0 s s ' d S ˜ T 0 T s s = E [˜ H ]

( t) HT

Vt= tSt

Vt+dt Vt= t(St+dt St) Vt=V0+Rt

0 sdSs

Vn =V0+ ( :S)n

Rt 0 sdSs

HT HT = V0+RT 0 0dSs t=^@H_@S

HT

V0 = E

"

H˜T

Z T 0

'0dS˜s

#

= E[ ˜HT] E"Z T 0

'sdS˜s

#

RT 0 'sdS˜s

(⌦,P,F)

(Xt)t2R⁺ Xt

X

X ⌦7!X(!) R⁺ !R^d X ⌦!F(R⁺,R^d)

! 7!X(!) R⁺!R^d t 7!Xt(!)

! R⁺

X

! Xt(!)

(F^t)t>0 R⁺ F^t⇢F^s8t6s

X X

X Ft^X = (Xs:s6t)

= X_s¹(A) 8A2B(R^d), s6t

F⁰ F^t=T

s>tF^s

X F t Xt F^t

F^t Xt

X

X Y 8t, Xt=Yt

X=Y 8n 8t1<· · ·< tn (Xt1,· · ·, Xtn)= (Y^L t1,· · ·, Ytn) Xt L

=Yt Xt+✏ L

=Yt+✏ ! t!Xt

t!Yt

F

M

M F

8t M E[Mt]<+1 8s6t E[Mt| F^s] =Ms

Mt2R

M E[Mt] =E[M0]

X Mt=E[X | F^t]

X 8t1 < t2 < t3 Xt3

Xt2qXt2 Xt1 F Mt=Xt E[Xt]

M F

E[Mt| F^s] = E[Xt| F^s] E[Xt]

= Ms Xs+E[Xs] +E[Xt| F^s] E[Xt]

= Ms+E[Xt Xs| F^s] E[Xt Xs]

= Ms+E[Xt Xs] E[Xt Xs]

= Ms

M E[M_t²]<+1 8t

M 2L² E[(Mt Ms)²| F^s] =E[M_t² M_s²| F] 8s6t M 2L²

E[(Mt Ms)²| F^s] = E[M_t²| F^s] 2E[MtMs| F^s] +E[M_s²| F^s]

= E[M_t²| F^s] M_s²

t1< t2< t3 E[(Mt3 Mt2)(Mt2 Mt1)] = 0 E[(Mt3 Mt2)(Mt2 Mt1)] =E[E[Mt3 Mt2 | F^t2]

| {z }

=0 M

⇥(Mt2 Mt1)]

M²([0, T])

M 2M²([0, T]) E

 sup

06t6T

(Mt)² 64E⇥

|MT|²⇤

⇣M^KT

N

⌘

K2[0,N]

M 2M²([0, T]) Mt=Rt 0'sds E[Mt] = E[M0] = 0

E[M_t²] = E"✓Z t 0

'sds

◆²#

= E 2 4

n 1

X

k=0

Z ^(k+1)t_n

kt n

'sds

!²3 5

= E 2 4

n 1

X

k=0

M^(k+1)t

n M^kt

n

!²3 5

= E

"_{n 1} X

k=0

⇣M^(k+1)t

n M^kt

n

⌘2#

= E 2 4

Xn k=0

Z ^(k+1)t_n

kt n

'sds

!²3 5

(Rb

af)²6(b a)Rb af² Pn

k=1 b a

n f(xk) ²6^{(b a)}_n ²P(f(x))² E[M_t²] 6 E

" _n X

k=0

t n

Z ^(k+1)t_n

kt n

'²_sds

#

6 t

nE

Z t

0

'²_sds !0 n!+1 E[M_t²] = 0 Mt= 0

⌧ R⁺ {⌧ 6t}2F^t 8t2R⁺ F⌧

⌧ F^⌧ = ({A⇢⌦, A\{⌧6t}2F^t 8t})

⌧ F^⌧

⌧ ⌫ ⌧^⌫

⌧6⌫ F^⌧ ⇢F^⌫

X F X⌧ F^⌧

X

a!lim+1(sup

t E[|Xt|1_|Xt|>a])<+1

Y 8t Xt6Y X

M ⌫6⌧

E[M⌧|F^⌫] =M⌫

M ⌫ 6⌧ < K K

M F M

E[M⌧] =E[M0] 8⌧ E[Mt] =E[M0] 8t M

M M0 = 0 X

Xt=Rt 0Mudu

>0 E[Xt] ^{F ubini}= Z t

0 E[Mu]du

= 0 ( Mu=M0= 0)

= X0

> s E[Xt|F^s] = E 2 66 64

Z t s

Mudu+ Z s

0

Mudu

| {z }

Fs mes

|F^s 3 77 75

= Xs+E

Z t

s

Mudu|F^s

= Xs+ Z t

s E[Mu|F^s]du

= Xs+ (t s)Ms

| {z }

6

=0

M

(⌧n)n2N +1 (Mt^⌧n)_t2R⁺

R

X

8t1<· · ·< tn,F E[F(Xs+t1,· · ·, Xs+tn)|Fs^X] =E[F(Xs+t1,· · ·, Xs+tn)| (Xs)]

X

8⌫ <⌧ f E[f(X⌧)|F⌫^X] =E[f(X⌧)|X⌫]

V˜n = E[f( ˜SN)|Fⁿ]

= E[f(SN)|Sn]

= v(n, Sn) X

X

⌫ 6⌧

E[f(X⌧)|F⌫^X] = E[f(X⌧ X⌫+X⌫|F⌫^X]

= (X⌫) X⌧ X⌫ F⌫^X

(x) =E[f(X⌧ X⌫+x)]

Nt=

0 t

Nt 0 t

Nt 0 t

Nt 0 t

Nt 0

t Nt

0 t

N Nt=

+1

X

n=1

1_{⌧n6t}=

+1

X

n=1

1_{^Pn_{i=0 i6}t}

i=⌧i ⌧i 1 ⇠E( ) (⌧0= 0)

X E( ) f(x) = e ^x1_R⁺(x)

X⇠E( ) P(X>t+s|X >t) =P(X>s)

p = P(X >t+s|X>t) = P(X >t+s;X>t)

P(X >t) = P(X >t+s) P(X>t)

= R+1

t+s e ^xdx R+1

t e ^xdx = e ^(t+s)

e ^t =e ^s=P(X >s)

X1,· · ·, Xn ⇠E( ) Sn=Pn

i=1Xi Sn⇠ (n, ) f(x) =e ^x _(n^nxn_1)!¹1_R⁺(x)

F(x) =P(X6x) (u) =E[e^iuX]

G(u) =E[e^uX]

Sn

G(u) = E⇥ e^uSn⇤

u <

= E

" _n Y

i=1

e^uXi

#

= E⇥

e^uX1⇤ ⁿ

Xi

=

✓Z +1

0

e^{(u )x}dx

◆ⁿ

=

✓ u

◆n

(n, ) G (u) =

Z

R⁺e^{(u )x}

nx^{n 1} (n 1)!dx

| {z }

In

06u6

= Z

R⁺

nx^{n 1} (n 1)!d

✓e^{(u )x} u

◆

=

n

u Z

R⁺e^{(u )x} x^{n 2} (n 2)!dx

= u In 1

=

✓ u

◆n

(N) N0= 0

8t, s>0

( Nt+s Nt q Ft^N

Nt+s Nt

Loi= Ns N0

Nt ⇠P( ) 8n2N P(Nt=n) = ⁽_n!^t)ne ^t

) N

N0=P+1

k=11_{⌧k60}= 0

P(Nt+s Nt = K, Nt = n) =^? P(Nt+s Nt = K)·P(Nt = n) Nt+s NtqNt

P(Nt+s Nt=K, Nt=n) = P(Nt+s=K+n, Nt=n)

= P

K+nX

i=1

i6t+s <

K+n+1X

i=1 i,

Xn i=1

i6t <

n+1X

i=1 i

!

= E[f( 1,· · ·, K+n+1)]

= Z

(R⁺)^K+n

K+n+1e ^{PK+n+1i=1 xi}f(x1,· · ·, xK+n+1)dx1· · ·xK+n+1

= · · · yj=

Xj i=1

xi

= P(Nt+s Nt=K)·P(Nt=n)

n2N P(Nt=n) = P(⌧n 6t ⌧n+1> t)

= P(⌧n 6t) P(⌧n+16t)

= Z t

0

e ^x

n

(n 1)!x^{n 1}dx Z t

0

e ^x

n+1

n! xⁿdx

= · · ·

= e ^t( t)ⁿ n!

(() N i)ii) iii) ⌧n= inf{s>0|Ns=n}

⌧n⇠ (n, )

⌧1= 1

P(⌧1> t+s) = P(Nt+s= 0)

= P(Nt+s= 0, Nt= 0)

= P(Nt+s Nt= 0, Nt= 0)

= P(Nt+s Nt= 0)·P(Nt= 0)

= P(Ns= 0)·P(Nt= 0)

= P(⌧1> s)·P(⌧1> t)

F(t) =P(⌧1> t))F(t+s) =F(t)·F(s) (⌧1)

⌧n+1 ⌧n ⇠E( )

⌧n+1 = inf{s>⌧n, Ns=n+ 1}

= inf{s>⌧n, Ns N⌧n= 1} s=⌧n+t

= inf{t>0, N⌧n+t N⌧n= 1}+⌧n

Nˆt = N⌧n+t N⌧n Nˆt i) ii) iii) F⌧^Nn

⌧n+1= inf{t>0,Nˆt= 1}

| {z }+⌧n

N (Nt t)_t>0

E[Nt+s (t+s)|Ft^N] = E[Nt+s Nt

| {z }

qFt

+ Nt

|{z}

F

|Ft^N] (t+s)

= Nt t+ E[Nt+s Nt] s

| {z }

E[Ns] s=0 Ns⇠P( s)

N0= 0

Nt⇠P⇣Rt 0 sds⌘

(Yi)i2N^⇤

( i) Nt=P+1

i=1Yi1_{⌧i6t}

X ⇠ N(m, ²)

f(x) = 1

p2⇡e ^{12(x m)2} m=E[X] ²=E(X m)²

E[e^tX] =e^{tm+t2 22}

E[e^tx] =R

R

e^{tx (x2 2m)2} p2⇡ dx

(Xi)i2N E[X1] =m V ar(X1) = ² p1

N PN

i=1(Xi m) ^Loi!N(0, ²)

(X1,· · ·, Xn) 8a1,· · · , an2Rⁿ Pn i=1aiXi

m= 0 B@

E[X1] E[Xn]

1 CA

C = (Cov(Xi, Xj))16i6j6n

(Y^⇤CY >0) C X

C X

8x2Rⁿ, f(x) = 1

pdet(C)⇥2⇡ⁿexp

 1

2(x m)^tC ¹(x m)

XiqXj Cov(Xi, Xj) = 0 Xi

↵i,j2R (Xi ↵ij·Xj)qXi A2Mn AX

Y qZ Y Z (Y, Z)

()) Cov(X, Y) =E[XY] E[X]E[Y] = 0

(() Xˆ = (Xi, Xj) Cˆ=

✓ ₂

i 0

0 _j²

◆

2

i = 0 _j²= 0

Xi Xi(!) =mi 8! XiqXj 2

i · j²6= 0

Xˆ :f(xi, xj) = 1

i j ·2⇡exp 1 2

(xi m)²

2i

1 2

(xj m)²

2j

!

= g(xi)·h(xj) XiqXj

8↵2 R,(Xi ↵Xj, Xj) ↵ Cov(Xi ↵Xj, Xj) = 0 Cov(Xi, Xj) ↵V ar(Xj) = 0

Xj ↵ ↵=Cov(Xi, Xj)

V ar(Xj)

X, Y (a, b)2R² aX+

bY ⇠N

G(t) = Eh

e^t(aX+bY)i

= E[e^taX]·E[e^tbY]

= exp

✓

t·amX+1

2t²⇥a²· X² +t·bmY +1

2t²⇥b²· Y²

◆

= exp

✓

t(amX+bmY) +1

2t²(a²· ²X+b²· ²Y)

◆

aX+bY ⇠N (amX+bmY), a^{2 2}_X+b^{2 2}_Y X

" { 1,1} ¹2

Y ="X

E[e^tY] = E[e^t"X]

= E[e^tX1_{"=1}] +E[e ^tX1_{"= 1}]

= 1 2

⇥E[e^tX] +E[e ^tX]⇤

= 1 2

⇣e^mt+t

2 2

2 +e ^mt+t

2 2 2

⌘

Y ⇠N(0, ²) Z=X+Y E[e^t⌧] = E[et(1+")X]

= 1

2 E[e^2tX] + 1 )Z 6⇠N

(X, Y) X

Y Y

(E[X|Y] =aY +b)

X 8n2N^⇤,8t16· · ·6tn2R⁺

(Xt1,· · · , Xtn)

Xt t

t!Xt

q(s, x, y) =P(Xt+s2[y, y+dy]|Xt=x)

f(s, x) =E[g(Xt+s)|Xt=x]

f

8<

:

@f

@t = 1

2

@²f

@x² f(0, x) = g(x)

B B

B 8t, s (Bt+s Bt)qFt^B

B Bt+s Bt=L Bs B0

B Bt B0 ⇠

N(mt, ²t)

Bt B0 =

n 1

X

k=0

Btk+1 Btk tk =kt n

=L n 1

X

k=0

Xk Xk B¹

n B0

=

n 1

X

k=0

X_k⁽ⁿ⁾ Xk L

⇠B¹

n B0 L

= 1

pn(B1 B0)

= 1

pn

nX1 k=0

Yk Yk=L B1 B0

8t2R^⇤, E[Bt] =tE[B1] n2N^⇤, E[Bnt B0] =E⇥Pn

k=1Bkt B_{(k 1)t}⇤

=nE[Bt B0] E[Bnt] =nE[Bt]

(p, q)2N⇥N^⇤ pE[Bt] =E[Bpt] =E[B_q⇥^p_qt] =qE[B^p_qt] 8r2Q⁺, E[Brt] =rE[Bt]

s2R⁺ (rn)n2N2(Q⁺)^N lim

n!+1rn=s E[Bs] = E[Brn] +E[Bs Brn]

= rnE[B1]

| {z }

n!+1! sE[B1]

+E[Bs Brn]

| {z }

n!+1! 0

V ar(Bt B0) = ²t n2N V ar(Bnt B0) = V ar

n 1

X

k=0

B(k+1)t Bkt

!

=

nX1 k=0

V ar(B(k+1)t Bkt)

= nV ar(Bt B0)

(p, q)2N⇥N^⇤ pV ar(Bt B0) = V ar(Bpt B0)

= V ar(B_q⇥^p_qt B0)

= qV ar(B^p_qt B0) t2R⁺ (rn)2Q^N rn!t r6t

V ar(Bt B0) = V ar(Bt Brn+Brn B0)

= V ar(Bt Brn) +rnV ar(B1 B0)

> ²t n!+1

rn>t, V ar(Bt B0) 6 V ar(Brn B0) 6 rn 2

t7!V ar(Bt B0)

k >0 V ar(Bt+k B0) = E[Bt+k B0 (m(t+k) E[B0]]²

= V ar(Bt+k Bt+Bt B0)

= V ar(Bt+k Bt) +V ar(Bt B0)

> V ar(Bt B0) Bt⇠N(mt, ²t)

L(x) = Eh

e^{x(Bt B0)}i

= E

"

exp x

nX1 k=0

Btk+1 Btk

!#

tk= kt n

=

nY1 k=0

E⇥

exp x(Btk+1 Btk) ⇤

= ⇣ Eh

exp⇣ x(B^t

n B0)⌘i⌘n

= E



1 +x(B^t

n B0) +x² 2 (B^t

n B0)²+

✓1 n

◆ ⁿ

=

✓

1 +xmt

n+x^{2 2} 2

t n+

✓1 n

◆◆ⁿ

= exp

 nln

✓ 1 +

✓

xmt+x^{2 2}t 2

◆1 n+

✓1 n

◆◆

= exp



xmt+x^{2 2}t 2 + (1)

N(mt, ²t) B

B0= 0 m=E[B1] = 0 ²=V ar(B1) = 1

B

B0= 0 t7!Bt(!) B

B Bt⇠N(0, t)

B Wt=W0+mt+ Bt

mt

E[f(BT)]

E

 f

✓1 n

Pn i=1Bti

◆

E

 f

✓1 T

RT 0 Budu

◆

)

Bt =

n 1

X

k=0

Btk+1 Btk

= X

Xk Xk N

✓ 0, t

n

◆

X1,· · ·, Xn

P(Xk= 1) =P(Xk = 1) = ¹₂ Sn=Pn k=1Xk 1

nSn !N(0,1) =B1 U^k

n =p¹

nSk (U^k

n)_k2[0,n]

U^(k+1)

n U^k

n =p¹nXk+1 F^{k Xpn}n

Un(t) =UK(t)

n +n

✓

t K(t) n

◆ ✓UK(t)+1

n

UK(t)

n

◆

K(t) =bntc

8p2N,8t1,· · · , tp2[0,1], (Un(t1),· · ·, Un(tp)) ^L! Bt1,· · · , Btp

Un(t) ^L! Bt Un(t)

n +1

Bt= p8

⇡

+1

X

n=1

sin(nt) n gn

gn N(0,1)

B 8<

:

(i) B0= 0 (ii) B

(iii) B 8t, s, Cov(Bt, Bs) =t^s ()) B

n2N t0= 0< tP1n<· · ·< tn2[0,+1] a1,· · ·, an2R

i=1aiBti

Xn i=1

ai

Xi j=1

Btj Btj 1 = Xn j=1

0

@ Xn i=j

ai

1

A Btj Btj 1

Btj Btj 1

E[Bt] = 0 s6t Cov(Bt, Bs) = E[Bt, Bs]

= E[(Bt Bs)Bs] +E[B²_s]

= E[Bt Bs]E[Bs] +s

= s

(() B Cov(Bt, Bs) =t^s

B0= 0

s6t Bt Bs

V ar(Bt Bs) = E(Bt Bs)²

= V ar(Bt) +V ar(Bs) 2Cov(Bt, Bs)

= t+s 2s

= t s Bt Bs⇠N(0, t s)⇠Bt s B0

u6s6t Bt Bs

Bu

(Bu, Bt Bs)

Cov(Bu, Bt Bs) = Cov(Bu, Bt) Cov(Bu, Bs)

= u u u6s u6t

= 0 Bt Bsq (Bu;u6s)

V ar(Bt Bs) =t s)E(Bt Bs)²6t s B

B B˜t= Bt

B¯t=cB^t

c2

Bˆt=tB¹

t

B˜0= B0= 0Pn B

i=1aiB˜t

Pn

i=1aiB˜t= Pn

i=1aiBt B

E[ ˜Bt] = E[Bt] = 0 B s6t

Cov( ˜Bt,B˜s) = Cov( Bt, Bs)

= Cov(Bt, Bs)

= s^t

= s

B¯0=cB0= 0 B Pn

i=1aiB¯t

Pn

i=1aiB¯t=cPn i=1aiB ^t

c2 B

E[ ¯Bt] =cE[B^t

c2] = 0 B s6t

Cov( ¯Bs,B¯t) = c²Cov(B^s

c2, B^t

c2)

= c²

✓s c² ^ t

c²

◆

= c²· s c²

= s Bˆ0P= 0n

i=1aiBˆt

Pn

i=1aiBˆt=Pn

i=1aitiB¹

t B

E[ ˆBt] =E[tB¹

t] = 0 B s6t

Cov( ˆBs,B¯t) = Cov(sB¹

s, tB¹

t)

= stCov(B¹

s, B¹

t)

= st

✓1 s^1

t

◆

= st⇥1

= s t

B B

(B_t² t)_t>0 8 2R

✓ exp

✓ Bt

2

2 t

◆◆

t>0

E[|B|]<+1 B s6t

E[Bt|F^s] = E[Bt Bs+Bs|F^s]

= E[Bt Bs|F^s] +E[Bs|F^s]

= E[Bt Bs] +Bs

= E[Bt s B0] +Bs

= Bs

E[|B²_t t|] 6 E[|B_t²|] +E[|t|] 6 V ar(Bt) +t 6 2t

< +1 s6t

E[B_t² t|F^s] = E[(Bt Bs+Bs)² t|F^s]

= E[(Bt Bs)²|F^s] +E[B_s²|F^s] + 2E[Bs(Bt Bs)|F] t

= E[(Bt Bs)²] +B_s²+ 2BsE[Bt Bs] t

= t s+B_s² t

= B_s² s s6t E

 exp

✓

(Bt Bs+Bs)

2

2 t

◆

|F^s = exp

✓ 2

2 t

◆

·E[exp( (Bt Bs+Bs))|F^s]

= exp

✓ 2

2 t

◆

·E[exp( Bs) exp( (Bt Bs))|F^s]

= exp

✓ 2

2 t

◆

·exp( Bs)E[exp( (Bt Bs))]

= exp

✓ 2

2 t

◆

·exp( Bs) exp

✓

2t s 2

◆

= exp

✓ Bs

2s 2

◆

X X0= 0

X

X (X_t² t)t>0

8 2R,

✓ exp

✓ Xt

2t 2

◆◆

t>0

(2))(1) (1))(2) (1))(3)

(3))(1) (3)

Xt XsqF^s N(0, t s)

E[exp( (Xt Xs))|F^s] = e ^XsE[e ^Xt|F^s]

= e ^Xs+

2t 2 ·Eh

e ^Xt

2t 2 |F^si

= e ^Xs+

2t 2 ·e ^Xs

2s 2

= e

2 2(t s)

| {z }

qFs

Xt XsqF^s Xt Xs⇠N(0, t s)

B B

8⌧ 8f E[f(Bt+⌧)|F⌧^B] =E[f(Bt+⌧)|B⌧]

Wt=Bt+⌧ B⌧ F^⌧

(s, t)2R⁺

E[f(Bt+s)|F^s] = E[f(Bt+s Bs+Bs)|F^s]

= (t, Bs)

= E[f(Bt+s)|Bs] (t, x) =E[f(Bt+s Bs+x)] = 1

p2⇡t Z

Rf(y+x)e ^y

2 2tdy

a2R Ta= inf{s>0, Bs=a}

Ta a B

Ta

Ta

8x2R⁺, E[e ^xTa] =e ^|a|p2x P(Ta<+1) = 1 E[Ta] = +1 Ta f(t) = |a|

p2⇡t³exp⇣

a² 2t

⌘1_{t>0}

a>0 {Ta6t} =

⇢ sup

s6t

Bs>a

= {8✏>0,9s2[0, t], Bs>a ✏}

= {8✏2Q^+⇤,9s2[0, t]\Q, Bs> a ✏}

= \

✏2Q^+⇤

[

s2[0,t]\Q

{Bs> a ✏}

| {z }

Ft mes

2F^t

E^t= exp(yBt y²t 2 )

⇣E[exp(yBTa

y²

2Ta)]= exp(yB^? 0 y²

2 ⇥0) = 1⌘

⌧ =Ta^t

E[e^{yB⌧ y22⌧}] = 1 1 =Eh

e^{yB⌧ y}

2

2⌧1_{Ta<+1}+E^⌧1_{Ta=+1}i

t!lim+1Eh e^{yB⌧ y}

2

2⌧1_{Ta<+1}

i

⇣e^{yB⌧ y}

2

2⌧6e^ya⌘

t!+1lim E[E^⌧1_{Ta<+1}] =Eh

e^{yBTa y22Ta}1_{Ta<+1}i

=e^yaEh

e ^y22Ta1_{Ta<+1}i 1 =e^yaEh

e ^y

2Ta

2 1_{Ta<+1}i + lim

t!+1E[E^⌧1_{Ta=+1}] 0 lim

t!+1E[E^⌧1_{Ta=+1}]6e^yaE[ lim

t!+1(e ^y

2

2⌧1_{Ta=+1})]60 8y >0 1 =e^yaEh

e ^y

2

2Ta1_{Ta<+1}

i

y!0 1 =P(Ta<+1) 1 =e^yaE[exp( ^y₂²Ta] x= ^y₂² E[e ^xTa] =e ^ap2x

X f F( ) =E[e ^X] =R

Re ^xf(x)dx f(x) =R

Re^{(↵+i )x}F(↵+i )d ↵

X Xt=

⇢ Bt t6Ta

2a Bt t > Ta

X a

X

X B

X (X_t² t)t>0 s < t E[Xt|F^s] = E[Xt|F^s]1_{Ta6s}+E[Xt|F^s]1_{Ta>s}

= E[(2a Bt)|F^s]1_{Ta6s}+E[Xt1_{Ta6t}+Xt1_{Ta>t}|F^s]1_{Ta>s}

= (2a Bs)1_{Ta6s}+E[(2a Bt)1_{Ta6t}+Bt1_{Ta>t}|F^s]1_{Ta>s}

= (2a Bs)1_{Ta6s}+E[E[(2a Bt)|F^Ta]1_{Ta6t}+E[Bt|F^Ta]1_{Ta>t}|F^s]1_{Ta>s}

= (2a Bs)1_{Ta6s}+E[(2a BTa)1_{Ta6t}+Bt1_{Ta>t}|F^s]1_{Ta>s}

= (2a Bs)1_{Ta6s}+Bs1_{Ta>s}+E[a1_{Ta6t} Bt1_{Ta6t}|F^s]1_{Ta>s}

= Xs 1_{Ta>s}E[(Bt a)1_{Ta6t}|F^s]

= Xs 1_{Ta>s}E[E[Bt BTa|F^Ta]1_{Ta6t}

| {z }

=0 Bt B_TaqFTa

|F^s]

(St, Bt) St= sup

06s6t

Bs

Ta

(St, Bt) F(a, b) = P(St> b, Bt< a)

=

Z +1 b

Z

1

f(x, y)dy dx f(b, a) = _b,a² F(a, b)

F(a, b) =P(Tb< t, Bt< a) T_b^B=T_b^Xb X^b b

F(a, b) = P(Tb< t, sb Bt>2b a)

= P T_b^B < t, X_t^b>2b a)

= P(T_b^X < t, Xt>2b a) ( T_b^B =T_b^X)

= P(T_b^X < t, Bt>2b a)

= P(Bt>2b a)

= Z +1

2b a

e ^x2t2 p2⇡tdx (St, Bt)

f(b, a) = 1_{b>0}1_{a<b} 2 a,b

Z a 2b

1

e ^x2t2 p2⇡tdx

= 1_{b>0}1_{a<b}⇥ 1 p2⇡t

d e^(a2t2b)2 db

= 1_{b>0}1_{a<b}⇥2(2b a) p2⇡t³ exp

 (a 2b)² 2t St

P(St> b) = P(St> b, Bt2R)

=

Z +1

b

Z

Rf(x, y)dy dx St

g(b) = Z

Rf(b, y)dy Tb

P(Tb6t) =P(St> b) hTb(t) = @

@t(P(St> b))

= b

p2⇡t³exp

✓ b² 2t

◆

1{t>0}

! R t!Bt(!) g(t)df(t) =R

g(t)f⁰(t)dt lim sup

t!+1

Bt

pt = lim

t!+1

✓ sup

t6s

Bs

ps

◆

= +1 lim inf

t!+1

Bt

pt = 1 lim sup

t!0

Bt

pt = 1 lim inf

t!0

Bt

pt = +1

R= lim sup

t!+1

Bt

pT 2R[{+1}

s2R⁺ R= lim sup

t!+1

Bt Bs

pt F^s= (Bu, u6s)

R F1= (Bu u>0) R F1

RqR R R[{+1}

R 0 =P

✓Bt

pt > R+ 1

◆

=P(B1> R+ 1)>0 R= +1

8 2R,P(9(tn)2(R⁺)^N, Btn =x) = 1 lim sup

+1 = +1 lim inf

+1 = 1 B

B

+1= lim sup

t!0⁺ Bt

pt = lim sup

t!0⁺ BtpB0

T 6lim sup

t!0⁺ Bt B0

t

B 0

B s2R^+⇤ Wt=Bs+t Bs

lim sup

t!0⁺

⇣_B

s+t Bs

t

⌘>lim sup

t!0⁺ Wt

t = +1

I B

n2N Z_tⁿ =P2ⁿ

j=1 Btj Btj 1

2

E(Z_tⁿ t)²_n!!₊₁0 lim

n!+1Z_tⁿ=t

L²

E(Z_tⁿ t)² = E 0

@

2ⁿ

X

j=1

(Btj Btj 1)² t 1 A

2

= V ar(Z_tⁿ) E[Z_tⁿ] =t

=

2ⁿ

X

j=1

V ar[(Btj Btj 1)²]

= 2ⁿV ar⇣

(B₂^t_n)²⌘

= 2ⁿ⇥ 2t² 2²ⁿ

= t²

2^{n 1} _n!+1! 0 EhP+1

n=1(Z_tⁿ t)²i

=t²P(1

2)^{n 1}<+1

(Z_tⁿ t)² (Z_tⁿ t)²!0

(St) (Vt)

dV˜t='tdS˜tV˜t=V0+Rt 0'sdS˜s

Rt

0f(s)ds= lim

n!1 t n

Pn 1

k=0f⇣_(k+1)t

n

⌘

Rt

0f(s)dBs= lim

n!1

Pn 1

k=0f⇣_(k+1)t

n

⌘ hB^(k+1)t

n B^kt

n

i

nlim!1

Pn 1 k=0

⇣B^(k+1)t

n B^(k+1)t

n

⌘

L= lim

n!1

Pn 1

k=0|(B^(k+1)t

n B^(k+1)t

n )| L= +1 t= lim

n!1

P2ⁿ

k=0(Btk+1 Btk)²6L⇥ lim

n!1max

k62ⁿ|Btk+1 Btk|

| {z }

=0

t60

Rt

0cdBs=c(Bt B0)

E ={f, 90 =t0< ... < tn= +1, (f0,· · ·, fn 1)2Rⁿ, f(x) =Pn 1

k=0fk1_[tk,tk+1[(x) R+1

0 f(x)dx=Pn 2

k=0fk(tk+1 tk) (fn= 0) f 2E I(f) =R+1

0 f(s)dBs=Pn 1

k=0fk(Btk+1 Btk)

I(f) f

I(f) B

V ar(I(f)) = E

nX1 k=0

fk(Btk+1 Btk)

!²

=

nX1 k=0

f_k²(tk+1 tk)

= Z +1

0

f(s)²ds

k I(f)kL²(⌦)=k f kL²(R⁺) L²(⌦) = {Xv.a.r., E[X²]<+1} L²(R⁺) = {f : R⁺ ! R, R+1

0 f² < +1} k X kL²(⌦)= E[X²]

1

2 k f kL²(R⁺)=

⇣R+1 0 f²⌘¹2

f, g2E E[I(f)I(g)] =R+1

0 f(s)g(s)ds L²(R⁺) f 2L²(R⁺) I(f) =R+1

0 f(s)dBs= lim

n!+1

R+1

0 fn(s)dBs

fn lim

n!+1kfn f kL²(R⁺)= 0 E L²(R⁺) k.k^L2(R⁺)

(fn)2E^N,kf fnkL²(R⁺)!0 In=I(fn)

In L²(⌦) L²(⌦)

kIn Ipk²L²(⌦)=kI(fn fp)k²L²(⌦)=kfn fpkL²(R⁺)

(fn) )In

(gn) (fn)2E !f L²(R⁺) limI(gn) = limI(fn)

f 2L²_loc(R⁺) ⇣

8T 2R⁺, RT

0 f²(s)ds <+1⌘ f [0, t] Rt

0f(s)dBs=R+1

0 f(s)1_{s6t}

| {z }

2L²(R⁺)

dBs

f 2L²(R⁺) Z2L²_loc(⌦) Z =R+1

0 f(s)dBs E[ZBt] =Rt

0f(s)ds 8t >0

()) t>0 E

 Bt

Z +1

0

f(s)dBs = E

Z +1

0

f(s)dBs

Z +1

0

1_{s6t}dBs

= Z +1

0

f(s)1_{s6t}ds

= Z t

0

f(s)ds f, g2L²_loc(R⁺)

E[Rt

0g(s)dBsRu

0 f(s)dBs] =Rt^u

0 g(s)f(s)ds (Xt=Rt

0f(s)dBs)t>0

(Xt) Cov(Xt, Xs) =Rt^s

0 f²(u)du

X (X_t² Rt

0f²(s)ds)t>0

X

f¯(s) =f(s)1_{s6t}2L²(R⁺) ( ¯fn)_n2N2E^N kf¯n f¯k^L2(R⁺)!0

E

✓Z t

0

f(s)dBs

◆²

= V ar(I( ¯f))

= V ar( lim

n!1I( ¯fn))

= lim

n!+1V ar(I( ¯fn))

= lim

n!+1kf¯nk^L2(R⁺)

= kf¯kL²(R⁺)

= kf kL²([0,t])

ab=¹₄((a+b)² (a b)²)

t1<· · ·< tn, a1,· · ·, an2R Y =PaiXti

X_t^(k)=Rt

0fk(s)dBs (fk)2E^N fk !f X^(k)

Y^(k) = PaiX_t^(k)_i = PaiPf_k^(j) Btj+1 Btj

Y = lim

k!+1Y^(k) L²(⌦) (Xt)

(u6t)

Cov(Xt+s Xt, Xu) = Cov(Xt+s, Xu) Cov(Xt, Xu)

= Z u

0

f²(v)dv Z

o

f²(v)dv

= 0

X^(k) X

Rt

0f(s)dBs=f(t)Bt Rt

0Bsf⁰(s)ds f 2L²_loc(R⁺)\C¹ E(X Y)²= 0,X =Y p.s.

ST = µt+ Bt+S0

| {z }

P(St60)>0 S

St+s St

St qF^t St+s St

St

✓Loi

= Ss S0

S0

◆

Xt = ln(St) ( St > 0) X

S 9b2R, 2

R⁺, St=S0exph

(b ₂²)t+ Bt

i

(e ^btSt)t>0

s < t St=Ssexph

(b ₂²)(t s) + (Bt Bs)i

=f(Ss, Bt Bs

| {z }

qSs

) S E[g(St)|F^s] =E[g·f(Ss, Bt Bs)|F^s] = (Ss)

(x) =E[g f(x, Bt Bs)] =· · ·=E[g(St)|Ss] E[St] =S0e^bt

V ar(St) =Eh

S₀²e^2bt(e ^{22t+ Bt} 1)²i

=· · ·=S₀²e^2bt(e ^2t 1)

E[f(St)|F^s] = E

 f

✓ Ssexp

✓ (b

2

2 )(t s) + (Bt Bs)

◆◆

|F^s

= E

 f

✓ xexp(b

2

2 )(t s) + (Bt Bs)

◆

|x=Ss

= E

 f

✓ xexp(b

2

2 )(t s) + p t sG

◆

|x=Ss

= Z

Rf⇣

Sse^{(b 22)(t s)+ pt sy}⌘e ^y

2

p 2

2⇡dy