∑

i=0

fi(ω)1(ti,ti+1](t) (4.2.1)Q4 0 = t0 < t1 < · · · < ti → ∞.

:MZ Ito B> 92o ofUE, ,^ t ≥ 0, +kk6 ti ∈ [0, t], |o fi.Jo Fti) Q5o. fLUEp^f"Z L0 "). T F = (Ft)t≥0 ∈L0, , F Jo Brown B = (Bt)t≥0 oL~|XwO6LUEI(F ),

I(F )t ≡∞∑

i=0

fi(Bt∧ti+1− Bt∧ti)Q4 >(4+kk.zo. v I(F ) = (I(F )t)t≥0 .o7NQ~ob/_o (Ft)t≥0.) 4.2.1 (I(F )t)t≥0 .6Y

E (I(F )t − I(F )s|Fs) = 0 , ∀t > s .3'. tj < t ≤ tj+1, tk < s ≤ tk+1 o'6 k, j ∈ N. , k ≤ j bI(F )t =

j−1∑

i=0

fi(Bti+1−Bti) + fj(Bt − Btj ) ;

I(F )s =

k−1∑

i=0

fi(Bti+1−Bti) + fk(Bs − Btk) .T k < j − 1, ,

I(F )t − I(F )s =

j−1∑

i=k+1

fi(Bti+1− Bti)

+fj(Bt −Btj ) + fk(Btk+1− Bs) . (4.2.2)T k + 1 ≤ i ≤ j − 1, , s ≤ ti G Fs ⊂ Fti . ZW

E(

fi(Bti+1− Bti)|Fs

)

= E

E(

fi(Bti+1− Bti)|Fti

|Fs

= E

fiE

Bti+1− Bti |Fti

|Fs

= 0 .tO6p(.ho fi ∈ Fti , tÆ6p(ho (Bt) .6Y. FoE(

fj(Bt −Btj )|Fs

)

= 0, t > tj ≥ s, fj ∈ Ftj ,

:MZ Ito B> 93

E(

fk(Btk+1− Bs)|Fs

)

= 0, tk+1 ≥ s > tk, fk ∈ Ftk ⊂ Fs .$EOU, nlE (I(F )t − I(F )s|Fs) = 0 .T k = j − 1, , tj−1 < s ≤ tj < t ≤ tj+1 b

I(F )t − I(F )s = fj−1(Btj − Bs) + fj(Bt − Btj )ZWkE (I(F )t − I(F )s|Fs) = 0 .) 4.2.2

(

I(F )2t −∫ t

0F 2s ds)

t≥0K.O6Y.3'. J" ^ t ≥ s

E

(

I(F )2t −∫ t

0

F 2udu

∣

∣

∣

∣

Fs

)

= I(F )2s −∫ s

0

F 2udu .rHmC, 2J"

E

(

I(F )2t − I(F )2s −∫ t

s

F 2udu

∣

∣

∣

∣

Fs

)

= 0 .vI(F )2t − I(F )2s = (I(F )t − I(F )s)

2 − 2I(F )tI(F )s − 2I(F )2s

= (I(F )t − I(F )s)2 − 2(I(F )t − I(F )s)I(F )s.Zw (I(F )t)t≥0 .6Y, G

E (I(F )t − I(F )s|Fs) = 0 .h. I(F )s ∈ Fs GkE I(F )s (I(F )t − I(F )s) |Fs

= I(F )sE I(F )t − I(F )s|Fs = 0 .

:MZ Ito B> 94ZW2J" E

(I(F )t − I(F )s)2 −

∫ t

s

F 2udu

∣

∣

∣

∣

Fs

= 0 .&f\e 4.2.1 o" 4Ofoe"Z. h (4.2.2) TNMTk < j − 1, ,

(I(F )t − I(F )s)2 =

j−1∑

i,l=k+1

fifl(Bti+1− Bti)(Btl+1

−Btl)

+

j−1∑

i=1

fifj(Bti+1− Bti)(Bt − Btj )

+

j−1∑

i=1

fifk(Bti+1− Bti)(Btk+1

−Bs)

+f 2j (Bt −Btj )

2 + f 2k (Btk+1

− Bs)2

+fkfj(Bt − Bti)(Btk+1−Bs) .if\e 4.4.1 S (Bt)t≥0 s (B2

t − t)t≥0 .Yo,#, nlE

(I(F )t − I(F )s)2∣

∣Fs

= E

(

j−1∑

j=k+1

f 2i (ti+1 − ti) + f 2

j (t− tj) + f 2k (tk+1 − s)

∣

∣

∣

∣

∣

Fs

)

vdE

(I(F )t − I(F )s)2|Fs

= E

(∫ t

s

F 2udu

∣

∣

∣

∣

Fs

)

.ZWT F .O6fUE, , Ito ~ I(F ) .6L(wzoo7NQ~Y, F → I(F ) .0o, b^ t ≥ 0,

E(

I(F )2t)

= E

(∫ t

0

F 2s ds

)

. (4.2.3)p( (4.2.3) Bw Ito pG, Q.|XL~oJ. L~o|X^lfUEo.

:MZ Ito B> 95ww Ito ~o|X^l:a o~UE . 62J\t69JoU. M 20 .o7oNQ~ (Ft)- YUE M = (Mt : t ≥ 0) p^, 6UoGdkt8|X(, O8.Qy||M ||

M 2 :=∑

n≥1

2−n[√

E[M2n ] ∧ 1],6).!o?# +2O^ T > 0,

√

EM2T QSw, # .%QyU. |oO8|X(.Q4 N supt>0 EM

2t < ∞ oo7Y*M_, |X

||M ||M 2 :=

√

supt>0

EM2t .8(:-=Ow, h Brown K) N6a, ).`Y. g1ftO8|X, I# M.O^ T > 0, !w [0, T ],

||M ||M 2 :=

√

EM2T .hYo03T ||M ||

M 2 = 0, M [0, T ] apoz a.s. ZW ||·||M 2 .

M 20 ou, TNMQ# .h1~mko.x) 4.2.1 o7NQ~YU M 2

0 . Hilbert U.3'. M (n) . M 20 4o Cauchy y, M (n)

T . L2 U o Cauchy y,hp0, \Lu ξ ∈ L2 &n E[(M (n) − ξ)2] → 0. |X Mt = E[ξ|Ft], M = (Mt : t ≤ T ) (zQoO61!) .NQ~lo7Y. _f Doob)p(E[ sup

t∈[0,T ]

|M (n)t −Mt|2] ≤ 4E(M (n)

T −MT )2,n supt∈[0,T ] |M (n)

t −Mt| S L2 Z P+4poz, ZW\Jy nk &nsup

t∈[0,T ]

|M (nk)t −Mt|eQQ4poz, K.C, eOkRm, M (nk) j [0, T ] O/4po M . ZW M .o7o, M ∈ M 20 .

:MZ Ito B> 96|O6U.~UEU. N (4.2.3) ol. Kvs, ^ F ∈ L0,|X||F ||2

L 2 :=

√

E

∫ T

0

F 2t dt. (4.2.4)v. L0 oO6h1~mko?, f L 2 ") L0 oJo ||·||

L 2 oplU, .O6 Hilbert U.TUE F = (Ft)t≥0 .f/_UE F (n) : n ∈ N i n iofoE

∫ T

0

|F (n)t − Ft|2dt→ 0! Ito ~o003s Ito pGVGi n,m→ ∞ !k

∣

∣

∣

∣I(F (n)), I(F (m))∣

∣

∣

∣

M 2 = E|I(F (n))T − I(F (m))T |2

= E

∫ T

0

|F (n)t − F

(m)t |2dt→ 0 I(F (n) . (M 2

0 , ||·||M 2) o Cauchy y. Zw M 20 .po, GUE6y

I(F (n)) M 20 4o\. Kv|X

I(F ) := limn→∞

I(F (n)),QBw F Jo Brown B o Ito ~. 9< I(F )t 'Cw~o.( ∫ t

0FsdBs zfs F.Bt.x) 4.2.2 e F → F.B .Y L 2 l M 2

0 oO60pGe, Q4 L 2D?||F ||

L 2 =

√

E

∫ T

0

F 2t dt . (4.2.5) M 2

0 .D? ||M ||M 2 =

√

E(M2T ) o Hilbert U. 5O+s, o

F ∈ L 2,(

(F.B)2t −∫ t

0

F 2s ds : t ∈ T

).Y.

:MZ Ito B> 97$/ 4.2.1 " |e4QdOHm.~UEoU L 2 .O6`aoU, o\e8MO6Ga,_CR9NBw.) 4.2.3 F = (Ft)t∈T ./_bRo7LUE, NE

[∫ T

0

F 2s ds

]

< +∞ . (4.2.6), F ∈ L 2 bP+kI(F )t = lim

m(D)→0

∑

l

Ftl−1

(

Btl − Btl−1

)Q4.o [0, t] Okkklo.3'. v L 2 ok/UEp^Q4K, G) F k/. on > 0,

Dn ≡ 0 = tn0 < tn1 < · · · < tnnk= T. [0, T ] O6kk6y&n

m(Dn) = supj

|tnj − tnj−1| → 0 as n→ ∞ .F

(n)t = F010(t) +

nk∑

l=1

Ftnl−1

1(tnl−1,t

nl](t) ; for t ≥ 0 . (4.2.7),6 F (n) .fUE, bZw F Ro7, G^ t, F

(n)t → Ft. ZW9Dk/4p|ekMi n iof!, k

E

[∫ T

0

|F (n)s − Fs|2ds

]

→ 0 .h|Xn F ∈ L 2.4) 4.2.1. a F = (Ft)t≥0 /_oh Brown Co (Ft)t≥0 |XIto ~!.3o. |O, UE t→ Ft oRo70.=0o, QQSQQQ50 _. i|X F = (Ft)t≥0 oFk)o7woYoL~

:MZ Ito B> 98!, QRo70C.2o. eh. F !R t oR! t % “”, T t → Ft Ro7, ,, ^! t, Ft o(QS!R t %o(Ox5Ft = lim

s↑tFs .4) 4.2.2. _))M'8.(on_Q50 (t, ω) → Ft(ω) .Jo, S" (4.2.5) kVX. AVl (4.2.5) QS'Cw

∫

Ω

∫ t

0

Fs(ω)2dsP(dω) < +∞.GKvoQ5a_).^ t > 0, X?

F (s, ω) ≡ Fs(ω)Tw [0, t] × Ω oX?Jo B([0, t]) ⊗ Ft Q5, Q4 B([0, t]) . [0, t] Borel σ- ?. # .LUEJo8|o<6Q50.T X = (Xt)t≥0 .o7LUE/_o (Ft)t≥0, f .6 Borel X?, b

E

∫ T

0

[f(Xt)]2dt <∞,LUE (f(Xt))t≥0 <o L 2.$/ 4.2.2 "6B).`TH"o, eH"oZ8dZ: ^

Borel Q5X? f &nE

∫ T

0

f [(Bt)]2dt <∞ (4.2.8), (f(Bt))t≥0 . L 2 4.a (4.2.8) ><."VD0? #

E

∫ T

0

[f(Bt)]2dt =

∫ T

0

E[f(Bt)2]dt

=

∫ T

0

Pt(f2)(0)dtQ4

Pt(f2)(0) =

1

(2πt)d/2

∫

Rd

f(x)2e−|x|2/2tdx

:MZ Ito B> 99

=1

(2π)d/2

∫

Rd

f(√tx)2e−|x|2/2dx .ZW, T f (, , (f(Bt) : t ≥ 0) <o L 2, b^<? α UE

(eαBt)t≥0 K<o L 2. LUE Ft =eαB2t I0? !

E

∫ T

0

F 2t dt =

1

(2π)d/2

∫ T

0

∫

Rd

e2αtx2

e−|x|2/2dxZWE

∫ T

0

F 2t dt <∞ if α ≤ 0 .i α > 0 !,

E

∫ T

0

F 2t dt <∞ ib4i T <

1

4α.$/ 4.2.3 f . [0, T ] oKo7X?. " :

∫ t

0

f(s)dBs = Btf(t)−∫ t

0

Bsdf(s),AVl.d<~.T F = (Ft)t≥0 ∈ L 2, ,t6UE∫ t

0

FsdBs and

(∫ t

0

FsdBs

)2

−∫ t

0

F 2s ds.L(wzoo7Y, Z#s

E

[∫ T

0

FsdBs

]2

= E

(∫ T

0

F 2s ds

)

.b^ t ≥ s,

E

(∫ t

s

FudBu

)2∣

∣

∣

∣

∣

Fs

= E

∫ t

s

F 2udu

∣

∣

∣

∣

Fs

..W, Ito oJo Brown oL~ps|Xw, Q4oJ.Ito pG, Ito pGoJ. Brown ktX8UE. K.C, TOoNQ~Y|XL~, 3JXQ.!FkÆX8UE.

:MZ Ito B> 101 bMQSlO6A,l6y τn &n^ n, M01τn>0 .k/o. ,# , +2lτn = n1|M0|<n. (4.3.1)h.+26+*g, lA,Yo! , Kv) ts'M_,h4QS|w 1τn>0 .'Ao, z M0 .k/o.$/ 4.3.1 1. A,Yot6A,l6yl!Pv.A,YoA,l6y.

2. T M .A,Y, ,\A,l6y τn &n^ n, M τn .O/Q~Y.

3. T M .o7A,Y, ,M.QSllO6A,l6y τn &n^ n, M τn .k/o7Y.

4. T M .'o7A,Y, , M . Y.

5. h (4.3.1) |Xo τn .A,l6y.^ ! *oYKv.A,Y, k$Q2v-5A,Y+JkQ~0.).CwY, Q#A,YY8n`~, Q~0)B, foO/Q~0K)B, 2J`_oO/Q~0/kQ2.$/ 4.3.2 O6UE X . (DL) o, T^ t > 0, LuOXt∧τ : τ l Ok !.O/Q~o. kO6A,Y M = (Mt)," : Q.Yib4iQ. (DL)o. O6/_lo7LUE ABw.ÆUE (zk/8UE), T A0 = 0

a.s. beOko ω ∈ Ω, Rm t 7→ At(ω) .fz o (_s, kj .k/8o). 6" Brown OI, O6<(o7A,Y)Q2.k/8o.x) 4.3.1 O6o7A,Y M = (Mt) .k/8oib4iQ.<(o,V t ≥ 0, Mt =M0 a.s.

:MZ Ito B> 103P+4po At, BUE X \kÆX8UE, UE A Bw. X oÆX8UE, 'w 〈X〉.$/ 4.3.3 " : T X \kÆX8UE, o^ s < t k 〈X〉s ≤〈X〉t a.s. |eo" 4, # " wO6o7k/8UEFkÆX8UE, QÆX8UE.zUE. Q'CwO6r , dvfl.$/ 4.3.4 (1) O6o7k/8UEoÆX8UE.z. (2) To7A,Y M oÆX8UE 〈M〉 apo 0, , M ≡ M0." o7A,YFkkÆX8, %8MÆX8UEoO6Rj.x) 4.3.2 M .O6o7A,Y, M FkkÆX8UE, QÆX8UE 〈M〉 .&n M2 − 〈M〉 Cwo7A,YovOoo7ÆUE 〈M〉.Z#s, T M .o7NQ~Y, , M2 − 〈M〉 .Y.3'. vO0h|e 4.3.1 kkM. ,# , Tkt6o7ÆUE A,A′ &n M2 − A s M2 − A′ .o7A,Y. ,

A−A′ = (M2 − A′)− (M2 − A),R.L'(wzok/8UE, l.o7A,Y, h|e 4.3.1, A = A′." ÆX8UEo\0. M .k/o7Y, !<? K . Mo/. D." M2 − TD(M) .Y, vd" QU M 20 kvOow. " '=, C?+.

(I) M2 − TD(M) .Y.TH"^I| t > 0 I|ok D, TDt (M) k/. ^ t >

s ≥ 0, \ k & tk < s ≤ tk+1, GE[TD

t (M)− TDs (M)|Fs]

= E[

∑

i>k

(Mti+1∧t −Mti∧t)2|Fs

]

+ E[

(Mtk+1∧t −Mtk)2 − (Ms −Mtk)

2|Fs

]

:MZ Ito B> 104

= E

[

∑

i>k

(Mti+1∧t −Mti∧t)2|Fs

]

+ E[

(Mtk+1∧t −Ms)2|Fs

]

=∑

i

E[(M(ti+1∧t)∨s −M(ti∧t)∨s)2|Fs]

=∑

i

E[M2(ti+1∧t)∨s −M2

(ti∧t)∨s|Fs]

= E[M2t −M2

s |Fs], M2 − TD(M) .O6o7Y.

(II) TDt (M) o8| t .k/o, h)2|.k/UE. QoÆ*BO6s t D Jo<?V1

E[TDt (M)2] ≤ 8K4.,# , si := ti ∧ t,

E[TDt (M)2] = E

[

∑

i

(Msi+1−Msi)

2]2

= E∑

i

(Msi+1−Msi)

4 + 2E∑

i>j

(Msi+1−Msi)

2(Msj+1−Msj)

2

= E∑

i

(Msi+1−Msi)

4 + 2E∑

i>j

(M2si+1

−M2si)(Msj+1

−Msj )2

= E∑

i

(Msi+1−Msi)

4 + 2E∑

j

(M2t −M2

sj+1)(Msj+1

−Msj )2

= E

[

∑

i

(Msi+1−Msi)

2 · ((Msi+1−Msi)

2 + 2(M2t −M2

si+1))

]

≤ 8K2ETDt (M) = 8K2E[M2

t −M20 ] ≤ 8K4,QO6sk D Jo<?OV1.

(III) T Dn .O6=iozoky, , M2 −TDn(M) .M 20 4o

Cauchy y, K.Clim

n,m→∞E[(TDn

t (M)− TDm

t (M))2] = 0. (4.3.2)

:MZ Ito B> 105lQ4t6k Dn s Dm, f D′ ")t_%dok, ZUETDn(M)− TDm(M).o7Y, GhS ", UE

(TDn

t (M)− TDm

t (M))2 − TD′

t (TDn(M)− TDm(M)), t ≥ 0.o7YbhLp)p( (a+ b)2 ≤ 2(a2 + b2) nTD′

t (TDn(M)− TDm(M)) ≤ 2TD′

t (TDn(M)) + 2TD′

t (TDm(M)). (4.3.3)+2J" limn E[TD′

t (TDn(M))] = 0 Bw.,# , t′i, s′i#. Dn, D′ okw, ti := t′i∧t, si := s′i∧t,!^ k, \vOo l &n tl ≤ sk ≤ sk+1 ≤ tl+1, ZW

TDn

sk+1(M)− TDn

sk(M) = (Msk+1

−Mtl)2 − (Msk −Mtl)

2

= (Msk+1−Msk)(Msk+1

+Msk − 2Mtl),Y TD′

t (TDn(M)) =∑

k

(TDn

sk+1(M)− TDn

sk(M))2

≤ TD′

t (M) · supk(Msk+1

+Msk − 2Mtl)2,h Cauchy-Schwarz )p(s (II) Q$,

E[TD′

t (TDn(M))]2 ≤ E[TD′

t (M)]2 · E[

supk(Msk+1

+Msk − 2Mtl)4

]

≤ 8K4E

[

supk(Msk+1

+Msk − 2Mtl)4

]

.i n→ ∞ !, sk+1 s sk io tl, Gh M o [0, t] oO/o70$supk(Msk+1

+Msk − 2Mtl)4o.zb<? 4K4 V1, hk/4p|ekM

lim|D|→0

E

[

supk(Msk+1

+Msk − 2Mtl)4

]

= 0.

:MZ Ito B> 106ZW (4.3.3) Y (4.3.2) Ck.

(IV) M ko7vÆoÆX8UE A, b M2 − A .Y. M2 − TDn(M) . M 20 4oO6 Cauchy y, h|e 4.2.1, Q

M 20 4k, w N . A =M2 −N . ZW^ t ≥ 0, TDn

t (M) . L2G K.P+4po At b M2 −A = N .O6Y.Z TDn

0 (M) = 0 a.s., G A0 = 0 a.s. |o^ t > s, s, t ∈ ⋃nDn, \Ga n &n,

TDn

t (M) ≥ TDn

s (M), a.s.ZW At ≥ As a.s., h A oo70S ⋃

nDn oK0kM A .O6o7ÆUE. hvO0, A s Dn o9lJ, I|ek/o7YCk.T M k/, 〈M〉 .ÆX8UE, M2 −〈M〉 .o7Y. lV !τ n *Q, n [M τ ]2 − 〈M〉τ .o7Y, hvO0Q$ 〈M〉τ . M τ oÆX8, k

〈M τ 〉 = 〈M〉τ . (4.3.4) M .O6A,Y, lQO6A,l6y τn, h (4.3.4) Q$ 〈M τn〉s 〈M τn+1〉 ! τn .OIo. V t ≥ 0, 〈M〉t = lim

n→∞〈M τn〉t.TH" 〈M〉 .O6YzMoo7ÆUEb 〈M〉τn = 〈M τn〉, ZW M2 −

〈M〉 .O6o7A,Y.

(V) 2J" |Xo 〈M〉 t#.o7A,Y M oÆX8UE. ^ t ≥ 0, i k −→ ∞ !,

TDk

t (M τn)p−→〈M〉τnt .l ε > 0, ^ n ≥ 1 k

limk→∞

P(|TDk

t (M)− 〈M〉t| > ε)

≤ limk

P(|TDk

t (M)− 〈M〉t| > ε, t < τn) + P(t ≥ τn)

:MZ Ito B> 107

≤ limk

P(|TDkt (M τn)− 〈M〉τnt | > ε) + P(t ≥ τn)

= P(t ≥ τn), limn P(t ≥ τn) = 0. kM TDkt (M)

p−→〈M〉t.(VI) Qd" : T M .o7NQ~Y, , M2−〈M〉 .Y. " 〈M〉Q~, lA,l6y τn, &n (M2 − 〈M〉)τn .Y, ,hYo Doob *|e,

E〈M〉t = limn

E〈M〉t∧τn = limn

E[M2t∧τn ] ≤ E[M2

t ].Zw 〈M〉 vÆ, GQ. (DL) o. +2" M2 K. (DL) o. h Dooba)p(E[max

s∈[0,t]M2

s ] ≤ 4E[M2t ], LuO M2

t∧τ : τ Q~Lu maxs∈[0,t]M2s V1, ZW M2 K. (DL) o.|e48MoRj<o9J, Q&n)M.dUlk oNoo_JYoÆX8, |e 4.2.2 # C

〈F.B〉t =∫ t

0

F 2s ds.^o7A,Y M,N |X

〈M,N〉 := 1

4(〈M +N〉 − 〈M −N〉),BQ. M,N o%8UE. Kvs, ^ t ≥ 0, ik ti ioz!,

∑

i

(Xti+1∧t −Xti∧t)(Yti+1∧t − Yti∧t)p−→〈X, Y 〉t. (4.3.5)ZWÆX%8kyfo03: M,N,M1,M2 .o7A,Y,

(1) (B0) 〈M,N〉 = 〈N,M〉;

(2) T a, b .<?, 〈aM1 + bM2, N〉 = a〈M1, N〉+ b〈M2, N〉;

(3) |〈M,N〉|2 ≤ 〈M〉〈N〉.

:MZ Ito B> 108$/ 4.3.5 " : i M,N .o7A,Y!, 〈M,N〉 . Nt6aovOoo7k/8UE: (i) 〈M,N〉0 = 0; (ii) MN − 〈M,N〉 .o7A,Y.o|eA,l!.<9Jo.x) 4.3.3 M,N .o7A,Y, τ . !. 〈M τ , N τ 〉 = 〈M,N〉τ = 〈M,N τ 〉.3'. ) M,N .k/o. Zw MN − 〈M,N〉 .Y, G M τN τ −

〈M,N〉τ K.Y, tO6pZCk. 2H" M τN τ − 〈M,N τ 〉 .Y.z" M τN τ −MN τ .Y, Zw MN τ − 〈M,N τ 〉 .Y. ^k/ !σ, h Doob *|e,

E(MσNτ∧σ) = E(Nτ∧σE(Mσ|Fτ∧σ)) = E(Nτ∧σMτ∧σ),C E[(M τN τ −MN τ )σ] = 0, kM M τN τ −MN τ .Y. OdZifA,l6yTH". # tÆ6pZh (4.3.5) :wv.+ 4.3.1 B = (B(1), · · · , B(d)) . d- x Brown , 6h|e 3.5.1 kM 〈B(i)〉t = t. " i i 6= j !, 〈B(i), B(j)〉 = 0. +2H" B(i)B(j).YNBw. ,# , t > s, hY0sk0nE[B(i)

t B(j)t −B(i)

s B(j)s |Fs] = E[(B(i)

t −B(i)s )(B

(j)t − B(j)

s )|Fs]

= E[(B(i)t −B(i)

s )(B(j)t − B(j)

s )] = 0.ZW 〈B(i), B(j)〉 = εi,jt.Os, T M,N .koo7A,Y, 〈M,N〉 ≡ 0. g)Q.k/Y_" , QKfA,l" OdZ. o.kfo, ZwOoYkkÆu0. ZWf|X_H". D = ti. [0, t] ok. hY0sk0,

E

(

∑

i

(Mti −Mti−1)(Nti −Nti−1

)

)2

= E∑

i

(Mti −Mti−1)2(Nti −Nti−1

)2

:MZ Ito B> 109

≤ E

[

TDt (M) sup

i(Nt1 −Nti−1

)2]

≤√

E[(TDt (M))2] · E[sup

i(Nti −Nti−1

)4],h|e 4.3.2 o" 4o (II) $m E[(TDt (M))2] O6s D Jo<?V1, ho70tÆwz, G M,N D o%8o.z,

〈M,N〉 = 0.$/ 4.3.6 fA,l" OdZ. AVA,l6yolFk0. M .o7A,Y, ' M2t = (M2

t − 〈M〉t) + 〈M〉t, M2 QvO.wO6o7A,YsO6o7ÆUEo. Jo6|e, 8Mo5>" 2JifA,lo+8. T M 1o7NQ~YodZ,,.).vkO6)2JfA,lo&)" 0? # , 6-T.>"o Doob-Meyer .oO6Zj, Doob-Meyer .~ " o|e_a, " KvK:w\/, Q.CO6 ( D o) lo7YQvOs.wO6lo7YsO6KvÆUEo, Q. Lo=J| .

§4.4 .usXR9|XwJo Brown o9xL~, +4, O++sQ^lJoo7YoL~, R9QSC.d<~okO.

§4.4.1 -t:"rW Fs, QS_f|XJo Browno Ito~oUE_|XJoo7NQ~Yo Ito ~. Joo7NQ~Yo~o|XE6: oo7NQ~Y M NO|Q~ao/_UE F , \vOoNQ~Y, w F.M , &n^o7NQ~Y N k〈F.M,N〉t =

∫ t

0

Fsd〈M,N〉s.6Rj<9J, .nÆd<~L~o:b.

:MZ Ito B> 110,# , M ∈ M 20 b F = (Ft)t≥0 .6fUEFt = f10(t) +

∑

i

fi1(ti,ti+1](t),|XIM(F ) =

∞∑

i=0

fi · (Mt∧ti+1−Mt∧ti) . Fo Brown , k69J03x) 4.4.1 1. IM(F ) ∈ M 2

0 .

2. ZUE 〈IM(F )〉t =∫ t

0F 2s d〈M〉s, IM(F )2t −

∫ t

0F 2s d〈M〉s .6Y.

3. (Ito pG) ^ T > 0, kE

(∫ T

0

FtdMt

)2

= E

∫ T

0

F 2t d 〈M〉t .ww" $03, 5>O6\e.) 4.4.1 M = (Mt)t≥0 .6o7NQ~Y, s < t ≤ u < v, f ∈ Fs,

g ∈ Ft, k/. ,E [g(Mv −Mu)(Mt −Ms)|Fs] = 0b

E[

f(Mt −Ms)2|Fs

]

= E [f (〈M〉t − 〈M〉s) |Fs] .3'. haPso03E (g(Mv −Mu)(Mt −Ms)|Fs)

= E E (g(Mv −Mu)(Mt −Ms)|Fu) |Fs

= E g(Mt −Ms)E(Mv −Mu|Fu)|Fs

= 0 .tÆ6p(`f, Zw f ∈ Fs, GQSQMaPs, _f|e 4.3.2$M2 − 〈M〉 .YQ.

:MZ Ito B> 111_H"|eo-. TwjJ, _" JotÆa03. I|t > s > 0, T2J, MQS t, s w4, G\ j, k &n tk = t,

tj = s. h\enE[(IM(F )2t − IM(F )2s)|Fs] = E[(IM (F )t − IM(F )s)

2|Fs]

= E

[

k−1∑

i=j

F 2ti(Mti+1

−Mti)2|Fs

]

= E

[

k−1∑

i=j

F 2ti(M2

ti+1−M2

ti)|Fs

]

= E

[

k−1∑

i=j

F 2ti(〈M〉ti+1

− 〈M〉ti)|Fs

]

= E

[∫ t

s

F 2ud〈M〉u

]

.xz 4.4.1 O6LUE F = (Ft)t≥0 Bw<o L 2(M), T\O6fUE6y F (n)t ) &nVI|! T > 0.

E

∫ T

0

|F (n)t − Ft|2d 〈M〉t

→ 0 as n→ ∞ .rHmC, L 2(M) .OkfUE?||F ||

L (M) =

E

(∫ T

0

F 2t d〈M〉t

)

12oopl, QivP`o! T sYM ∈ M 2

0 ,ZW L 2(M).6 BanachU. ,# , o?.h1~mko, &n L 2(M) .6 Hilbert U.T F ∈ L 2(M), b ∣

∣

∣

∣F − F (n)∣

∣

∣

∣→ 0o'6O6fUE6y, hok ItopG√

E IM(F )2T = ||F || ,kMIM(F ) ≡ lim

n→∞IM(F (n)) , in M

20\. f F.M z ∫ t

0FsdMs &Ks") IM(F ). ko|e.

:MZ Ito B> 112x) 4.4.2 F ∈ L 2(M), ,1. IM(F ) ∈ M 2

0 .

2. (Ito pG) ^ T > 0, kE

(∫ T

0

FtdMt

)2

= E

∫ T

0

F 2t d 〈M〉t . |e4OCo Ito pG:Oo.S|e.x) 4.4.3 M,N ∈ M 2

0 b F ∈ L 2(M), G ∈ L 2(N). E

(∫ T

0

FtdMt

)(∫ T

0

GtdNt

)

= E

∫ T

0

FtGtd 〈M,N〉t , (4.4.1) b F Jo M oL~ F.M hy03vORj: F.M ∈ M 20 b^

N ∈ M 20 k

E〈F.M,N〉T = E[(F.M)TNT ] = E

∫ T

0

Fsd〈M,N〉s. (4.4.2)vORj03.vo. o (4.4.1) o" , F , G .fUE" % (r ), vdOo F , G, l#. F G ofUEy F (n) G(n), , F (n).M s G(n).N o#. F.M G.N , ZWRo4pk, lo4p2JO6>"o)p(, O+0.x) 4.4.4 # , Rj (4.4.5) po:_:foB: ^ N ∈ M 2

0 ,k(KtNt −

∫ t

0

Fsd〈M,N〉s : t ∈ [0, T ]).Y, zC〈K,N〉t =

∫ t

0

Fsd〈M,N〉s.J" 6,#, uW Doob |eC: O6Q~UE X .Yib4i^k/ ! τ kE[Xτ ] = E[X0].

:MZ Ito B> 113OS2JH" (4.4.5) 4o T fk/ ! _KOICkQSw, N nkw)Q2, h# Zw (4.4.5).oOkoo7NQ~Y N Ck,Gl^k/ ! τ ≤ T , N τ K.NQ~Y, _fk (4.3.3) kE[KτNτ ] = E[NτE[KT |Fτ ]] = E[KTN

τT ]

= E

∫ T

0

Ftd〈M,N τ 〉t

= E

∫ T

0

Ftd〈M,N〉τt

= E

∫ τ

0

Ftd〈M,N〉t.TVk/8UE V , f FoZ F.V ")~UE(F.V )t(ω) =

∫ t

0

Fs(ω)dVs(ω)., o (4.4.2) QS'CwE〈F.M,N〉T = EF.〈M,N〉T , (4.4.3)l F.〈M,N〉T .)j [0, T ] F Jo 〈M,N〉 o~. AVQoR.L~, l.d<~, GdU6Rj, QS4d<~o03lL~ n.`6 4.4.1 M ∈ M 2

0 ,

1. G ∈ L 2(F.M), G.(F.M) = (GF ).M ;

2. F,G ∈ L 2(M), (G+ F ).M = G.M + F.M ;

3. σ . !, , (F.M)σ = F.Mσ = F σ.Mσ.3'.v OCot603i M .k/8UE!.Cko. ^N ∈ M 2

0 ,

E〈G.(F.M), N〉T = EG.〈F.M,N〉T= EG.F.〈M,N〉T = EGF.〈M,N〉T .

:MZ Ito B> 114 FskE〈(F.M)σ, N〉T = E〈F.M,N〉σT = E(F.〈M,N〉)σT

= EF.〈M,N〉σT = EF.〈Mσ, N〉T ,h oL~Rj|ekM 1 s 3 otO6pZCk, 2 s 3 otÆ6pZ F" .

§4.4.2 Kunita-Watanabe dtMNw Ito o9x|X, Kunita-Watanabe o)p(QS8M|oO8:&)o|X, PoD. M 20 |XO6k/0X, vdif Riesz ")|e_|XML~. M ∈ M 0

0 , F ∈ L 2(M), ^ N ∈ M 20 , |XX

φ(N) := E

(∫ T

0

Ftd〈M,N〉t)

, (4.4.4)T2B" φ . M 20 ok/0X, ,h Riesz ")|ekM, QkO6vOo"), w K ∈ M 2

0 , Nφ(N) = E[NT ·KT ],^ N ∈ M 0

0 kE[NTKT ] = E

∫ T

0

Ftd〈M,N〉. (4.4.5), K .O+4|XoL~ F.M , ZwQY.|e 4.4.3 4oRj.x) 4.4.5 0X φ o").L~ F.M .Qdq2JO69Jo)p(C 0X φ ok/0. 6h%8UEo|X Cauchy-Schwarz )p(, eOkIRm k|〈M,N〉t − 〈M,N〉s|2 ≤ (〈M〉t − 〈M〉s)(〈N〉t − 〈N〉s); (4.4.6),ofUE F,G, eQQsk

∣

∣

∣

∣

∫ T

0

FsGsd〈M,N〉s∣

∣

∣

∣

2

≤∫ T

0

F 2s d〈M〉s

∫ T

0

G2sd〈N〉s. (4.4.7)

:MZ Ito B> 115lkM)p(oQ5oLUE F,G Ck. 6)p(Bw Kunita-

Watanabe )p(. _f Cauchy-Schwarz )p(kMo)p(, ivQKOIBw Kunita-Watanabe )p(:

E

∣

∣

∣

∣

∫ T

0

FsGsd〈M,N〉s∣

∣

∣

∣

≤

E

(∫ T

0

F 2s d〈M〉s

)

E

(∫ T

0

G2sd〈N〉s

)1/2

.

(4.4.8)f%o φ n|φ(N)| ≤ ||F ||

L 2(M) · ||N ||M 2. (4.4.9)Qd, 6)p(!YQSpC|e 4.4.3 o" 42JpCoUE (~Tr ). d<W>(TwL~oRj.$/ 4.4.1 " (4.4.6).

§4.4.3 &-teJoNQ~Yo~e`Jv, h.oUE F M 1'. Ito ~^.A,k/UEJoo7A,Yo~, <J. M = (Mt)t≥0 .L(wzoo7A,Y, ,QS9 !6yτn &n τn ↑ ∞ a.s. b^ n, M τn = (Mt∧τn)t≥0 .L(zoo7NQ~Y (TJ, .QS.k/o). F = (Ft)t≥0 ./_UE, \ !y σn &n σn ↑ +∞ b^ n,

F σn k/. !C F .A,k/o. vRol, zloRo/_UE.A,k/o.|X τn = τn ∧ σn. , τn ↑ ∞ a.s., b^ n, M τn ∈ M 20 .

F(n)t = Ft1t≤τn .

∫ ∞

0

(F (n)s )2d〈M〉s =

∫ τn

0

F 2s d〈M〉s ≤ n〈M〉τn, F (n) ∈ L 2(M τn). QS|X

(F.M)t =

∫ t

0

F (n)s dM τn

s if t ≤ τn ↑ ∞

:MZ Ito B> 116Bw F Joo7A,Yo Ito ~. QS" F.M o|X.SXob)P`o !y τn o9. ,# , hk 4.4.1 $^ ! σ ≤ τ kF σ.Mσ = (F τ .M τ )σ. (4.4.10)G t ≤ τn,

∫ t

0

F (n+1)s dM τn+1 =

∫ t∧τn

0

F (n+1)s dM τn+1

=

∫ t

0

F (n)s dM τn .h|X, F.M s

(F.M)2t −∫ t

0

F 2s d〈M〉s.L(zoo7A,Y. bs|e 4.4.3 OIQS" , F.M .h03RjovOo7A,Y: ^o7A,Y N , k

〈F.M,N〉 = F.〈M,N〉. (4.4.11)Nw$, 6|e8MwL~oV14p|eb" wL~v. Riemann P+4po. 64p|e" Ito >(!K.kfo.x) 4.4.6 M .o7A,Y, F (n) .6 t QeQQ4pozoA,k/UEy%bO6A,k//_UE F V1, (F (n).M) ^k/j P+O/s4plz. Z#s, T F .A,k/oRo7/_UE, ^ t ≥ 0, i [0, t] ok D = ti o=ioz!, L~(F.M)t . Riemann-Stieltjes

∑

i

Fti(Mti+1−Mti)P+4po.3'. F M .k/UE. ! F (n) ∈ L 2(M) %bh LebesgueV14p|e, Q L 2(M) 4ou4plz. ZL~.o7e, G

:MZ Ito B> 117

F (n).M M 20 4ou4poz, vd Doob Y)p(kM (F (n).M) ^k/j P+O/s4plz.Os, hA,l, +2" ^UEy X(n), T\iofo !y τk &n^ k, (X(n))τk kj P+O/s4po

0, , X(n) Kkj P+O/s4po 0. ,# , ^ ε > 0,

P(maxs∈[0,t]

|X(n)s | > ε) ≤ P(max

s∈[0,t]|X(n)

s | > ε, τk ≥ t) + P(τk ≤ t)

≤ P(maxs∈[0,t]

|(X(n))τks | > ε) + P(τk ≤ t),ZWlimnP(max

s∈[0,t]|X(n)

s | > ε) ≤ P(τk ≤ t). τk ↑ +∞. Gk limn P(maxs∈[0,t] |X(n)s | > ε) = 0.dO6-, Zw F A,k/Ro7, G F .*[UEy

FD = F010 +n∑

i=1

Fti−11(ti−1,ti]oeQQ4po, F k/, if o-T" -Ck, vdifA,l" F A,k/!-KCk.

§4.4.4 &-t[QdL~e^.QkfoY . O6/_o7LUE X = (Xt)t≥0 .6o7Y, T X Fk.Xt =Mt + VtQ4 (Mt)t≥0 .o7A,Y, (Vt)t≥0 .L(zoo7/_k/8UE. h|e 4.3.1 6..vOo, Bw X oY.. YoU.0U, dQSNl, YD~ (,~>(), YKÆXQtX?o&_ (Ito >().T g . [0,+∞) oo7X?, ^k/j k/8

supD

∑

l

|g(tl)− g(tl−1)| < +∞

:MZ Ito B> 118Q4 D l [0, t] o^kk (^I|o t ≥ 0), , [0,∞) ^ Borel Q5X? f ,∫ t

0

f(s)dg(s)e.w Lebesgue-Stieltjes ~. T:s s→ f(s) .Ro7o, ,∫ t

0

f(s)dg(s) = limm(D)→0

∑

l

f(tl−1)(g(tl)− g(tl−1))lTw Riemann-Stieltjes o\. ZW, T V = (Vt)t≥0 .6o7k/8UE, F .Ro7UE, ,∫ t

0

FsdVs.O6RmS Riemann-Stieltjes ~VX|XoLUE(∫ t

0

FsdVs

)

(ω) ≡∫ t

0

Fs(ω)dVs(ω)

= limm(D)→0

∑

l

Ftl−1(ω)(Vtl(ω)− Vtl−1

(ω)) .T (Ft : t ≥ 0) .6Ro7/_UE, ,L~o|XQSvo(^.o7Y, ∫ t

0

FsdXs =

∫ t

0

FsdMs +

∫ t

0

FsdVsQ4ltOoo7A,Yo Ito ~, QK.o7A,Y, h|e4.4.6, Q.lRw(o Riemann P+4po; tÆ.d<oLebesgue-Stieltjes ~, Q. Riemann eQQ4po. hWP+4poVX, k

∫ t

0

FsdXs = limm(D)→0

∑

l

Ftl−1

(

Xtl −Xtl−1

)

,AVl6O).eQQo, L~)2e.wRm_|Xo. |oo7YoÆX8UEK.\o.8\ 4.4.1 X =M + V oÆX8UE 〈X〉 \b 〈X〉 = 〈M〉.

:MZ Ito B> 1193'. [0, t] ^k D kTD(X)t = TD(M)t + TD(V )t + 2

n∑

i=1

(Mti −Mti−1)(Vti − Vti−1

),i m(D) = max1≤i≤n |ti − ti−1| → 0 !, Q4tOP+io 〈M〉t, tÆeQQZWP+io 0, h Cauchy-Schwarz )p(, t√

TD(M)t · TD(V )tV1, Z KP+io 0. ZW#o-Ck.hWkMo7Y X = M + V s Y = N +W o%8\bpoQY,o%8 〈X, Y 〉 = 〈M,N〉.

§4.5 lp1. M .o7A,Y. " : M ≡ M0 ib4i 〈M〉 ≡ 0.

2. f . [0, 1] o(uw.Qyolo7X?, , f .k/8oib4i∑

t |∆f(t)| <∞, f kÆX8UEib4i ∑

t |∆f(t)|2 <∞,Q4 ∆f(t) = f(t)− f(t−).

3. ÆX8o|X, ^ p > 0, BLUE X kk/ p X8, Tik=ioz!,∑

i |Xti+1∧t −Xti∧t|p P+4p. p X8UEw V p(X). X .o7A,Y. " : p > 2, V p(X) ≡ 0; 0 < p < 2, 〈X〉 > 0 , V pt (X) = ∞.

4. X .O6o7NQ~YbkNkÆu, X0 = 0. " : 〈X〉t =t · EX2

1 .

5. Z .k/Lu, A .Y 0 Mok/o7ÆUE. " :

E[ZA∞] = E

[∫ ∞

0

E(Z|Ft)]dAt

]

.

:MZ Ito B> 120

6. M .o7A,Y. " : M2 FkÆX8UEb〈M2〉t = 4

∫ t

0

M2s d〈M〉s.

7. M . Gauss UEb.o7Y, " : 〈M〉 .t|0UE (+s!kJ).

8. M .o7A,Y, K ∈ L2(M), ξ . Fs Q5Lu. " : t > s ≥ 0,

∫ t

s

ξKdM = ξ

∫ t

s

KdM.

9. T M .o7A,Y, M0 = 0, (1) " : T 〈M〉T Q~, , M =

(Mt : t ∈ [0, T ]) .NQ~Y. (2) " : M supt 〈M〉t < ∞ i tiof!eQQ4p.3'.lA,l6y&n M τn , (M2 − 〈M〉)τn .k/o7Y, ,E[M2

t∧τn ] = E〈M〉t∧τn ≤ E〈M〉t <∞.6h Fatou \e, w n iof, E[M2t ] ≤ E〈M〉t. )p(qC

Mt∧τn : n ≥ 1 .O/Q~o. VG M .Y.

10. M,N .o7A,Y, K ∈ L2loc(M) ∩ L2

loc(N), " : K.(aM + bN) =

aK.M + bK.N, a, b ∈ R.

11. fQ-H": o7Y X kXn

t = Xn0 + n

∫ t

0

Xn−1dX +1

2n(n− 1)

∫ t

0

Xn−2d〈X〉.

12. f . [0, T ] o7X? (s ω J), " : f = (f(t)) .o7Yib4i f o7bk/8.

13. M .o7A,Y&n5 d〈M〉t eQQs Lebesgue 5p, " : \<6Q5UE V Brown B &n Mt =M0 +∫ t

0VsdBs.

:MZ Ito B> 121

14. M .o7A,Y, σ ≤ τ . !. (1) TKt := ξ1(σ,τ ](t),Q4 ξ .k/ Fσ Q5o, K.M = ξ(M τ−Mσ); (2)" : 〈M〉σ = 〈M〉τib4i M [σ, τ ] .<(.

vj Ito N=~Ito >(. K. Ito 1944 6ko. Zw Ito Po!X4Q5>CwO6\e, G Ito >(K<Bw Ito \e. Ito \e# .Lt~o||e.

§5.1 Ito N4;_4_foLp>(X2

tj−X2

tj−1=(

Xtj −Xtj−1

)2+ 2Xtj−1

(

Xtj −Xtj−1

)

. (Xt)t≥0 .6A,Y, j = 1, · · · , nh, Q4 0 = t0 < t1 < · · · < tn = t.Vkk, nlX2

t −X20 = 2

n∑

j=1

Xtj−1

(

Xtj −Xtj−1

)

+

n∑

j=1

(

Xtj −Xtj−1

)2.w m(D) → 0, h|e 4.4.6 |e 4.3.2, lP+4poVX, k

X2t −X2

0 = 2

∫ t

0

XsdXs + 〈X〉t ..Y (Xt)t≥0 oN_f Ito >(. 9DlsA,lS# 4.4.1, ko,~>(. Q. Ito t~4oJp+.) 5.1.1 (,~) X s Y .o7Y. , XY .o7YbXtYt −X0Y0 =

∫ t

0

XsdYs +

∫ t

0

YsdXs + 〈X, Y 〉t. (5.1.1)AVlQdo%8.QoA,Y,o%8. .Co7YUoD~.o, T X =M+V , Y = N+W .QoY., ,XY oA,Y,. X.N +Y.M , Qok/8,. X.W +Y.V + 〈M,N〉..Lt~o||e.

122

:RZ Ito ?LCIVW 123x) 5.1.1 (Ito>() X = (X1t , · · · , Xd

t ). d6o7Yb f ∈ C2(Rd).,f(Xt)− f(X0) =

d∑

i=1

∫ t

0

∂f

∂xi(Xs)dX

is

+1

2

d∑

i,j=1

∫ t

0

∂2f

∂xi∂xj(Xs)d〈X i, Xj〉s . (5.1.2)>( (5.1.2) _)e.wV t eQQCk, Zwo70, KQSe.weOko ω, Oko t ≥ 0 Ck. T X Fk. X i

t =M it + V i

t , Q4 M1t , · · · ,Md

t .o7A,Y, V 1t , · · · , V d

t .o7/_ok/8UE, ,QoltOQSCwd∑

j=1

∫ t

0

∂f

∂xi(Xs)dM

js +

d∑

j=1

∫ t

0

∂f

∂xi(Xs)dV

js , (5.1.3)G &_oUE f(Xt) q.o7Y, Qo.h Ito >( 8M, Qo7A,Y,.

Mft = f(X0) +

d∑

j=1

∫ t

0

∂f

∂xi(Xs)dM

js . Qo7k/8,w

d∑

i=1

∫ t

0

∂f

∂xi(Xs)dV

is +

1

2

d∑

i,j=1

∫ t

0

∂2f

∂xi∂xj(Xs)d〈M i,M j〉s .$/ 5.1.1 " :

〈Mf ,Mg〉t =∫ t

0

d∑

i,j=1

∂f

∂xi(Xs)

∂g

∂xj(Xs)d〈M i,M j〉s,T B = (B1

t , · · · , Bdt )t≥0 . Rd Brown , ,o f ∈ C2(Rd),

f(Bt) .o7YbQY.f(Bt)− f(B0) =

∫ t

0

∇f(Bs).dBs +

∫ t

0

1

2∆f(Bs)ds . (5.1.4)

:RZ Ito ?LCIVW 124Mf

t = f(Bt)− f(B0)−∫ t

0

1

2∆f(Bs)ds ., Mf .6A,Yb


0

〈∇f,∇g〉(Bs)ds .3'.(Ito >() " Oxd.i X = M .o7A,Y!o Ito >(._fA,l=, +2Joo7NQ~Y M = (Mt)t≥0 5/" . G2J" f(Mt)− f(M0) = (f ′(M).M)t +

1

2(f ′′(M).〈M〉)t . (5.1.5)Zwk! t, G%. R9Nlo,I, >(o f(x) =

x2 (f ′(x) = 2x and f ′′(x) = 2) .!to,

M2 −M20 = 2M.M + 〈M〉 . (5.1.5) o f(x) = xn Ck:

Mn −Mn0 = nMn−1.M +

n(n− 1)

2Mn−2.〈M〉 ,_f,~>(o Mn s M , nl

Mn+1 −Mn+10 =Mn.M +M.Mn + 〈M,Mn〉

=Mn.M +M.

(

nMn−1.M +n(n− 1)

2Mn−2.〈M〉

)

+ nMn−1.〈M〉

= (n+ 1)Mn.M +(n+ 1)n

2Mn−1.〈M〉,VG (5.1.5) X? xn+1 Ck. ZW Ito >(o^(Ck. f ∈ C2(R), M k/ |M | ≤ a. l(y fn &n fn, f

′n, f

′′n [−a, a] #O/4po f, f ′, f ′′. , (5.1.2) fn CkbQRlot6d<~ (tÆs (5.1.3) otÆ) eQQ4po (5.1.2) o_. hL~4p|e 4.4.6 Q$ f ′

n(X).M P+4po f ′(X).M , ZW(5.1.5) f Ck. QdOoo7A,Y M , QS_f9<&foA,l_pC" .

:RZ Ito ?LCIVW 125$/ 5.1.2 f . [a, b] o7QkX?, " : \(y fn &n fn sf ′n #O/4po f s f ′.T f +.O6ju C2 o, ,kA,o Ito >(. D . Rd oO6ju, f ∈ C2(D), X0 ∈ D, , Ito >( (5.1.2) o X dM D %vCk, tasC, ζ = inft > 0 : Xt ∈ Dc, ,i t < ζ !eQQsk

f(Xt)− f(X0) =

d∑

i=1

∫ t

0

∂f

∂xi(Xs)dX

is

+1

2

d∑

i,j=1

∫ t

0

∂2f

∂xi∂xj(Xs)d〈M i,M j〉s . (5.1.6)BwA, Ito >(. ,# , Dn = x ∈ Rd : d(x,Dc) ≥ 1/n, τn . Dc

n o64!, , τn ↑ ζ . Zw Dn oX? f QS^Cw Rd o C2 X?, G_f Ito >( (5.1.2) nQ Xτn Ck, Qdw n→ ∞ Q.+ 5.1.1 B = (Bt) . R3 o!F Brown , y 6= 0. h(x) := |x− y|−1,

x ∈ R3 \ 0, h(B) = h(Bt) .A,Y, hQ).Y.6h (3.4.3), fw., K.C Brown oeOkRm)vFl y w. ^ k ≥ 1, lDk := x ∈ R3 : |x− y| ≥ 1

k. Tk . Dc

k o64!, Zw 〈Bi, Bj〉t = εi,jt, b h R3 \ y z, GhA, Ito >(h(BTk

t )− h(0) =

∫ t

0

∇h(Bs)1s<Tk · dBs, Dk ,

|∇h|2 = 1

h4≤ k4,ZW h(BTk) .Y, Tk ↑ Ty = ∞ a.s., GkM h(B) .A,Y. )/H"

limt↑+∞

E[h(Bt)] = 0, (5.1.7)ZW h(B) ).Y. h(B) .Q~o'A,Y, Q. Y, h Doob o Y4p|e (5.1.7), h(Bt) eQQ4po 0, oeOkRm, |Bt| i

:RZ Ito ?LCIVW 126of, K.C 3- x Brown Q7vdM^8|ok/ju. $m 1- x Brown .jo.$/ 5.1.3 " (5.1.7).Lt~eTO6fo+>.

1. Ito ~: Jo Brown oL~.

2. NQ~YoÆX8Q~e.

3. A,l=: fOyvÆiofo !y_5/A,l.O6Jo`, dUW`QSk/o7Yo4-kOlo7A,Yd..

4. o7A,Y.Yo<9JokO. <(oo7A,Y)Q2.k/8o, ~)2eQQoVX|X. T M .o7A,Y, QkÆX8UE 〈M〉, .&n M2 − 〈M〉 .o7A,YovOo7ÆUE.

5. YU: UE X BwY, TQ.O6o7A,YsO6o7k/8UEo: X = M + V , 6..vOo, BwY.. Q4o=J%.o7A,Y, k/8UE.wwwYUk`Yo0.

6. Y.60U. YkÆX8, QoÆX8poQY,oÆX8, Zwo7ÆUEoÆX8apoz. Y%k%8, K.QY,%o%8.

7. YD~, bk,~>(:

XtYt = X0Y0 +

∫ t

0

XdY +

∫ t

0

Y dX + 〈X, Y 〉.

8. Y&_O6ÆXo7QtX?dPv.Y, bk Ito >(:

f(Xt) = f(X0) +

∫ t

0

f ′(X)dX +1

2

∫ t

0

f ′′(Xs)d〈X〉s.

:RZ Ito ?LCIVW 127

§5.2 Ito Ns~+4, kM Ito >(oO$9Jo_f, SW8MLe4Q9JQJvoO$-Ts>(, QQQukRo_f, Z#1?:.

§5.2.1 WS+4, OfoLtEdZt = ZtdXt , Z0 = 1 (5.2.1)Q4 Xt = Mt + At o7Y. E (5.2.1) o.Bw X oL)?. E

(5.2.1) _)e.w~EZt = 1 +

∫ t

0

ZsdXs (5.2.2)Q4~. Ito ~oVX. wwl (5.2.2) o., QS3Zt = exp(Xt + Vt)Q4 (Vt)t≥0 e|wO6 “1!” (Q.k/8o). _f Ito >(, k

Zt = 1 +

∫ t

0

Zsd(Xs + Vs) +1

2

∫ t

0

Zsd〈M〉swwGEE (5.2.2) 3w Vt = −12〈M〉t.) 5.2.1 Xt =Mt +At (Q4 M .6o7A,Y, A /_oo7k/8UE), b X0 = 0. ,

E (X)t = exp

(

Xt −1

2〈M〉t

).E (5.2.2) o..UE E (X) Bw X = (Xt)t≥0 oL)?.8\ 5.2.1 (Mt)t≥0 .L(zoo7A,Y. ,QL)? E (M) .6o7'A,Y.

:RZ Ito ?LCIVW 1284) 5.2.1. 9D Ito ~o|X, ^ T > 0 TE

∫ T

0

e2Mt−〈M〉td〈M〉t < +∞ (5.2.3),L)?E (M)t = exp

(

Mt −1

2〈M〉t

).'o7NQ~Y.O69Jo,#.7L E (M) K4).Y, hQ|. Y.) 5.2.2 X = (Xt)t≥0 .6'o7A,Y. , X = (Xt)t≥0 .6 Y: ^ t > s ≥ 0, E(Xt|Fs) ≤ Xs. Z#s t → EXt .vo, ZW^ t > 0, EXt ≤ EX0.3'. " aPso Fatou \e: T Xn .+U (Ω,F ,P) 'Q~L6y, ,E [limn→∞Xn|G ] ≤ limn→∞E [Xn|G ]Q4 G . F oJ σ- ? (~T ).h|X, \A,l6y τn &n Xτn = (Xt∧τn)t≥0 .Y. ZW

E (Xt∧τn |Fs) = Xs∧τn , ∀t ≥ s, n = 1, 2, · · · .Z#skE (Xt∧τn) = EX0.h Fatou \e, Xt = limn→∞Xt∧τn .Q~o. _f Fatou \eo Xt∧τn s

G = Fs, nE[Xt|Fs] = E

[

limn→∞

Xt∧τn |Fs

]

≤ limn→∞E[Xt∧τn |Fs]

= limn→∞Xs∧τn = Xs9D|X X = (Xt)t≥0 .6 Y.

:RZ Ito ?LCIVW 129v, O6o7 Y X = (Xt)t≥0 .6Yib4iQoPs t→ E(Xt) .<?. ZWkok.`6 5.2.1 M = (Mt)t≥0 .L(zoo7A,Y. , E (M) .6 Y.Z#s,

E

[

exp

(

Mt −1

2〈M〉t

)]

≤ 1 for all t ≥ 0 .5O+s E (M) ! [0, T ] .Yib4iE

[

exp

(

MT − 1

2〈M〉T

)]

= 1 . (5.2.4)A,YoL)?+r4k`9Jo0. 4_f4, $mO68|YoL)?.!.O6!oY.O<Jo,d. " (5.2.4)oO6Ga.O~o Novikov a5>.x) 5.2.1 (Novikov) M = (Mt)t≥0 .6L(zoo7A,Y. TE

[

exp

(

1

2〈M〉∞

)]

< +∞ , (5.2.5), E (M) .O/Q~Y.3'. o" .?8Mo (3O [16]). D, Novikov a(5.2.5) , " ^ 0 < α < 1

E (αM)τ ≡ exp

(

αMτ −1

2α2 〈M〉τ

)

: τ wk/ !.O/Q~o, ZW (E (αM)t) .O6Y, vdw α ↑ 1.ww" O/Q~0, ^8|o α, E (αM) .A,Y αM oL)?, G E (αM) .'o7A,Y, E (E (αM)t) ≤ 1. ko03E (αM)t ≡ exp

α

(

Mt −1

2〈M〉t

)

− 1

2α (α− 1) 〈M〉t

= (E (M)t)α exp

1

2α (1− α) 〈M〉t

.^k/ ! τ s^ A ∈ F∞

E [1AE (αM)τ ] = E

1A (E (M)τ )α exp

[

1

2α (1− α) 〈M〉τ

]

. (5.2.6)

:RZ Ito ?LCIVW 130 (5.2.6) _f Holder )p(o 1α> 1 s 1

1−α, n

E 1AE (αM)τ = E

(E (M)τ )α 1A exp

[

1

2α (1− α) 〈M〉τ

]

≤ E [E (M)τ ]α

E

[

1A exp

(

1

2α 〈M〉τ

)]1−α

≤

E

[

1A exp

(

1

2α 〈M〉∞

)]1−α

≤

E

[

1A exp

(

1

2〈M〉∞

)]1−α

. (5.2.7)Y

E (αM)τ : wk/ ! τ kO/Ko70, 9D|e 1.2.4 kMQ.O/Q~o, Z E (αM) |.Y, b.O/Q~Y. ZW

E (αM)∞ = limt→∞

E (αM)tkE [E (αM)∞] = E [E (αM)0] = 1, ∀α ∈ (0, 1). (5.2.7) otO6)p(4QSl A = Ω s τ = ∞, nlo)p(

1 = E [E (αM)∞]

≤ E [E (M)∞]α

E

(

exp

(

1

2〈M〉∞

))1−α

, ∀α ∈ (0, 1). α ↑ 1 nlE (E (M)∞) ≥ 1 E [E (M)∞] = 1, hWkM E (M)t O/Q~Y.Y|eo" QSkM, T^ t ≥ 0,

E

[

exp

(

1

2〈M〉t

)]

< +∞ , (5.2.8), E (M) .O6Y, O/Q~.OO6!F Brown B = (Bt), b F = (Ft)t≥0 ∈ L2. TE

[

exp

(

1

2

∫ T

0

F 2t dt

)]

<∞

:RZ Ito ?LCIVW 131,Xt = exp

∫ t

0

FsdBs −1

2

∫ t

0

F 2s ds

(5.2.9).O6 [0, T ] o!Y.

Novikov a`), QS4dZQ).,TH". jww$mY ∫ t

0BsdBs oL)?.!.O6Y, Q Novikov aJhC~

E

exp

[

1

2

∫ T

0

B2t dt

]).6To;T, QSTwr CQ.T (Xt)t≥0 .'o7 Y, ,P

supt∈[0,T ]

Xt ≥ λ

≤ 1

λE(X0) . (5.2.10)-_)?Y>(, QS" o)?)p(.$/ 5.2.1 B = (Bt) .!F Brown , ,^ T > 0

P

supt∈[0,T ]

Bt ≥ λT

≤ e−λ2

2T . (5.2.11)

§5.2.2 Levy r Brown z$ tO6_f. Levy o Brown YRj. (Ω,F ,Ft,P) .Fk Nd<aoo+U.x) 5.2.2 (Levy) Mt = (M1t , · · · ,Md

t ) . (Ω,F ,Ft,P) l(o Rd oL(zoLUE. , (Mt)t≥0 .6 Brown ib4i(1) 6 M i

t .o7A,Y.

(2) M itM

jt − δijt .6Y, ^= (i, j), 〈M i,M j〉t = δijt.3'. +2" G0,. uW, |eo (Mt)t≥0 .6 Brownib4i^ t > s ξ = (ξj) ∈ Rd, k

E(

ei〈ξ,Mt−Ms〉∣

∣Fs

)

= exp

−|ξ|22

(t− s)

(5.2.12)

:RZ Ito ?LCIVW 132ZWO/_UEZt = exp

(

i

d∑

j=1

ξjMjt +

|ξ|22t

)

J" Q.6Y. wW, _f Ito>(oX? f(x) = ex (! f ′ = f ′′ = f)SYXt = i

d∑

j=1

ξjMjt +

|ξ|22t ,,

Zt = Z0 +

∫ t

0

Zsd

(

i

d∑

j=1

ξjMjs +

|ξ|22s

)

+1

2

∫ t

0

Zsd〈id∑

j=1

ξjMj〉s

= 1 + i

d∑

j=1

ξj

∫ t

0

ZsdMjs +

|ξ|22

∫ t

0

Zsds−1

2

∫ t

0

d∑

k,j=1

ξkξjZsd〈Mk,M j〉s

= 1 + i

d∑

j=1

ξj

∫ t

0

ZsdMjsQdO6pZ_K

1

2

∫ t

0

d∑

k,j=1

ξkξjZsd〈Mk,M j〉s =1

2|ξ|2

∫ t

0

Zsds ..9D 〈M i,M j〉s = δijs. Zw Z .o7A,Yb |Zs| = e|ξ|2s/2, G^ T > 0

E〈Z〉T = E

∫ T

0

|Zs|2ds =∫ T

0

e|ξ|2sds < +∞.h OoO6 kM Z .L( 1 oo7NQ~Y. >( (5.2.12) hY0kM^ t > s,

E

(

ei〈ξ,Mt〉+ |ξ|2

2t

∣

∣

∣

∣

Fs

)

= ei〈ξ,Ms〉+ |ξ|2

2s.

:RZ Ito ?LCIVW 133

§5.2.3 -teP Brown zrIaÆ# , O6o7A,Y9U'6!rd. Brown .x) 5.2.3 (Dambis, Dubins, Schwarz) +U (Ω,F ,Ft,P) L(zoLUE. N 〈M〉∞ = ∞ oo7A,Y, τt = infs : 〈M〉s > t .,^ t ≥ 0, τt . !, Bt =Mτt .6 (Fτt)-Brown, bMt = B〈M〉t .3'. vÆo !O (τt)t≥0 Bw!r, Zw6 τt . !, bv

t → τt .vÆo. 6 τt .eQQko, Zw 〈M〉∞ = ∞ a.s. h 〈M〉too70〈M〉τt = t P-a.s.f Doob k/ *|eoNQ~ (bO/Q~) Y (Ms∧τt)s≥0 S !

τt ≥ τs (t ≥ s), nlE [Mτt |Fτs ] =Mτs

i.e. Bt .6 (Fτt)- A,Y. Fo" _foY (M2s∧τt − 〈M〉s∧τt)s≥0 kM

E[

M2τt − 〈M〉τt |FTs

]

=M2τs − 〈M〉τs .ZW (B2

t −t).6 (Fτt)-A,Y. VNM t→ Bt .o7o, G B = (Bt)t≥0.6 (Fτt) Brown .$/ 5.2.2 f . [0,∞) L(zoo7vÆX?, b f(∞) = ∞. |Xf−1(x) = infy : f(y) > x." (1) f−1 lo7b^ x ≥ 0 k f(f−1(x)) = x. (2) f(y) > x ib4i

y > f−1(x).|o, Zw〈M〉t ≤ s = t ≤ τs ∈ Fτs ,G 〈M〉t . (Fτs)- !bU_Co7A,Y M .'6 Brown o!r.

M = B〈M〉.

:RZ Ito ?LCIVW 134wSW_" Novikov |e. ul Brown B = (Bt), k > 0 a > 0

σa = inft : Bt = kt− a,,E[

e−sσa]

= ea(k−√k2+2s). (5.2.13)hWkM σa .)?Q~obk

E[

ek2σa/2

]

= eak. (5.2.14)$/ 5.2.3 uH (5.2.13) o" UE, H"E[

ek2σa/2

]

<∞hW" (5.2.14).w k = 1, kE[

eBσa−σa/2]

= 1.,QSkM(

eBσa∧t−(σa∧t)/2 : t ≥ 1).O6Y.$/ 5.2.4 " oUE.Y, # Q.6 Doob Y

eBσa∧t−(σa∧t)/2 = E[

eBσa−σa/2|Ft

]

. Bt =Mτt , Zw 〈M〉t . (Fτt) !, Gk1 = E

[

eBσa∧〈M〉t−(σa∧〈M〉t/2

]

= E[

1σa<〈M〉teσa/2−a

]

+ E[

1σa≥〈M〉teMt− 1

2〈M〉t

]

.w a ↑ +∞, hV14p|ekME[

eMt− 12〈M〉t

]

= 1.

:RZ Ito ?LCIVW 135

§5.2.4 Girsanov x)I|O6bo+U (Ω,F ,Ft,P). T > 0, Q . (Ω,FT ) +5&ndQ

dP

∣

∣

∣

∣

FT

= ξ'6Lu ξ ∈ L1(Ω,FT ,P). h|X, ok/o FT - Q5Lu X∫

Ω

X(ω)Q(dω) =

∫

Ω

X(ω)ξ(ω)P(dω)zf'CwEQ(X) = EP(ξX) . T X . Ft- Q5o, t ≤ T , ,

EQ(X) = EP(EP (ξX|Ft))

= EP(EP (ξ|Ft)X) .o t ≤ TdQ

dP

∣

∣

∣

∣

Ft

= EP (ξ|Ft)Q+ P .! T %o'Y.U_, T T > 0 b Z = (Zt)t≥0 . (Ω,F ,Ft,P) L( 1 ! T%oo7?5!Y. (Ω,FT ) |XO65 Q

Q(A) = E (ZTA) if A ∈ FT . (5.2.15)K.CdQ

dP

∣

∣

∣

∣

FT

= ZT .Zw E (ZT ) = 1, G Q . (Ω,FT ) +5. Zw Z = (Zt)t≤T .Y, G^ t ≤ T ,dQ

dP

∣

∣

∣

∣

F t

= Zt.$/ 5.2.5 T (Zt)t≥0 .L( 1 oo7!Y, , (Ω,F∞) \+5 Q, Q4 F∞ ≡ σFt : t ≥ 0, &n^ t ≥ 0, dQdP

∣

∣

Ft= Zt.

:RZ Ito ?LCIVW 136\e4o-TH", ~T .) 5.2.3 (1) O6/_UE X .O6 Q- Yib4i XZ .O6 P- Y. (2)T XZ . P- A,Y, X .O6 Q- A,Y. (3) Q 5eQQVX, Z M.?5!o.$/ 5.2.6 τ . !, " : T (XZ)τ . P- Y, , Xτ . Q- Y.QS5>%" Girsanov |ew.x) 5.2.4 (Girsanov) (Mt)t≥0 .+U (Ω,F ,Ft,P) ! T %oo7A,Y, (Zt)t≥0 .L( 1 oo7!Y. ,Xt =Mt −

∫ t

0

1

Zs

d 〈M,Z〉s.+U (Ω,F ,Ft,Q) ! T %oo7A,Y.3'. _fA,lQS M,Z, 1/Z .k/o. ! M,Z .k/Y. 2J" X + Q %.Y:

EQ Xt|Fs = Xs for all s < t ≤ T ,K.C,

EQ 1A (Xt −Xs) = 0 for all s < t ≤ T , A ∈ Fs .h|XEQ 1A (Xt −Xs) = EP (ZtXt − ZsXs)1AZW+2" (ZtXt) +5 P ! T %.Y. 9D*~>(, k

ZtXt = Z0X0 +

∫ t

0

ZsdXs +

∫ t

0

XsdZs + 〈Z,X〉t

= Z0X0 +

∫ t

0

Zs

(

dMs −1

Zs

d 〈M,Z〉s)

+

∫ t

0

XsdZs + 〈Z,X〉t

= Z0X0 +

∫ t

0

ZsdMs +

∫ t

0

XsdZsQv.O6Y.

:RZ Ito ?LCIVW 137Zw Z = (Zt : t ≤ T ) .O6!Y, QS_f Ito >(o logZt, nllogZt − logZ0 =

∫ t

0

1

Zs

dZs −1

2

∫ t

0

1

Z2s

d〈Z〉s ,, Zt = E (N)t Q4Nt =

∫ t

0

1

Zs

dZs.O6o7A,Y. ZW Zt = E (N)t .w Ito ~EZt = 1 +

∫ t

0

ZsdNs ,Y 〈M,Z〉t = 〈

∫ t

0

dMs,

∫ t

0

ZsdNs〉 =∫ t

0

Zsd〈N,M〉s .hWkM∫ t

0

1

Zs

d 〈M,Z〉s = 〈N,M〉t,ZWkok.`6 5.2.2 Nt . (Ω,F ,Ft,P) L(zoo7A,Y, &nQL)?(E (N)t : t ≤ T ).o7Y. Q5U (Ω,FT ) |X+5 Q

dQ

dP

∣

∣

∣

∣

Ft

= E (N)t for all t ≤ T .T M = (Mt)t≥0 .+ P oo7A,Y, ,Xt =Mt − 〈N,M〉t.+ Q ! [0, T ] oo7A,Y. (_)I|X! [0, T ] oA,Yo+8.)Zw P s Q t65p, OSQkeoz+. P+4p603t65K.po (eK" %).

:RZ Ito ?LCIVW 138`6 5.2.3 Q Jo P p, P- o7Y X peo Q- o7Yb〈X〉Q = 〈X〉P. |ok/<6Q5UE H Jo X o Q- L~s P- L~O/._fo Brown , F .o7/_UE&n

Zt = exp

(∫ t

0

FsdBs −1

2

∫ t

0

F 2s ds

)

, t ≥ 0.Y, ,)5 Q ,

Bt := Bt −∫ t

0

Fsds, t ≥ 0.o7Y, bQoÆX8UE. t, ZW B .)+. Brown .QdCC Girsanov |eo+. Girsanov |e_|oNQrd5ol6, jkxU o Lebesgue 5.NQ)o, ,Rd o Gauss 50? `T, Gauss 5

µ(dx) =1

(2πt)d/2exp

(

−|x|22t

)

dx. Rd o+5, Q&nQ oU!Lu x .!F!S*o. ^ y ∈ Rd

µ(dx− y) =1

(2πt)d/2exp

(

−|x− y|22t

)

dx

= exp

(

x · y − 1

2|y|2)

µ(dx),QSftHmC, 6 Gauss 59UNQdJoz_o Gauss 5Ko7, KBw3)0, zCLu x 7→ x− y +5exp

(

x · y − 1

2|y|2)

µ(dx).!F!S*o.fxU )\ σ- koNQ)z3)o5, h.QSOEGoNQ3)5. wO Wiener 5, Girsanov |e_

:RZ Ito ?LCIVW 139fo Brown .>"o Cameron-Martin |e. O! t ∈ [0, 1], B.IewIRm Bω(t) = Bt(ω), ω ∈ Ω. W . [0, 1] o7X?bL(w 0 oo7X?p^, µ . W o Wiener 5, f . W o'Q5X?, E[f(B)] =

∫

W

f(w)µ(dw).KQSC, + µ %, U!UE. Brown .w\ Cameron-Martin U,

H = h ∈ W : h ∈ L2[0, 1],K.CQkbk?NQ~oyHp^. , h(t) =∫ t

0h(s)ds, |X5

Q = exp

(∫ 1

0

h(t)dBt −1

2

∫ 1

0

|h(t)|2dt)

· P,vEQ[f(B)] = E

[

exp

(∫ 1

0

h(t)dBt −1

2

∫ 1

0

|h(t)|2dt)

; f(B)

]

=

∫

W

exp

(∫ 1

0

h(t)dw(t)− 1

2

∫ 1

0

|h(t)|2dt)

f(w)µ(dw)

=

∫

W

f(w)µh(dw),Q4µh(dw) = exp

(∫ 1

0

h(t)dw(t)− 1

2

∫ 1

0

|h(t)|2dt)

µ(dw)., Girsanov |eCUE B = (Bt − h(t) : t ∈ [0, 1]) + Q %K.Brown ,

∫

W

f(w − h)µh(dw) = EQ[f(B − h)] = E[f(B)] =

∫

W

f(w)µ(dw). W oNQ θh : w 7→ w + h, ,o W o'Q5XX? f k∫

f(w)µh(dw) =

∫

f(w − h+ h)µh(dw)

=

∫

f(w + h)µ(dw) =

∫

f θh(w)µ(dw).f Cameron-Martin |e5>.

:RZ Ito ?LCIVW 140x) 5.2.5 (Cameron-Martin) o h ∈ H , Wiener 5 µ W oNQe θh %oJo µ Ko7, µθ−1

h = µh.zCEG Cameron-Martin UoX?NQ, Wiener 5k3)0.$/ 5.2.7 B = (Bt) . Brown . 0 < s < t < 1 haPsE

[

exp

(∫ 1

0

h(s)dBs −1

2

∫ 1

0

|h(s)|2ds) ∣

∣

∣

∣

Bs, Bt

]

.

§5.2.5 Ox)Y")|e. Brown oO6R-T. x Brown ;_kO6Kvo , hgwwfU, +OOx;_. B = (Bt)t≥0 p+U (Ω,F ,P) Ox!F Brown. (F 0t )

(qk F 0∞ = σ

(⋃

t>0 F 0t

)

) .h Brown (Bt)t≥0 Co. F∞ . F 0∞oplb Ft . F 0

t F∞ 4oOkz+dCo σ- ?. (,# (Ft) .lo7o.)x) 5.2.6 M = (Mt)t≥0 b+U (Ω,F ,Ft,P) oNQ~Y. ,\LUE F = (Ft)t≥0 ∈ L 2, &n^ t ≥ 0,

Mt = E(M0) +

∫ t

0

FsdBs a.s.Z#s, ^ Brown (Ft)t≥0 YkO6o71!.|eo" P`oo6\e. T > 0 .VI|!?.) 5.2.4 +U (Ω,FT ,P) oLu_

φ(Bt1 , · · · , Btk) : ∀k ∈ Z+, tj ∈ [0, T ] and φ ∈ C∞0 (Rk)

L2(Ω,FT ,P) 4K.3'. T X ∈ L2(Ω,FT ,P), ,h|X, \O6 F 0T - Q5X?s X eQQp. ZW)O0, X ∈ L2(Ω,F 0

T ,P). 9D|X, F 0T =

σBt : t ≤ T. D = Q ∩ [0, T ], [0, T ] ke?_. Zw D [0, T ] 4K

:RZ Ito ?LCIVW 141, F 0T = σBt : t ∈ D. bZw D Q?, GQS' D = t1, · · · , tn, · · · . Dn = t1, · · · , tn, b Gn = σ Bt1 , · · · , Btn. , Gn vÆb Gn ↑ F 0

T . Xn = E (X|Gn). , (Xn)n≥1 .NUYbhY4p|eXn → X a.s.b Xn → X in L2. ^ n, Xn . Gn Q5o, ,\'6 Borel Q5X? fn : Rn → R &n

Xn = fn(Bt1 , · · · , Btn).Zw Xn ∈ L2, G fn ∈ L2(Rn, µ) Q4 µ . (Bt1 , · · · , Btn) o*, E[X2

n] =

∫

Rn

fn(x)2µ(dx) .Z C∞

0 (Rn) L2(Rn, µ) K, G^ n, \6y φnk in C∞0 (Rn) &n φnk → fn in L2(Rn, µ). hWkM

φnn(Bt1 , · · · , Btn) → X

in L2.T I ⊂ R .6j, ,f L2(I) ") I Jo Lebesgue 5NQ~oX? h PCo Hilbert U.) 5.2.5 T > 0. ^ h ∈ L2([0, T ]), oO6)?Y:

M(h)t = exp

∫ t

0

h(s)dBs −1

2

∫ t

0

h(s)2ds

; t ∈ [0, T ]. (5.2.16),L := spanM(h)T : h ∈ L2([0, T ]) L2(Ω,FT ,P) 4K, Q4 span ")uCo0JU.3'. +2" , H ∈ L2(Ω,FT ,P) &n^ Φ ∈ L, k E[H · Φ] = 0, , H = 0.^ 0 = t0 < t1 < · · · < tn = T s ci ∈ R, O*[X? h(t) = ci for

t ∈ (ti, ti+1]. ,M(h)T = exp

∑

i

ci(Bti+1− Bti)−

1

2

∑

i

c2i (ti+1 − ti)

.

:RZ Ito ?LCIVW 142Zw^ Φ ∈ L k E[HΦ] = 0, GE

[

H exp

(

∑

i

ci(Bti+1− Bti)−

1

2

∑

i

c2i (ti+1 − ti)

)]

= 0 .,6t|0o! exp−12

∑

i c2i (ti+1 − ti) QSY~4ny, ZWkM

E

[

H exp

(

∑

i

ci(Bti+1− Bti)

)]

= 0 .Zw ci .V?, GkE

[

H exp

(

∑

i

ciBti

)]

= 0^ ci s ti ∈ [0, T ] Ck. vR5Jo ci ., 6p(^&? ciCk. o φ ∈ C∞0 (Rn), k

φ(x) =1

(2π)n

∫

Rn

φ(z)e−i〈z,x〉dzQ4φ(z) =

∫

Rn

φ(x)ei〈z,x〉dx. φ o Fourier r. ZWE[Hφ(Bt1 , · · · , Btn)] =

1

(2π)nE

[

H

∫

Rn

φ(z) exp

(

i∑

j

zjBtj

)

dz

]

=1

(2π)n

∫

Rn

φ(z)E

[

H exp

(

i∑

j

zjBtj

)]

dz

= 0 .vd^ φ ∈ C∞0 (Rn),

E[H · φ(Bt1 , · · · , Btn)] = 0 . (5.2.17)# g|Xw Rn O6k"Z5 µ wµ(A) = E [H · 1A(Bt1 , · · · , Btn)] , A ∈ B(Rn),

:RZ Ito ?LCIVW 143h Fourier rovO0, µ = 0 ib4iQ Fourier rwz.9D\e 5.2.4, . φ(Bt1 , · · · , Btn) oX?p^ L2(Ω,FT ,P)4K,G ^ G ∈ L2(Ω,FT ,P) kE[H ·G] = 0.Z#s, E[H2] = 0 kM H = 0.x) 5.2.7 (Ito) ξ ∈ L2(Ω,FT ,P). ,\ F = (Ft)t≥0 ∈ L 2, &n

ξ = E[ξ] +

∫ T

0

FtdBt .3'. h\e 5.2.5 +2" o (5.2.16) 4|Xo ξ = X(h)T (Q4h ∈ L2([0, T ])) " |eo-. Zw X(h)t .)?YGQ No~E

X(h)T = 1 +

∫ T

0

X(h)td

(∫ t

0

h(s)dBs

)

= E[X(h)T ] +

∫ T

0

X(h)th(t)dBt .ZW8d., l Ft = X(h)th(t) QSw.Y")|e 5.2.6 Y Ito o")|esY0kRkM.

§5.3 ;yR (*)2; kt8gH(, O8.gHo H9, H9olP`oi r; |O8.gHo H9, H9l.LUE S = (St). (Ft). S oKv+.kO6gH, Po0(UE.LUE X = (Xt), Q4 Xt .P!R t oH9M(. O~gH4.)O6LUE H = (Ht), Q4 Ht .A H9o?, QpoH9g H9. K.CgHoH9EXt = HtSt + (Xt −HtSt),

:RZ Ito ?LCIVW 144lt,#.g 2;s 2;oH1u, _eo. H .(Ft) /_o, K.CgH)$m_o+.CgH.KHo, .)QH9oÆu+_K2;H9oÆu,

dXt = HtdSt + (Xt −HtSt)rdt,ltÆv. H9oÆ=(. pod(e−rtXt) = Htd(e

−rtSt).Kvs, e−rtXt . t !RH9o, ZWe−rTXT = X0 +

∫ T

0

Htd(e−rtSt),dH9poL'H9( gH4Jod H9oL~,zCKHdZ, gHo0(UEhL'gHgH4Oppt|. 2;Yi.)O8gH4 H = (Ht) &n_ogHOn X = (Xt) N (1) X0 = 0, XT ≥ 0; (2) E[XT ] > 0.x) 5.3.1 (tO||e) 2;Yiib4i\po P o+5 Q&ndoH95 (e−rtSt : 0 ≤ t ≤ T ) .Jo Q oY.iIO65\!, C2;\pY5, pY5\kM2;Yi.`To, U_o" )T. Yio2;Bwk#2;.2;k#08).C+G>M, OkgH.Bo.".PoP? PozP.<oF"qoJ. QfoP.)|t|o!RSt|5A'8 H9 (M: i, #, owp). QfoPo.<(Po, <( (M) Po.)8|!wS|5A (zM) O H9ooi. PPoQaoj#.P_lP3/, Po_lPQS)/, OSP_Z!(wz, Po_.k o. O~Po|.)6__)(Æ[. PoPIwF"q, O~lP!w T oF"q.)O6 FTQ5oLu V , K.CF"qo(.Lo, h.llP!

:RZ Ito ?LCIVW 145r tw. lP! T , |5 K oP_olP(w ST −K, _oPo(w (ST −K)+, Zw ST < K !gHQS9)/_SÆM.F"qo|k`8, Q4Q>"o.O~o |, QoD`f, O6C;2;, gHgHFoOn_)QSdUgHz' H9_#, gHFKPoOn_)QSdUgHFKo(#.tasC, V . T !RlPoF"q, ,_)\ X0 4 H = (Ht)&nQt|o0(UE X = (Xt) lP! N V = XT . Y.Y")|eO">o. ,# , T2;Yi, , Q 5 (e−rtSt) .Y, V. FT Q5o, TY")|eCk, ,\ H &ne−rTV = e−rTEQV +

∫ T

0

Htd(e−rtSt).C V o |. V pY5oPs(. 8|(BwB|: q8gHOF"qd, q2JkR*lo[SY")|e8Mo45/gH, QdoOnYYHygHo&n, k^ . ^F"qQSI5/ |o2;Bwp2;. 2;pm8).C2;kNBogHv, &nHQS#.9:4kO6>"oO|, .C_(OIoMo5K_).OIo. vO|.2;YioO6k. hO|QSNMT2;k |om, pY5v.vOo.x) 5.3.2 (tÆ||e) O6k#2;4, 6F"qk |ib4ipY5vO. (St) oÆ=.O6<? h Brown kox

dSt

St

= bdt+ σdBt,Q4 b, σ > 0 .<?. 6EBw Black-Scholes E, _o$-BwBlack-Scholes $-, h)?Y>(

St = S0 exp

(

(b− 1

2σ2)t + σBt

)

,

:RZ Ito ?LCIVW 146Q4 S0 .L'5. 2;i. r, ,de−rtSt = S0 exp

(

(b− r − 1

2σ2)t+ σBt

)

= S0 exp

(

−1

2σ2t+ σ(Bt + t

b− r

σ)

)

. Wt = Bt + t(b − r)/σ h Girsanov |e, \p5 Q &n W = (Wt). Brown , !e−rtSt = S0 exp

(

−1

2σ2t + σWt

).Y.oO6F"q V , .!QSfJo (e−rtSt) oL~")? 6(St) Co. B Co, K. W Co, hY")|e\ F =

(Ft) &ne−rTV = e−rTEQV +

∫ T

0

FtdWt, (e−rtSt) . σW o)?Y, ZWσdWt =

de−rtSt

e−rtSt

,e−rTV = e−rTEQV +

∫ T

0

Ft

σe−rtSt

d(e−rtSt).! e−rTEQV . V o | V0.Z#s, o<(PoV = (ST −K)+ =

(

S0 exp

σWT + (r − 1

2σ2)T

−K

)+

,Zw Q 5, WT .!S*, OS o|QSf!S*X?(s"^M_, Bw. Black-Scholes >(V0 = S0Φ(d+ σ

√T )−Ke−rTΦ(d),

:RZ Ito ?LCIVW 147Q4 Φ .!F!S*X?,

d =log(S0/K) + (r − σ2/2)T

σ√T

.AV$-4o b . H9oNLÆ=, σ .&, 6>(s b J,K.Cs H9oNL"J, +sH9o&EkJ.

§5.4 lp1. (1) J E

∫ t

0sdB2

s . (2) J E(

∫ t

0sdB2

s

)2

. (3) J: E(

exp(∫ t

0sdBs)

)

.

2. kk/fodju D ⊂ Rd, f . D oo7X?, " : f .zoib4i^ x ∈ D, f(B+ x)τ .Y, Q4 B .!F Brown , τ .Yx Mo Brown B + x Jo_ Dc o64!.

3. X = (Xt) .o7A,YbQÆX8UE.t|0o, X0 = 0, " :

X . Gauss UE.

4. X, Y .o7Y, " 〈XY 〉 = X2.〈Y 〉+ Y 2.〈X〉.

5. (Tanaka) f .iX?, M .A,Y, " : f(M) = (f(Mt)) .Ybf(Mt)−

∫ t

0f ′−(M)dM : t ≥ 0 .6ÆUE, Q4 f ′

− .Rk?.3'.|X f oRk?f ′−(x) = lim

y↑0

f(x+ y)− f(x)

y, x ∈ R.Zwi0, lJo y vÆ, OS\, hkQ2.f, v f ′

− .vÆRo7o. l'o j ∈ C∞0 (−∞, 0) N ∫

Rj(x)dx = 1.

fn(x) =

∫

R

f(x+ y/n)j(y)dy., f ∈ C∞(R), Zw j .3#Ao, f .o7o, Ghk/4p|ekM, ^ x ∈ R k fn(x) → f(x). hfz4p|ef ′n(x) = (fn)

′−(x) =

∫ 0

−∞f ′−(x+ y/n)j(y)dy ↑ f ′

−(x).

:RZ Ito ?LCIVW 148Zw f ′n vÆ, G f ′′

n .'o. vd fn _f Ito >(%b_fL~o4p|e 4.4.6 kM-.

6. X .o7Y, " : |X| K.o7Y, b 〈|X|〉 = 〈X〉. hWT B. Brown , ,(1) |B| .o7YbQoY,.6 Brown .

(2) |B| oo7ÆUE,w L, L + t : Bt = 0 Æ, Bw BzwoA,!, h Levy o.3'.|X| .Y# R9 o 4" w, ,|Xt| − |X0| −

∫ t

0

sgn(X)dX.ÆUE, Q4 sgn(x) = 1x>0 − 1x≤0 .K(X?oRk?, ZW〈|X|〉 = sgn(X)2.〈X〉 = 〈X〉. T X . Brown , |B| oY,.M , o7ÆUE,. L, , 〈M〉t = 〈|B|〉t = 〈B〉t = t, ZWh LevyoRj, M . Brown . h Ito >( B2 = |Bt|2 = 2|B|.|B| + t, ZW|B|.|B| .o7Y, kM |B|.L = 0. " weOkoRm, dLt o#AV t : Bt = 0 4.

7. M .o7A,Y, M0 = 0 b 〈M〉 .t|0o. " : M . Gauss UEb.kÆuUE.

8. B .!F Brown , X .ko B o!Lu. Mt := BtX ,

t ≥ 0,

(a) " : M .o7A,YbQ.O6Yib4i EX12 <∞.

(b) J 〈M〉.

(c) -TkOl Mt = BAt, Q4 A .O6ko B oY 0 Moo7ÆUE.3'. F 0 . Brown o, Ft = F 0

t ∨ σ(X). H"• B . (Ft)-BM

:RZ Ito ?LCIVW 149

• tX . (Ft)- !• Kv BtX .o7A,Y• T X k/, , Doob |ekM (BtX) .Y•

E[|BtX −Bt(X∧n)|] =√tE

√X −X ∧ n" 〈M〉 = tX .

v3 X ko=J)o._f4Q9JoO LtEkU.o\vO0|e.

§6.1 |7<tE.>t|0oE, LtEd<NCw.O6<tE O6h Brown koLx. jdSt

St= rdt.O6>io<tE, nlo.t|0o.. QfoLtE Black-Scholes E

dSt

St= rdt+ σdBt")Æ=.O6<? O6Lx. LtEK.h Ito 6\o. fs0y.(oLtE

dXt = b(t, X)dt+ σ(t, X)dBt, (6.1.1)Q4 B .O6 r- x!F Brown , X .$oo7 d- xUE. LtE.O6.(, Zw# LtkVX, Lo~/kVX. ZW oE.)yoL~EXt −X0 =

∫ t

0

b(s,X)ds+

∫ t

0

σ(s,X)dBs. (6.1.2)Æ? (b, σ) T6C . W d . [0,+∞) l Rd oo7ep^, D3O/4ponO, fB(W d) ") W d Borel σ- ?, Bt(W

d) ")h w 7→ w(s), s ∈ [0, t] CoJ σ- ?, Rd ⊗Rr ") d× r Bp^, A d,r ") NyaoQ5e α : [0,∞)×W d −→ Rd⊗Rr p^: ^ t ≥ 0, α(t, ·). (W d,Bt(Wd))l

150

:EZ NAP>=8 151

Rd ⊗Rr Q5o. Kvs, Jh σ ∈ A d,r, b ∈ A d,1. !, X . (Ft)- /_oVG σ(t, X). (Ft)-<6Q5o. i σ(t, X) = σ(t, Xt), b(t, X) = b(t, Xt)! (!lo σ : R+ × Rd −→ Rd ⊗ Rr, b : R+ × Rd −→ Rd ⊗ R), BE. Markov -o; +OoE. Markov -odXt = b(t, Xt)dt+ σ(t, Xt)dBt, (6.1.3)Qo~.(.

Xt −X0 =

∫ t

0

b(s,Xs)ds+

∫ t

0

σ(s,Xs)dBs. (6.1.4):Z#od..i σ(t, X) = σ(Xt), b(t, X) = b(Xt)! (!lo σ : Rd −→Rd ⊗ Rr, b : Rd −→ Rd ⊗ R), BE.!T Markov -z Ito -o.|o2J9w.0OLtEo.o\vO0VX, LtEo.J<tE/Se.n, ZwgqkO6 Brown . CLtEo.# .EU[NC.O68MÆ?X? b, σ o.(.xz 6.1.1 +8|t6X? b, σ , LtE (6.1.3) (KBwE(b, σ)) o..)'6bk (Ft) o+U (Ω,F ,P) ot6o7/_UE (X,B), N:

1. B . Rr !F (Ft)-Brown ;

2. (6.1.4) Ck.K.C, +UQ o Brown ..oPC,, +kÆ?.8|o. .ovO0kt8)eo.0.xz 6.1.2 C (6.1.3) o.kRmvO0.)o8|+U oL( Brown B, +kvOoO6LUE NLtE, eO6bo+U ot6. (X,B) (X ′, B′) b B = B′, X0 = X ′0,

X = X ′. |oC.k*vO0.)t6FeL'*o. X X ′.po, keokx*O. d<CoLtE.ovO0.)*vO0.

:EZ NAP>=8 152Nw OCo.o+8%o, qkO6_.o+8KOI9J. CE(6.1.3) o. (X,B) ._., T X Jo B Cobplo (FB

t ) /_._.o.Bw.. d5>oO69J|eC , _.\QSkME+U Brown V8|!k.. # , gQC_...)C.O6_. .)VO6L( Brown _ (.Æ) kO6., !TkRmvO0, ,_vOO6...LtEoO6fh>"ojJ, QV0w |X4o,$+8, k.hk_., k*vO0hkRmvO0.+ 6.1.1 (Tanaka) OOxLtE:

Xt =

∫ t

0

sgn(Xs)dBs , 0 ≤ t <∞Q4 sgn(x) = 1x≥0 − 1x<0. zC.L(wzoLtEdXt = sgn(Xt)dBt.

1. *vO0Ck, Zw. X M.!F Brown (h Levy oRj|e).

2. kO6.. Wt .Ox!F Brown b Bt =∫ t

0sgn(Ws)dWs. ,

B K.Ox!F Brown bWt =

∫ t

0

sgn(Ws)dBs , (W,B) ...

3. T (X,B) .., , (−X,B) K... ZWRmvO0)Ck.

4. k_.. 6o" '\/, e3O [10].

§6.2 X ksyq,§6.2.1 pr Gauss &F0LtEQS(s.M_. O

dXjt =

n∑

i=1

σji dB

it +

N∑

k=1

βjkX

kt dt (6.2.1)

:EZ NAP>=8 153

(j = 1, · · · , N), Q4 B .O6 n- x Brown , σ = (σji ) .<? N × n B, β = (βj

k) .<? N ×N B. (6.2.1) QS'CwdXt = σdBt + βXtdt .

eβt =∞∑

k=0

tk

k!βk. β o)?. _f Ito >(, k

e−βtXt −X0 =

∫ t

0

e−βsdXs −∫ t

0

e−βsβXsds

=

∫ t

0

e−βs(dXs − βXsds)

=

∫ t

0

e−βsσdBsZWXt = eβtX0 +

∫ t

0

eβ(t−s)σdBs .Z#s, T X0 = x, , Xt .L( eβtx o!S*. j, n = N = 1,Xt ∼ N(eβtx,

σ2

2β

(

e2βt − 1)

) .QS" (Xt) .^UE (o7o1UE), QCQ+X? p(t, x, z) wPtf(x) :=

∫

RN

f(z)p(t, x, z)dz

= E (f(Xt)|X0 = x) ,ZW(Ptf)(x) = E (f(Xt)|X0 = x)

=

∫

R

f(z)1

√

2π σ2

2(e2βt − 1)

exp

(

− |z − eβtx|2σ2

β(e2βt − 1)

)

dz,

:EZ NAP>=8 154ZWp(t, x, z) =

1√

2π σ2

2(e2βt − 1)

exp

(

− |z − eβtx|2σ2

2(e2βt − 1)

)

.ww" (Xt) . Markov UE, 2JH"p(t, x, y) : t > 0.CQX?, N Chapman-Kolmogorov E

∫

Rn

p(t, x, y)p(s, y, z)dy = p(s+ t, x, z), s, t ≥ 0, x, z ∈ Rn. (6.2.2)$/ 6.2.1 H" p(t, x, y) N Chapman-Kolmogorov E.$/ 6.2.2 `TY o"^(NMd

dx(Ptf) = eβtPt

(

d

dxf

)

.T B = (B1t , · · · , Bn

t )t≥0 . n- x Brown , ,yEo. Xt:

dXt = dBt − (AXt) dt,Q4 A ≥ 0 . d× d B, B X wJQB A o Ornstein-Ulenbeck UE. bkXt = e−AtX0 +

∫ t

0

e−(t−s)AdBs .$/ 6.2.3 X0 = x ∈ Rn, J Ef(Xt), Q4 Xt .JQBw A oOrnstein-Uhlenbeck UE.

§6.2.2 Brown zO Black-Scholes $-dSt = St (µdt+ σdBt) . (6.2.3)E (6.2.3) o..∫ t

0

µds +

∫ t

0

σdBs ,

:EZ NAP>=8 155oL)?St = S0 exp

∫ t

0

σdBs +

∫ t

0

(

µ− 1

2σ2

)

ds

.i σ s µ w<?!,

St = S0 exp

σBt +

(

µ− 1

2σ2

)

t

QBw^ Brown . S0 = x > 0, St q.!o, blogSt = log x+ σBt +

(

µ− 1

2σ2

)

t.L( log x +(

µ− 12σ2)

t 8 σ2 o!S*. Fs, TwLtE(6.2.3) o., (St)t≥0 .O6^UE, Qo*hCQX? Pt(x, dz) J|, b|X

∫

R

f(z)Pt(x, dz) = E (f(Xt)|X0 = x)

= E(

f(xeσBt+(µ− 12σ2)t)

)

=

∫

R

f(xeσz+(µ−12σ2)t)

1√2πt

e−z2

2πtdz

=

∫ ∞

0

f(y)1

σy√2πt

e−1

2πt(1σlog y

x−(µ

σ− 1

2σ))

2

dyQ4 σ > 0, Su_rxeσz+(µ−

12σ2)t = y .<|X (Ptf)(x) =

∫

Rf(z)Pt(x, dz). hO>(t/n

(Ptf)(x) =

∫

R

f(xeσz+(µ−12σ2)t)

1√2πt

e−z2

2πtdz

=

∫

R

f(xeσ√ty+(µ− 1

2σ2)t)

1√2πe−

y2

2π dy

= E(

f(xeσ√tξ+(µ− 1

2σ2)t)

)Q4 ξ ∼ N(0, 1). ' Pt(x, dy) n|X, nPt(x, dy) =

1√2πt

1

σye−

12πt(

1σlog y

x−(µ

σ− 1

2σ))

2

dy on (0,+∞)

:EZ NAP>=8 156 (St) kCQ+p(t, x, y) =

1√2πt

1

σye−

12πt(

1σlog y

x−(µ

σ− 1

2σ))

2

on (0,+∞) .ZWo^ Brown (Ptf)(x) =

∫ ∞

0

1√2πt

1

σye−

12πt(

1σlog y

x−(µ

σ− 1

2σ))

2

f(y)dy,Q4 x > 0.

§6.2.3 Cameron-Martin MOfLtEdXt = dBt + b(t, Xt)dt (6.2.4)Q4 b(t, x) . [0,+∞)×R k/ Borel Q5X?. QSf+ro5. (6.2.4). (Wt)t≥0 . (Ω,F ,Ft,P) !F Brown , |X (Ω,F∞) +5 Q w

dQ

dP

∣

∣

∣

∣

Ft

= E (N)t for all t ≥ 0Q4 Nt =∫ t

0b(s,Ws)dWs Jo5 P .Y, b 〈N〉t =

∫ t

0b(s,Ws)

2ds, Q^kj o7. ZWE (N)t = exp

(∫ t

0

b(s,Ws)dWs −1

2

∫ t

0

b(s,Ws)2ds

).O6Y. Girsanov |eBt ≡Wt −W0 − 〈W,N〉t)5 Q .Y, b 〈B〉t = 〈W 〉t = t. h Levy oYRj|ekM (Bt)t≥0. Brown . 5O+

〈W,N〉t = 〈∫ t

0

dWs,

∫ t

0

b(s,Ws)dWs〉

:EZ NAP>=8 157

=

∫ t

0

b(s,Ws)dsGWt −W0 −

∫ t

0

b(s,Ws)ds = Bt. (Ω,F , Q) !F Brown . ZWWt = W0 +Bt +

∫ t

0

b(s,Ws)ds (6.2.5), (Ω,F∞, Q) o (Wt)t≥0 . (6.2.4) o.. 6.. SDE (6.2.4) oO6..x) 6.2.1 (Cameron-Martin-Girsanov) b(t, x) = (b1(t, x), · · · , bn(t, x)) .[0,+∞) × Rn k/ Borel X?, Wt = (W 1, · · · ,W n

t ) . (Ω,F ,Ft,P) O!F Brown , |X (Ω,F∞) +5 Q : ^ t > 0,

dQ

dP

∣

∣

∣

∣

Ft

= exp

n∑

k=1

∫ t

0

bk(s,Ws)dWks − 1

2

n∑

k=1

∫ t

0

∣

∣bk(s,Ws)∣

∣

2ds

.,\Jo+ Q o Brown (B1t , · · · , Bn

t )t≥0 &n (Wt)t≥0 + Q.dXj

t = dBjt + bj(t, Xt)dt (6.2.6)o..|O, T (Xt) . SDE (6.2.6) '6+U (Ω,F ,Ft,P) o., |X P

dP

dP

∣

∣

∣

∣

∣

Ft

= exp

−n∑

k=1

∫ t

0

bk(s,Xs)dBks −

1

2

n∑

k=1

∫ t

0

∣

∣bk(s,Xs)∣

∣

2ds

,QS" (Xt)t≥0 + P . Brown . ZW SDE (6.2.6) o.k*vO0: QOk.e*.

§6.3 X k`y*LtEo.o\vO0)eoVX)eo;_&f, TJ*Rmz Brown .x8|o, ,2JO_., %T

:EZ NAP>=8 159

§6.4 >: odys+4, 0_.\vO0o|-T. h|X, _.. ().. " Æ? N ^ Lipschitz aoO6\vO0-, " |ot6)p(: Gronwall )p(s Doob o Lp- )p(.) 6.4.1 (Gronwall) TO6'X? N~Eg(t) ≤ h(t) + α

∫ t

0

g(s)ds , 0 ≤ t ≤ TQ4 α ≥ 0 .<?b h : [0, T ] → R .Q~X?, ,g(t) ≤ h(t) + α

∫ t

0

eα(t−s)h(s)ds , 0 ≤ t ≤ T .3'. F (t) =∫ t

0g(s)ds. , F (0) = 0 b

F ′(t) ≤ h(t) + αF (t)G(

e−αtF (t))′ ≤ e−αth(t) .6t)p(~, nl

∫ t

0

(

e−αsF (s))′ds ≤

∫ t

0

e−αsh(s)ds,ZWkF (t) ≤

∫ t

0

eα(t−s)h(s)dskM Gronwall )p(.OoLtEdXj

t =

n∑

l=1

f jl (t, Xt)dB

lt + f j

0 (t, Xt)dt ; j = 1, · · · , N (6.4.1)Q4 f jk(t, x) . R+ × RN Borel Q5X?, b^ RN o3J k/.if Picard _" \vO0. " 4o=J;F. Doob o

:EZ NAP>=8 160

Lp- )p(oO6Z8dZ: T (Mt)t≥0 .NQ~o7Y, b M0 = 0, ,^ t > 0

E

sups≤t

|Ms|2

≤ 4 sups≤t

E(

|Ms|2)

= 4E〈M〉t . (6.4.2)) 6.4.2 (Bt)t≥0 . (Ω,Ft,F ,P) #(!F Brown , b (Zt)t≥0 s(Zt)t≥0 .o7/_UE. f(t, x) .O6 Lipschitz X?

|f(t, x)− f(t, y)| ≤ C|x− y| ; ∀t ≥ 0, x, y ∈ RQ4 C .<?.

1) TMt =

∫ t

0

f(s, Zs)dBs −∫ t

0

f(s, Zs)dBs ∀t ≥ 0,,E sup

s≤t|Ms|2 ≤ 4C2

∫ t

0

E∣

∣

∣Zs − Zs

∣

∣

∣

2

ds, ∀t ≥ 0.

2) TNt =

∫ t

0

f(s, Zs)ds−∫ t

0

f(s, Zs)ds ∀t ≥ 0,E sup

s≤t|Ns|2 ≤ C2t

∫ t

0

E∣

∣

∣Zs − Zs

∣

∣

∣

2

ds ∀t ≥ 0 .3'. w" tO6-, AVlsups≤t

|Ms|2 = sups≤t

∣

∣

∣

∣

∫ s

0

(

f(u, Zu)− f(u, Zu))

dBu

∣

∣

∣

∣

2

.,h Doob o L2- )p(E sup

s≤t|Ms|2 = E sup

s≤t

∣

∣

∣

∣

∫ s

0

(


dBu

∣

∣

∣

∣

2

≤ 4E

∣

∣

∣

∣

∫ t

0

(

f(s, Zs)− f(s, Zs))

dBs

∣

∣

∣

∣

2

:EZ NAP>=8 161

= 4E

∫ t

0

∣

∣

∣f(s, Zs)− f(s, Zs)

∣

∣

∣

2

ds

≤ 4C2

∫ t

0

E∣

∣

∣Zs − Zs

∣

∣

∣

2

ds .)G" tÆ6-. ,# sups≤t

|Ns|2 = sups≤t

∣

∣

∣

∣

∫ s

0

(


du

∣

∣

∣

∣

2

≤(∫ t

0

∣

∣


∣

∣

∣ds

)2

≤ t

∫ t

0

∣

∣


∣

∣

∣

2

ds

≤ C2t

∫ t

0

∣

∣

∣Zs − Zs

∣

∣

∣

2

dsQ4tÆ6)p(_K Schwartz )p(.x) 6.4.1 O SDE (6.4.1). T f ji N Lipschitz a:

∣

∣f ji (t, x)− f j

i (t, y)∣

∣ ≤ C|x− y| (6.4.3)s0Æ=a∣

∣f ji (t, x)

∣

∣ ≤ C(1 + |x|) (6.4.4)Q4 t ∈ R+, x, y ∈ RN . ,^ η ∈ L2(Ω,F0,P) s Rn- (o!F Brown Bt = (Bit), \vOo_. (Xt) of (6.4.1) NL(a X0 = η.3'. ww"ZfU, wOxdZLtEo_-T

dXt = f(t, Xt)dBt , X0 = η.e<tEOI, dU Picard 6.:

Y0(t) = ηbYn+1(t) = η +

∫ t

0

f(s, Yn(s))dBs ,

:EZ NAP>=8 163QSkM∞∑

n=0

E supt≤T

|Yn+1(t)− Yn(t)|2 ≤∞∑

n=0

(4C2)nT n

n!E sup

t≤T|Y1(t)− Y0(t)|2

< ∞ .ZW Yn : n ≥ 1 .U M 20 4o Cauchy y, b

Yn(t) → Xt uniformly on [0, T ] , P-a.s.TNM (Xt) .LtEo_.." vO0. Y s Z .eO6 Brown ot6.. ,Yt = η +

∫ t

0

f(s, Ys)dBsbZt = η +

∫ t

0

f(s, Zs)dBs .e\0o" 4,IE(

|Yt − Zt|2)

≤ 4C2

∫ t

0

E|Ys − Zs|2dsh Gronwall )p(kME(

|Yt − Zt|2)

= 0 .4) 6.4.1. |e 6.4.1 4o Yn . Brown oX?, zC Yn(t) 4P`oL( η s (Bs : 0 ≤ s ≤ t).

§6.5 CO!TXoLtEdX i

t =m∑

j=1

σij(Xt)dB

jt + bi(Xt)dt (6.5.1)

:EZ NAP>=8 164Q4 σij , b

i ∈ C∞(Rn) .kk/k?oNiX?, b B = (Bt) . (Ω,F ,Ft,P) m- x Brown . X = (Xt)≥0 .FkL( X0 ovO_.. Tf ∈ C2

b (Rn,R), ,h Ito >(

f(Xt)− f(X0) =

∫ t

0

n∑

k=1

∂f

∂xk(Xs)dX

ks

+1

2

∫ t

0

n∑

k,l=1

∂2f

∂xk∂xl(Xs)d〈Xk, X l〉s . (6.5.1)

〈Xk, X l〉t =∫ t

0

m∑

j=1

σkj (Xs)σ

lj(Xs)dsnl

f(Xt)− f(X0) =

∫ t

0

m∑

j=1

(

n∑

k=1

σkj

∂

∂xk

)

f(Xs)dBjs

+

∫ t

0

(

1

2

n∑

k,l=1

m∑

j=1

σkj σ

lj

∂2

∂xk∂xl+

n∑

k=1

bk∂

∂xk

)

f(Xs)ds .|X a = (akl)k,l≤n Q4akl =

m∑

j=1

σkj σ

lj . (akl)k,l≤n Bb'|o.

L =1

2

n∑

k,l=1

akl∂2

∂xk∂xl+

n∑

k=1

bk∂

∂xk(6.5.2)Q.O6 Rn O6Æ*m-tJJ. ,

f(Xt)− f(X0) =

∫ t

0

m∑

j=1

(

n∑

k=1

σkj

∂

∂xk

)

f(Xs)dBjs +

∫ t

0

(Lf)(Xs)ds .^ f |XMf

t = f(Xt)− f(X0)−∫ t

0

(Lf)(Xs)ds .

:EZ NAP>=8 165^ f ∈ C2b (R)

Mft =

∫ t

0

m∑

j=1

(

n∑

k=1

σkj

∂

∂xk

)

f(Xs)dBjs. (Ω,F ,Ft,P) oO6Y, b


0

m∑

j=1

(

n∑

k,l=1

σljσ

kj

∂f

∂xk∂g

∂xl

)

(Xs)ds

=

∫ t

0

(

n∑

k,l=1

akl∂f

∂xk∂g

∂xl

)

(Xs)ds .ZW" w#.8\ 6.5.1 T (Xt)t≥0 . SDE (6.5.1) (Ω,F ,Ft,P) (8| Brown )o_., ,^ f ∈ C2b (R)

Mft = f(Xt)− f(X0)−

∫ t

0

(Lf)(Xs)ds+ P .Y, Q4 L h (6.5.2) |X.j, σij = δij b bi = 0 (! L = 1

2∆), (Bt)t≥0 .

dXt = dBto_.bMf

t = f(Bt)− f(B0)−1

2

∫ t

0

(∆f)(Bs)ds P .O6Y. |O, Levy oYRj|e" wQ03f(Bt)− f(B0)−

1

2

∫ t

0

(∆f)(Bs)ds.YVG Xjt s Xj

iXit − δijt .Y, ppRjw Brown . ZW+ Mf Ok f .Y6Z0_)QSppRj SDE (6.5.1) o. (Xt)t≥0 o*, ZWRjw (6.5.1) o.. 86|X.

:EZ NAP>=8 166xz 6.5.1 L . C∞(Rn) 0JJ. (Xt)t≥0 . (Ω,F ,Ft,P) o7LUE. ,C (Xt)t≥0 s+ P OU. L- Yo., T^f ∈ C∞

b (Rn)

Mft ≡ f(Xt)− f(X0)−

∫ t

0

Lf(Xs)ds+ P .A,Y.ZW SDE (6.5.1) +U (Ω,F ,P) o_. (Xt)t≥0 . L- Yo., Q4 L h (6.5.2) 8MbMf

t = f(Xt)− f(X0)−∫ t

0

Lf(Xs)ds+ P %.Y. :5O+, ZwL(fg)− f (Lg)− g (Lf) =

n∑

k,l=1

akl∂f

∂xk∂g

∂xlGk〈Mf ,Mg〉t =

∫ t

0

L(fg)− f (Lg)− g (Lf) (Xs)ds .U_, 2B" Yo.. SDE o.. wOOxdZ.x) 6.5.1 b(·) s σ(·) . R Borel Q5X?b^3 k/, \<? λ > 0, &n λ−1 ≤ σ(·) ≤ λ. L =

1

2σ(x)2

d2

dx2+ b(x)

d

dx.T (Ω,F ,P) oo7LUE (Xt)t≥0 .Yo.: ^ f ∈ C2

b (R)

Mft = f(Xt)− f(X0)−

∫ t

0

Lf(Xs)ds.o7A,Y, , (Ω,F ,P) o (Xt)t≥0 . SDE

dXt = σ(Xt)dBt + b(Xt)dt . (6.5.3)o..

:EZ NAP>=8 167g++>Q" , O6:o" Oo additional top-

ics, ,gQexdZ. ww" (Ω,F ,P) oUE (Xt)t≥0 .O6., 2JO6 Brown B = (Bt)t≥0 &nXt = X0 +

∫ t

0

σ(Xs)dBs +

∫ t

0

b(Xs)ds . (6.5.4)" oJ.J 〈X〉t, -Tw⟨

Mf ,Mg⟩

t=

∫ t

0

(L(fg)− fLg − gLf)(Xs)ds

=

∫ t

0

(

σ2∂f

∂x

∂g

∂x

)

(Xs)ds .Z#s, T9U!X? f(x) = x (!' Mf w M), ,〈M〉t =

∫ t

0

(σ(Xs))2 ds!h Levy oYRj|ekM

Bt =

∫ t

0

1

σ(Xs)dMs.O6 Brown . v (Xt, Bt) NL~E (6.5.4), ZW (Xt)t≥0 .

(6.5.3) o.. k \1. .E dXt = rdt + αXtdBt. (\): ifZJ exp(−αBt +

12α2t).)

2. (0E) U, V .o7Y, Z := exp(V − V0 − 12〈V 〉). " : E

dX = dU +XdV kvO. X = Z(X0 + Z−1.(U − 〈U, V 〉)).

g!go[1] Bauer, H., Probability Theory and Elements of Measure Theory,

Academic Press, 1981

[2] Billingsley, P., Probability and Measure, John Wiley & Sons, 1986

[3] Chung, K.L., A Course in Probability Theory, Academic Press, New

York, 1974

[4] Chung, K.L., Williams, R.J., Introduction to Stochastic Integration,

Birkhauser Boston, Inc., 1983

[5] Doob, J.L., Stochastic Processes, Wiley, New York, 1953

[6] Durrett, R., Brownian motion and Martingale in Analysis, Wadsworth

Inc., 1985

[7] Dynkin, E.B., Theory of Markov Processes, Prentice-Hall, Inc., Engle-

wood Cliffs, New Jersey, 1961

[8] Feller, W., An Introduction to Probability Theory and its Applica-

tions, Vol. 1(1959), 2(1970), Wiley & Son

[9] Halmos, P.R., Measure Theory, Springer-Verlag, 1974

[10] Ikeda, N., Watanabe, S., Stochastic Differential Equations and Dif-

fusion Processes, North-Holland, 1981

[11] Ito, K., Mckean Jr., H. P., Diffusion Processes and Their Sample paths,

Sringer-Verlag, 1965

[12] Novikov, A.A., On moment inequalities and identities for stochastic integrals,

Proc. second Japan-USSR Symp. Prob. Theor., Lecture Notes in Math., 330,

333-339, Springer-Verlag, Berlin 1973

168

7DQS 169

[13] Parthasarathy, K.R., Probability measures on Metric spaces, Academic

Press, New York, 1967

[14] Revuz, D., Yor, M., Continuous Martingales and Brownian Motion,

Springer, 1991

[15] Wiener, N., Differential space”, J. Math. Phys. 2, 132-174 (1924)

[16] Yan, J.A., Criteres d’integrabilite uniforme des martingales exponentielles,

Acta. Math. Sinica 23, 311-318 (1980)

[17] Yosida, K., Functional Analysis, Springer-Verlag, 1980

[18] rX[, *!,&, 1,2, 8a:M , 1996

[19] q/, 0%& , &ga:M , 1988

[20] f., , *!#, 2p&wM , 1987

[21] ?, &"1, P:M , 1998

[22] ^, q/, ?, 52*!., P:M , 1995

[23] _0, 1y, *! , &ga:M , 2005

Z|Borel ?, 2

Brown , 63

Brown oYRj, 1318UE, 83)Q2,, 11u_r>(, 8v,, 11Y, 117Y., 117

Cameron-Martin-Girsanov>(, 1575, 45U, 4D~5U, 10

Dambis, Dubins-Schwarz |e, 133

Doob Z)p(, 35

Doob ., 48

Doob k/ *|e, 30

Doob k/ *|e, o7!, 44

Doob Y)p(, 33

Doob Ya)p(, o7!, 47fz4p|e, 7kÆuUE, 60xUE, 58 *UE, 42xU, 58ÆX8UE, 78, 102

Follmer \e, 51

Fatou \e, 8

Fubini |e, 10,~>(, 122*vO0, 152ze, 72

Girsanov |e, 136

Gronwall )p(, 159RmvO0, 152+5, 4+U, 11

Ito >(, 122

Ito -E, 151

Ito pG, 91

Jensen )p(, 22A,lUE, 42A,!, 148fX?, 6Ko7, 9~, 6^Brown , 154eQQ, 8eQQ4p, 14

Kolmogorov 0-1 , 23

Kolmogorov )p(, o7!, 47

Kolmogorov |e, 58

Kolmogorov _a?|, 37Q5U, 2

170

OU 171Q5e, 6

Levy UE, 78

Levy )?, 78

Levy 5, 78

Levy-Khinchin >(, 78

Lebesgue V14p|e, 8

λ- , 3

Lr-4p, 14, 1

Novikov a, 129

Ornstein-Uhlenbeck UE, 154

π- , 3_., 152

Radon 5, 4

Radon-Nikodym k?, 9

Riesz ., 48, 53y\u, 63., 152

σ- ?, 1

σ-k5, 4 Z, 35LUE, 40LtE, 150L)?, 127#, 13?:Ps, 13

Tanaka >(, 147a?:Ps, 19 !, 38

Wald Y, 47p5U, 45, 6, 13okx*O, 56<6Q5, 42Y4p|e, 51

Yamada-Watanabe a, 158

Yamada-WatanabevO0|e, 152Y, Y, Y, 25IRm, 40Y")|e, 140P+4p, 14k/8UE, 101k/4p|e, 9kx*, 54kx*O, 54O/Q~0, 15ÆUE, 101)?Y, 127Y!0, 51

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