Christian Bayer Thesis

7/24/2019 Christian Bayer Thesis

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Prof. Raul Tempone:

Prof. Josef Teichmann:

DISSERTATION

Selected Topics in Numerics

of Stochastic DifferentialEquations

ausgefhrt zum Zwecke der Erlangung des akademischen Grades eines

Doktors der technischen Wissenschaften unter der Leitung von

ao.Univ.-Prof. Mag. rer. nat. Dr. rer. nat. Josef Teichmann

eingereicht an der Technischen Universitt Wien

bei der Fakultt fr Mathematik und Geoinformation

von

Dipl.-Ing. Christian Bayer

Matrikelnummer: 9926114

Burggasse 112/37

1070 Wien

Wien, im Mrz 2008


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1/2

1

1/2

1/2

1

Rn


7/191


8/191


9/191


10/191

1/2

1

Rn


11/191


12/191


13/191

D(A)

Gmd,1

Zt


14/191


15/191

B = (Bt)

t[0,[= (B1

t

, . . . , Bd

t

)t[0,[

d

, F, (Ft)t[0,[, P

T > 0

[0, T]

V, V1, . . . , V d : R

n Rn

dXxt =V(Xxt)dt +

di=1

Vi(Xxt)dBit, t[0, T],

Xx0 =x Rn

(Ft)t[0,T]

(Xxt)t[0,T]

Xxt =x +

t0

V(Xxs)ds +d

i=1

t0

Vi(Xxs)dB

is


16/191

t [0, T]

L2

E

t0

V(Xxs) +

di=1

Vi(Xxs)2

ds


17/191

V, V1, . . . , V d C

x Rn

Xx

=(Xxt)t[0,T] x Xx

M Rn M

xM

x

XxtM, 0tx, M Rn

Vi(x)TxM, xM, i= 0, . . . , d .

Lf(x) =V0f(x) +

1

2

d

i=1V2i f(x)

=Vi(Vif)(x), x Rn,

tu(t, x) =Lu(t, x), (t, x)[0, T] Rn,u(0, x) =f(x), x Rn,

f : Rn R

L

x

u

f

u(t, x) =E(f(Xxt)).

pt(x, y) n Xx

x Rn

t >0

y Rn

E(f(Xxt)) =

Rn

f(y)pt(x, y)dy

f : Rn R

(Xxt)P Xxt


18/191

Xx

x

(t,x,y)

{V1, . . . , V d, [V0, V1], . . . , [V0, Vd]}

Rn x

Rn V0

Xx

(y, t)pt(x, y)

x

H

H

, H

ddt u(t) =Au(t) + f(t), t[0, T],u(0) =x,

x H

A

A :D(A) H H

f : [0, T] H

u(t) D(A)

t[0, T]


19/191

H

A

H

S = (St)t[0,[

HH

C0

StSs = St+s s, t[0, [

S0= idH H

limt0

Stx xH= 0 xH

A C0 S A

Ax= limt0

Stx xt

, x D(A),

D(A)H

xH

C0

S= (St)t[0,[

C0

H

A

A

H

x D(A)

Stx D(A) t0

ddt Stx= AStx= StAx x D(A) t0

c R (A)

]c, [

StL(H)M ect t[0, [

M > 0

L(H)

L(H)

H

t St L(H)

A

St = etA =

k=0

1

k!tkAk

t0

A


20/191

A

C0 (A)

AL(H) Ax= A( A)1x, xH.

C0 (etA )t[0,[

limnAnx= Ax, x D(A),

limn e

tAn x= Stx, xH,

t

[0,

[

n(A)

n N

A

C0 S

u : [0, T] H

u

u(t) D(A)

t [0, T]

t]0, T]

xH

f 0

x D(A)

u(t) =Stx, t[0, T], xH\ D(A)

H

u

t[0, T]

u(t) , yH=x , yH+ t

0u(s) , AyHds+

t0

f(s) , yHds, y D(A),

A

A

u : [0, T] H

u(t) =Stx + t0

Stsf(s)ds, t[0, T].

u

u

A

f

u


21/191

H

B

d

d N

dXxt =

AXxt + (X

xt)

dt +d

i=1

i(Xxt)dB

it, t[0, T],

Xx0 =xH.

, 1, . . . , d : H H

B = (Bit)iN, t[0,[

U

(ei)iN

Xt=

i=1

Bitei

U

L2

EXt2U

=

i=1

E

Bit2

=.


22/191

tu(t, x) =Au(t, x) + F(u(t, x)) W(t, x),

t [0, T]

x Rn

W

[0, T]Rn

A

C0 S = (St)t[0,[

H

, 1, . . . , d: HH C

C

H

H

xH

Xx = (Xxt)t[0,T] H

Xx

Xxt D(A)

Xxt =x +

t

0 AXxs + (X

xs)ds +

d

i=1 t

0

i(Xxs)dB

is,

t[0, T]

Xx

y D(A)

Xxt , yH=x , yH+

t0

Xxs , AyH+ (Xxs) , yHds+

di=1

t0

i(Xxs) , yHdBis, t[0, T].


23/191

Xx

Xxt =Stx +

t0

Sts(Xxs)ds +d

i=1

t0

Stsi(Xxs)dBis,

t[0, T]

Xx

Xx

f(Xxt)

f

Xx

xH

Y

P T0

Ys2Hds


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X

Xx,n

n N

An A

limn supt[0,T]

EXxt Xx,nt 2H = 0

xH

A

Lt= (L

1t , . . . , L

et ) t[0, [ e

j >0 j j

R

j= 1, . . . , e

Ljt =

Njt

k=1Zjk, t[0, [,

Njt j (Z

jk)kN

j

j = 1, . . . , e

j j = 1, . . . , e

1, . . . , e : H H C

dXxt =

AXxt+ (Xxt)

dt +d

i=1

i(Xxt)dB

it+

ej=1

j(Xxt)dL

jt , t[0, T],

Xx0 =xH.

Xx

Xxt =x +

t0

(AXxs+ (Xxs))ds +

di=1

t0

i(Xxs)dB

is

+e

j=1

t0

j(Xxs)dL

js,


25/191

t[0, [

Xx

Xxt =Stx +

t0

Sts(Xxs)ds +d

i=1

t0

Stsi(Xxs)dBis

+e

j=1

t0

Stsj(Xxs)dLjs,

t[0, [

D(A)

H

Xxt =Stx

x D(A)

D(A)

Stx A :D(A) D(A) x

{y

D(A)

|Ay

D(A)

} D(A)

H

(A, D(A)) C0

(St)t[0,[ H D(A)

x2D(A)=x2H+ Ax2H.

A

(

D(A),

D(A))

D(An)

nN

D(An+1) ={ x D(An)|Ax D(An) } , n N.

D(A) =

nND(An).


26/191

D(An)

D(A)

H

H=C([0, 1])

B

D(B) =C1([0, 1])

Bx(u) =f(u) d

dux(u), u[0, 1],

f

D(B2) ={0}

D(An

)

n N

x2D(An)=x2H+n

k=1

Akx2H

.

(D(An), D(An))

(StD(An))t[0,[ S D(An) C0

D(An)

(A, D(An+1))

n = 1

D(A)

St t

[0, [

St D(A) D(A)

St H St

D(A)

limt0

Stx xD(A)= 0, x D(A),

(St) D(A) C0

(A, D(A))

(A, D(A2))

x D(A)

Ax= limt0

Stx xt

D(A),

(

D(A),

D(A))

D(A)

Ax= limt

Stx xt

=Ax H,

x D(A)

(A, D(A))(A, D(A2))

x D(A2)

x D(A)

Ax D(A)

limt0

Stx xt

=Ax H,

limt0

StAx Axt

=A2x H,


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Stxxt Ax D(A) D(A2)

D(A)

(A, D(A2))

D(A)

C0

(D(A2), D(A2))

(D(An), D(An)) n

S

n

n N

(A)

R(, A) = ( A)1 : H D(A)

H

D(A)

( A) :D(A)H

D(An)

D(An+1)

H St

R(,A)

H

D(A) St R(,A)

D(A)A

D(A2

)

St

D(A2

)

A

D(An) St R(,A)

D(An)

D(An+1) St D(An+1)A

D(A)

D(A)

D(A) D(An)

D(A)

D(A)

pn(x) =xD(An) , x D(A), n N.


28/191

d(x, y) =

n=0

1

2npn(x y)

max(1, pn(x y)) , x, y D(A),

(xn)nN

pk k N

D(Ak)

limnxn= x

(k) D(Ak)

x(k) =x(l) lk

x D(A)

x = limn xn D(A)

S

D(A)

A

D(A)

D(A)

F

G

f :F G

f

xF

hF

Df(x) h= limt

f(x + th) f(x)t

,

f

x

h

f

x

F

h

F

Df :F F G

Df

FF G

F

F


29/191

D2f(x) (h, k) = limt0

Df(x + tk) h Df(x) ht

,

DfC1(FF; G)

F ={ fC(R;R)| x /[0, 1] :f(x) = 0 } .

ddx :F F

d

dtft=

d

dxft, f0= fF.

(ft)0t


30/191

n N x D(An

)

D(An)

D(An)

D(An+1) D(An) Xx

D(An1)

x D(An) D(An1)

x D(A)

D(A)

x D(An)

D(An)

Xx,n =

(Xx,nt )t[0,T] x D(An)

X

x,n

C0

S

n

(A, D(An+1))

S

D(An)

Sn

D(An+1)

Sn+1

Xx,n = Xx,n+1

x D(An+1)

n

x D(An+1)

Xx

D(An)

Xxt =Stx +

t0

Sts(Xxs)ds +d

i=1 t

0Stsi(Xxs)dB

is

Xxt =x +

t0

AXxs + (X

xs)

ds +d

i=1

t0

i(Xxs)dB

is,

Xx

D(An)

x D(A)

Xx

D(An)

n N

Xx,n

Xx,1

n N Xx = Xx,1

Xx

D(An)

Xxt D(A)

Xx

D(An)

x D(An+1)

0:D(An+1) D(An) D(An)

0(x) =Ax + (x) 1

2

di=1

Di(x) i(x), x D(An+1).


31/191

0 V0 x

D(An+1

) D(Am+1

)

n, m N i :D(An

) D(An

)

i :D(Am) D(Am) i = 1, . . . , d

0 D(An+1) D(An) 0 :D(A) H D(An+1) x D(An+1)

Xx

D(An)

dXxt =0(X

xt)dt +

di=1

i(Xxt) dBit, t[0, T]

Xx0 =x

Xx

D(An)

x

D(An+1)

Xx

E(f(XxT))

f : Rn R

f

E(f(XxT))

tu(t, x) =Lu(t, x)

u(0, x) = f(x)

L

u(t, x) =E(f(Xxt))

XxT B

(XxT)P X

xT

Xx

0 =t0 < t1


32/191

X

Nk X

xtk

tk =

tk+1 tk k = 0, . . . , N 1 Bk

B

ik =B

itk+1 Bitk k= 0, . . . , N 1 i= 1, . . . , d

Bk = Bk

Bk

tkY Y

d

Y N(0, Id) B

XNk

(k)

N1k=0

d

0 3

k

d

Bk =

tkk, k= 0, . . . , N 1.

Bk

Bk k

ik =

+1

1/2

1 1/2 , i= 1, . . . , d, k = 0, . . . , N 1,

ik i= 1, . . . , d k= 0, . . . , N 1

(Bk)

N1k=0

B

(, F, P)

B

B

B

XxT XNN


33/191

B

XN

Xx

XN

Xx

X

NN X

xT

(tNk)

Nk=0

(tN) = maxk=0,...,N1

tNk

0

N

(XNN)NN

XxT

limN

EXxT XNN = 0.

>0

C >0

EXxT XNN

C(tN).

G

Rn R

(X

NN)NN XxT

limN

E

f

XNN

=E(f(XxT)), f G.

> 0

f G

C =

C(f)> 0

E(f(XxT)) EfXNN C(tN), N N.

G = Cb(Rn)

G

1/2


34/191

1

1

C4+

C2+

(k + )

k

k

f

C2

X

NN X

xT

XNN

1/2 E(f(XxT)) EfXNN f EXxT XNN f(tN) 12 .

1/2

f

C

1

f


35/191

d


36/191

Bt = (B

1t , . . . , B

dt) t0 d

(, F, (Ft), P)

f : [0, T] Rd f : [0, [ Rd f(t) =(f1(t), . . . , f d(t))

0

f

f0(t) =t, t[0, T] t[0, [,

B0t =t

V0, . . . , V d : Rn Rn

C

Rn

V2

V

V2f(x) =D2f(x) (V(x), V(x)) + Df(x) DV(x) V(x), x Rn.

Xx = (Xxt)t[0,T]

V0, V1, . . . , V d x Rn

A

{0, 1, . . . , d}

A=

k=0

{0, 1, . . . , d}k ,

A

I= (i1, . . . , ik) deg A

deg(I) = deg((i1, . . . , ik)) =k+ # {j {1, . . . , k} |ij = 0 } ,

Am={ I A |deg(I)m } , m N.

f : [0, T]Rd f : [0, [ Rd

d


37/191

I A \ {}

f

fI(t) =f(i1,...,ik)(t) =

0


38/191

deg m

Xx

n= 1

V(y) =

y

y R

x= 0

Xxt =t

f(t) =f(0)+f(0)t+1

2f)(0)t2 + + 1m/2! f

(m/2)(0)tm/2+Rm(t, 0, f),

y

y

B(0,0,...,0)t =

1

|(0, 0, . . . , 0)|! t|(0,0,...,0)|,

||

m

Ad,1 d + 1

e0, . . . , ed Ad,1

e0, . . . , ed

R


39/191

Ad,1 d+ 1

C

f :{e0, e1, . . . , ed} C

f : Ad+1C f

Ad,1

{0, . . . , d} A Ad,1 I = (i1, . . . , ik) eI := ei1 eik e = 1 Ad,1

A

(A, )

A

Ad,1

deg(ei1 eik ) = deg((i1, . . . , ik)),

e0

Ad,1

Ad,1 e0

Ad,1

e0

t

e0 Ad

m N

m

d

1

2

e0, . . . , ed

m

Amd,1

Amd,1 Ad,1

m

eIeJ= 0 deg(I J)> m

m

Amd,1 m

m

C

f :{e0, . . . , ed} C

f : Amd,1C f m


40/191

deg

Wk

k

Amd,1= R W1 Wm,

R W0={1}R 1

AR A R Wk Wl Wk+l k, l N Wk ={0} k > m

x= x0+ +xm xAmd,1 xiWi

x

Wi i = 0, . . . , m t R

t : Amd,1 Amd,1

t(x) =x0+ tx1+ t

2

x2+ + tm

xm, x Am

d,1.

t

Amd,1

exp(x) =

k=0

xk

k!.

Amd,1

Am

d,1

Ad,1

Ad,1

x Amd,1 x0> 0

log(x) = log(x0) +

mk=1

(1)k1k

x x0x0

k,

log(x0) x0

(x x0)m+1 = 0 x0

[x, y] = xy yx

e0, . . . , ed

m

d

1

2

gmd,1

m

Gmd,1 gmd,1


41/191

gmd,1

Amd,1

{e0, . . . , ed} Amd,1

gmd,1 e[] :A Ad,1 e[] = 0 e[(i)] = ei i {0, . . . , d}

e[(i1,i2,...,ik)] = [ei1 , e[(i2,...,ik)]],

(i1, . . . , ik) A k >1

gmd,1=

e[I] I Am

R.

{ eI|I Am} Amd,1

e[I]

I Am

e[(1,2)] = [e1, e2] =

[e2, e1] =

e[(2,1)]

Gmd,1 = exp(gmd,1) Amd,1

exp(y) exp(z) = exp

y +z +

1

2[y, z]+

1

12([y, [y, z]] [z, [z, y]])+

,

y, z gmd,1 Gmd,1 1W1 Wm

Gmd,1

gmd,1

TxG

md,1=

xw

wgmd,1

, xGmd,1.gmd,1 Uk = g

md,1Wk k = 1, . . . , m

gmd,1= U1 Um.

z0= 0 zgmd,1 x0= 1 xGmd,1 exp :gmd,1Gmd,1

Gmd,1

d = 2 m = 2

e0 = 0 A22

1

e1 e2 e

21 e1e2 e2e1 e

22

g22 e1 e2 [e1, e2] G22

G22

1 a c0 1 b0 0 1

a,b,c R


42/191

I3 = 1 0 00 1 00 0 1

g220 x z0 0 y

0 0 0

x,y ,z R

g22

R3

(x1, x2, x3) (y1, y2, y3) =

(x1+ y1, x2+ y2, x3+ y3+ x1y2)

e1 e2 E1 =

0 1 00 0 00 0 0

E2 =

0 0 00 0 10 0 0

[E1, E2] = 0 0 10 0 00 0 0

E1 E2 [E1, E2]

G2d

Gmd,1

Yy = (Yyt )t[0,[ y Amd,1

Yyt =y IAm BIt eI.

y

y = 1

Yy

m

Amd,1

eI A

md,1 Y

y

Amd,1

dYyt =y

IA

deg(I)m2

BIt dt eIe0+d

i=1

IA

deg(I)m1

BIt dBiteIei

=y

IAdeg(I)m

BIt eIe0 dt + yd

i=1

IA

deg(I)m

BIt eIei dBit,

Amd,1

Yy

dYyt =Yy

t e0dt +d

i=1

Yyt ei dBit,

Yy0 =y Amd,1.


43/191

Di(y) = yei i = 0, . . . , d A

md,1

D0, . . . , Dd Di

Gmd,1 Di G

md,1

eigmd,1

Di G

md,1 A

md,1

BIt

t

deg(I)

Y1t t(Y11).

Gmd,1

y Gmd,1 Yyt Gmd,1

t0 Y1t Gmd,1

Yyt = yY

1t y =

1 Gmd,1 Y1

Di(y) = yei i = 0, 1, . . . , d Gmd,1

y Gmd,1

y

Yy

t Gmd,1

t y

Gmd,1 Amd,1

gmd,1 Amd,1 log Amd,1 \ Gmd,1 Y1Gmd,1 {


44/191

Sk

k

Sk

e() = # {j {1, . . . , k 1} |(j)> (j+ 1) }

Zt= log(Y1

t ) =

IAm\{}It e[I],

I= (i1, . . . , ik) A \ {}

I

t = Sk

(

1)e()

k2k1e()B1(I)

t

((i1, . . . , ik)) = (i(1), . . . , i(k)) Sk

(d + 1)

(t, B1t , . . . , B

dt)t[0,[

gmd,1

Zt

Y1t {e[I]| I A\{}}

gmd,1

gmd,1

g22 {e1, e2, [e1, e2]} Zt= B

1t e1+ B

2t e2+ At[e1, e2]

At=1

2

t0

B1s dB2s1

2

t0

B2s dB1s = 1

2

t0

B1s dB2s

1

2

t0

B2s dB1s

Amd,1

t0 y Amd,1

E

Yyt

=y exp

te0+ t

2

di=1

e2i

.

Yy

Yyt L= D0+12

di=1 D

2i

Di(y) = yei


45/191

u(t, y) = E((Yyt ))

tu(t, y) =Lu(t, y)

u(0, y) =(y)

: Amd,1 R

u(t, y) =etL()(y)

v(t, y) =

y exp

te0+ t

2

di=1

e2i

.

v(t, y)

v

t(t, y) =

y exp

te0+

t

2

di=1

e2i

e0+1

2

di=1

e2i

=Lv(t, y)

E(Yyt ) /Gmd,1

f

V0, . . . , V d


46/191

supp

Rn

Rn O Rn

(O) = 0

Rn

m N

m

Rnxk (dx)0

m

Rnp(x)(dx) =

Ni=1

ip(xi)

p

m

Rn

f : Rn

R

Rn

f(x)(dx)N

i=1if(xi).

f

m

f

supp

f d

supp

N

Amn,com m

n e1, . . . , en

Amn,com

Amn

{ eiej ejei|1i, jn } Amn,com Rn m

Rn

m N

m

1N dimAmn,com m N


47/191

(m + 1)

A Rn

conv A

convA

conv A

A

A

A

A

A Rn cone A

cone A

C Rn

xC

nx

{nx= nx(x)}={ y Rn | nx , y=nx , x }

x

{nxnx(x)}={ y Rn | nx , y nx , x } .

C

C

C Rn int C= xC

x

C

C

0

xC

C

yC >0 :x (y x)C.

C

ri C

int Cri C

int C

C

C

Rn

ri C

C

m= 1

Rn

x (dx) 0,

(A\(A{nx = 0})) = 0

(B(0, )c) 0

= (x, nx)> 0

0< ({ yA| nx , y })0 p 1

Rn

p(x)(dx) =

Ni=1

ip(xi).

bary() =N

i=1

ixi,


49/191

(, F)

A F : Rn

() (d)0

p(())(d) =N

i=1

i(i)

p: Rn

R

1

n

n= 1

n1

B F

B A (B) Rn (A \ B) = 0

(n 1)

A

B

x (cone A)

nx

nx , bary()=

nx , () (d)0,

bary()cone (A)

bary()(cone (A)) x= bary()

nx

0 =nx , x=nx , bary()=

Anx , () (d).

1A nx , = 0

(A \ { | nx , ()= 0 }) = 0.

bary()(cone (A))

{nbary() = 0} 0

bary() int(cone (A))cone (A)


50/191

(A)

A

(A)

0() 1

1Nn+1

1, . . . , N 1, . . . , N >0

1+ + N =()

Rn

bary()conv A

: Rn Amn,com

(x1, . . . , xn) =m

k=0

(i1,...,ik){1,...,n}k

xi1 xik ei1 eik .

Amn,com

Amn,com

= Rn

A = supp

A Rn

ci1,...,ik (i1, . . . , ik) {1, . . . , n}k

km m

supp A

m

k=0

(i1,...,ik){1,...,n}k

ci1,...,ik ei1 eikconv (K).

A

conv (A)

(A)


51/191

Xx = (Xxt)t[0,T]

Xx0 =x

V0, . . . , V d R

n

f : Rn R

E(f(XxT)) = f XxT()P(d).

(, F, P)

C0([0, T];R

d)

0

P

C0([0, T];Rd)

C0([0, T];R

d)

C0([0, T];R

d)

d= 2

T > 0

m N

1, . . . , N > 0 1, . . . , N C0([0, T];Rd)

m

1, . . . , N

E(BIT) =

Ni=1

iIi(T), I Am,


52/191

T >0

T = 1

T

[0, T]

I(T)

BIT()

C0([0, T];Rd)

E(BIT) =N

i=1IB

IT(i),

ZI

1p m.

k = 1

BI1T BIkT

m

deg(I1) + + deg(Ik) m k 1

k

(ZI : I Am) (BIT : I Am) Gmd,1


53/191

m

Gmd,1Amd,1 (Y1t )t[0,[

C0([0, T];Rd)

t()1t

()1t =

IAm

I(t)eI, t[0, T].

yt= ()1t

dyt=d

i=0

yteidi(t),

y0= 1,

()1t Gmd,1 t[0, T]

E(Y1T) =

Ni=1

i(i)1T.

i i

i = 1, . . . , N

Gmd,1

Gmd,1

1

T >0

N N

y1, . . . , yNGmd,1

1, . . . , N

E(Y1T) = exp

T e0+T

2

dj=1

e2j

=

Ni=1

iyi.

(Y1T)

P

Y1T

Amd,1

supp((Y1T)P) =Gmd,1 Amd,1.


54/191

M

n

d < n

D1, . . . , Dd

TxM x M D1(x), . . . , Dd(x)

[Di, Dj](x) [Di, [Dj, Dk]](x) i ,j,k = 1, . . . , d

x M x, y M T > 0

: [0, T] Rd

z

dzt=

di=1

Di(zt)di(t), t[0, T],

z0= x,

zT = y M

D1, . . . , Dd

T

m

d

yiGmd,1

yi= (i)1T

i i = 1, . . . , N

Gmd,1 Dj (y) =yej j = 0, . . . , d

i : [0, T] Rd+1 i = 1, . . . , N 1 xi

0i(t) = t Rd 0

t

Amd,1

1

m = 3

1, . . . , N

x1, . . . , xN Rd 3 Rd N = 2d {x1, . . . , xN}={+1, 1}d

i = 2d

i= 1, . . . , N

m= 3

d

i(t) =tzi t[0, 1]

i i= 1, . . . , N


55/191

m = 5

d = 2

m= 5

d

m

m

fCm(Rn) I= (i1, . . . , ik) Am

VIf(x) =V(i1,...,ik)f(x) =Vi1 Vik f(x), x Rn,

V[I]f(x) =V[(i1,...,ik)]f(x) = [Vi1 , [Vi2 , . . . , [Vik1 , Vik ] ]]f(x),

V

0, . . . , V

d

Xx = (Xxt)t[0,T]

x Rn T 1}

1, . . . , N >0 1=

(T)1 , . . . , N =

(T)N : [0, T] Rd

[0, T]

m

T

T

T


56/191

E(f(XxT)) =

IAmVIf(x)E(B

IT) + O(T

m+12 )

=N

j=1

j

IAmVIf(x)

Ij (T) + O(T

m+12 )

f Cm+1(Rn;R)

XxT() xT

dxt=d

i=0Vi(xt)d

i(t), t[0, T],

x0= x Rn,

: [0, T] Rd

f(XxT()) m deg

f(XxT()) =

IAm

VIf(x)I(T) + O(T m+12 )

= (T)

(T),IT =

T

deg(I)

(1),I1 .

1, . . . , N

E(f(XxT)) =

Ni=1

if(XxT(i)) + O(T

m+12 ).

N

Xx(i) i = 1, . . . , N

Xx

[0, T]


57/191

0 =t0< t1


58/191

f(XxT((i1,...,iM)))

i1 iM (i1, . . . , iM) {1, . . . , N }M

f

f

f

C

V0, . . . , V d l N

I A \ {, (0)} : V[I]

V[J]|J Al\ {, (0)}

Cb (R

n),

ACb (Rn) Cb (Rn) A

m

> m 1

C >0

E(f(XxT)) =

(i1,...,iM){1,...,N}Mi1 iMf(XxT((i1,...,iM))) + C

fM(m1)/2

f

ti= iT

M, i= 0, . . . , M .

E(Y1t ) = exp

te0+

t

2

di=1

e2i

,


59/191

Y1

Gmd,1

m

deg

k

W1, . . . , W k

k N

W1 Wk

: W1 WkW1 Wk, (w1, . . . , wk)w1 wk

W1 Wk W1 Wk F

: W1 WkF

k

: W1 WkF

=

W1 Wk

F

W1 Wk

Am,kd,1 = A

md,1 Amd,1,

k Amd,1 Yt = Y1t Amd,1

p : Amd,1 R k

p: Am,kd,1 R

p(y) =p(yk), y Amd,1,

yk =(y , . . . , y) =y y Am,kd,1 .


60/191

p

l N

j1, . . . ,

jk :

Amd,1 R a

j

j = 1, . . . , l

p(y) =l

j=1

ajj1(y) jk(y)

y Amd,1 p k

p: Amd,1 Amd,1 R,

p(y1, . . . , yk) =l

j=1

ajj1(y1) jk(yk).

p(y) = p(y , . . . , y)

p

A

m,kd,1 F = R

E

Ykt

=E

Yt Yt

k Yt

Am,kd,1

(y1 yk) (z1 zk) = (y1z1) (ykzk)

y1, . . . , yk, z1, . . . , zk Amd,1 Am,kd,1 Am,kd,1

exp : Am,kd,1 Am,kd,1

exp(y) =

k=0

yk

k!, y Am,kd,1 ,

y0 = 1 1

Amd,1

y Am,kd,1

0


61/191

Y1t =Yt

y= 1 Amd,1

E

Ykt

= exp

t(e0 1 1 + + 1 1 e0)

+ t

2

di=1

(e2i 1 1 + + 1 1 e2i )

+ td

i=1

(ei ei 1 1 + ei 1 ei 1 1 +

+ 1

1

ei

ei).

p

k

Amd,1

p

Am,kd,1

E(p(Yt)) =p(exp(Ht)),

Ht

Ykt

dYkt =

Ytd

i=0

ei dBit Yt Yt

+ Yt

Yt

di=0

ei dBit

Yt+

+ Yt Yt

Yt

di=0

ei dBit

=d

i=0Ytei

Yt Yt

+ Yt Ytei Yt+ + Yt Yt

Ytei

dBit.

Dki : Am,kd,1 Am,kd,1

Dki (y1 yk) = (y1ei) y2 yk+ y1 (y2ei) yk+ + y1 y2 (ykei),


62/191

i= 0, . . . , d

dYkt =

di=0

Dki

Ykt dBit,

Yt D

ki

Di

yyk,i= 0, . . . , d

Lk

Ykt

Lk =Dk0 +1

2

di=1

Dki

2.

p : Am,kd,1 R

Lkp(yk)

y Am,kd,1

Dkip Dkip(y1 yk) =

=0

py1 yk+ Dki (y1 yk)=pD

ki (y1 yk),

i = 0, . . . , d y1, . . . , yk Amd,1

Dki

2p

A

m,kd,1

Dki2p(y1 yk) =pDki 2(y1 yk),

i = 0, . . . , d

y1, . . . , yk Amd,1

(r; y) = 1 y

r

1,

(r, r; y) = 1 y2

r 1,

(r, s; y) = 1 yr

ys

1,

y Amd,1 r {1, . . . , k} s {1, . . . , k} r < s r

s

Dki (y1 yk) =y1 yk k

r=1

(r; ei)

,


63/191

Dki

2(y1 yk) =y1 yk

kr=1

(r, r; ei)+ 2

r


64/191

tv(t, z) =pz

texp

t

kr=1

(r; e0) + t

2

di=1

kr=1

(r, r; ei)

+ td

i=1

r


65/191

ZT

Y1t

Zt

log(E(Y1t ) gmd,1

Zt

Y1

t

gmd,1(=id)

exp

h

expH

gmd,1

expm

Amd,1 Gmd,1

R(=id)H A

m,md,1

R Amd,1

gmd,1 H

h gmd,1 h H Amd,1 H

Amd,1 m > m

Zt R(Y

1t )

Zt E(R(Y

1t ))

expH

Zt

m

gmd,1 d {e1, . . . , ed} 1 e0 2

gmd,1

gmd,1

gmd,1

A

A={e0, e1, . . . , ed} .


66/191

A

A

(ei1 eik ) (ej1 ejr ) =ei1 eik ej1 ejr .

A

e0, . . . , ed

m Uk

m m Jm,m =

mn=m+1 Un

Amd,1

m

Amd,1 Jm,m

gmd,1/g

md,1

Amd,1

A

m,md,1 = A

md,1/Jm,m.

gmd,1 Amd,1 g

md,1

Jm,m m pr,: Am,d,1 Am,d,1

x Ad,1 [x]m,=x + Jm, Am,d,1

, : A

d,1=

i=0

Wi Ad,1=

i=0

Wi

pr,

[x]m,

=

,x

m, Am,d,1 .

pr, ,(Jm,)Jm, ,

pr,

m

pr, pr,= pr,, m.

Am,d,1 = proj lim

mA

m,md,1

prm: A

m,d,1 Am,md,1 m m


70/191

U(gmd,1)

Am,d,1

C

pm : C Am,md,1 p= pr, p m

: C Am,d,1 pm= prm m m m

deg> m={ fi1 fik| k N, i1 ik I, deg(fi1) + + deg(fik )> m }R

U(gmd,1) Amd,1

deg> m ker(m) m

m:U(gmd,1)/ deg> m Am,md,1


71/191

ZT

A = {e0, e1, . . . , ed} gmd,1 U(gmd,1)/ deg> m Amd,1

A

m

: Amd,1U(gmd,1)/ deg> m

A

gmd,1

Amd,1

U(gmd,1)/ deg> m

Jm,m ker() U(gmd,1)

e0, . . . , ed

m

Jm,m 0 U(gmd,1)

A

m,md,1

U(gmd,1)/ deg> m m : Am,md,1 Am,md,1

(ei) =ei

[ei]m,m

= [ei]deg>m,

[ei]

deg>m

= ei+

deg> m

ei

U(gmd,1)

m(ei) = [ei]m,m

m

[ei]m,m

= m

[ei]deg>m

= [ei]m,m

i= 0, . . . , d

m

[fi]deg>m

= ([fi]m,m) = [fi]deg>m.

A

m,md,1 U(gmd,1)/ deg> m

m

= id

Am,m

d,1

,

m= id

U(gmd,1)/

deg>m

,

m

fi1 fik

m = deg(fi1) + + deg(fik )

fi1 fik= 0 U(gmd,1)/ deg> m

m

fi1 fik = 0 Am,md,1 ,

fi1 fik = 0 Am,d,1 .


72/191

{ [fi1 ]m,m [fik ]m,m|k N, Ni1. . .ik1, deg(fi1) + + deg(fik ) m }

A

m,md,1

Yy

m

Amd,1 m

dYy,mt =d

i=0

Di(Yy,m

t ) dBit, Yy,m0 =y,

Di(y) =yei yAmd,1 i= 0, . . . , d

dB0t = dt

A

Am

m

Zmt = log(Y

1,mt ) =

N

i=1Zi,mt figmd,1, t0,

PH ={f1, . . . , f N} gmd,1

Zmt

Zi,m

i= 1, . . . , N

Zm

e[I] I Am

A=

r=0{1, . . . , N }r .

A J = (j1, . . . , jr)

j1, . . . , jr {1, . . . , N } r N

A

deg() = 0

deg(J) =deg((j1, . . . , jr)) = deg(fj1)+ +deg(fjr ), J A\{} .

Am=

J Adeg(J)m , m N.


73/191

ZT

m, m N

1 m m

Y1,mt

m,m

=

J Am

1

r!Zj1,mt Zjr,mt [fj1 ]m,m [fjr ]m,m.

f

gmd,1 [f]m,m

f Am,md,1

(H,


74/191

Amd,1 A

md,1

m m y Amd,1

y = y0 + y1 + + ym

y Amd,1 y= y0+ y1+ + ym+ ym+1+ + ym ym+1= 0Wm+1 ym=0Wm m,m(y) =y y Amd,1 m,m : A

md,1 Amd,1

y y

expm: W1 Wm Amd,1 Am,md,1

expm(y) = [1]m,m+

mk=1

1

k![y]km,m, yW1 Wm Amd,1.

m, m N

mm2

t > 0

E

Y1,mt

m,m

= expm

te0+

t

2

di=1

e2i

.

Amd,1 Am,md,1

EY1,mt m,m = EY1,mt m,m=

exp

te0+ t

2

di=1

(e2i )

m,m

= expm

te0+

t

2

di=1

e2i

,

Amd,1

expm(y) = [exp(y)]m,m

y Amd,1 y = te0+ t2

di=1 e

2i Amd,1

Zt Z

1,mt , . . . , Z

N,mt

Y1,mt

m,m

m

Zm

Y1,mt

m,m

1


75/191

H

Xx

H

dXxt = (AXxt + (X

xt))dt +d

i=1

i(Xxt)dB

it

Xx

H

dXxt = (AXxt+ (X

xt))dt +

d

i=1i(X

xt)dB

it+

e

j=1j(X

xt)dL

jt .

A : D(A) H H

, 1, . . . , d 1, . . . , e : H H C

1

(Bt)t0 =

(B1t , . . . , Bdt)t0

(, F, P)

(Ljt )t0 j

j = 1, . . . , e

(Ft)t[0,[ A C0

(St)t0


76/191

Ptf(x) =E(f(Xxt))

f : H R

x H

(t, x) Ptf(x)

HhH A Ah

Hh

Xxt()


77/191

dBit d

i(t)

dXxt() = (AX

xt() + (X

xt())

1

2

di=1

Di(Xxt()) i(Xxt()))dt

+d

i=1

i(Xxt())d

i(t)

= (1, . . . , d) : R0 Rd

Xxt()

Ptf(x) t > 0 x

H

f(Xxt())

X0= x Xn= (AXn1+ (Xn1))t

n+

di=1

i(Xn1)nBi,

n 1

Xn / D(A)

D(A)

S

Xxt

Sts

Xxt


78/191

C

, 1, . . . , d D(An)

D(An)

n0

C

(D(An), D(An))

Xx

H

D(A)

m1

[0, 1]

1, . . . ,

N

l(s) =

l(s/

t)

s [0, t]

l = 1, . . . , N

[0, t]

0

0 = Ax + (x) 1

2

di=1

Di(x) i(x).

0 x D(A) D(An)

D(An+1)

x D(An+1)

D(An)


79/191

D(Ar(m))

r(m) 0 m 1

esupp(Xxt; 1, . . . , r)

esupp(Xxt; 1, . . . , r) = supp(X

xt) {Xxt(1), . . . , X xt(r)},

t >0

xH

1, . . . , r

[0, t]

(t)1 , . . . ,

(t)N

[0, t]

ST(x) = 0stT

esupp(Xxs; (t)1 , . . . , (t)r ).

Xxt

P(Xxt /supp(Xxt)) = 0

m 1

r(m) 0

fC(H;R) x D(Ar(m)

)

0< t


80/191

(if)(X

xs)

i {1, . . . , d} i= 0

(0f)(x) =Df(x) Ax + Df(x)

(x) 12

di=1

Di(x) i(x)

.

(0f)(Xxs) = (0f)(x) +

s

0(20 f)(X

xu )du +

d

i=1 s

0(i0f)(X

xu ) dBiu,

(20 f)(x) =D

2f(x)(Ax,Ax) + Df(x) (A2x + A(x) + ) + ,

(i0f)(Xxu ) x

D(A2)

D(Ak+1) D(Ak) xAx

C

k N

f(Xxt) = (i1,...,ik)Adeg(i1,...,ik)m

(i1

ik f)(x)B

(i1,...,ik)t + Rm(t ,f ,x)

Rm(t,x,f)

=

(i1,...,ik)A, i0{0,...,d}deg(i1,...,ik)m


81/191

T > 0

m 1

r(m)

x D(Ar(m))

m

0 = t0 < t1


82/191

C

x

Rm(t,x,f) C sup0st, l=1,...,N

maxmdeg(i0,...,ik)m+2

|i0 ik f(Xxs(l))| tm+12 .

Ptf(x) Q(t)f(x) C supySt(x)

|i0 ik f(y)| tm+12 .

PTf(x) Q(tp) Q(t1)f(x) =p

r=1

Q(tp) Q(tr+1)(Ptr f(x) Q(tr )Ptr1f(x)).

Ptr f(x) Q(tr)Ptr1f(x) = (Ptr Q(tr ))Ptr1f(x),

f(x)

Ptr1f(x)

|PTf(x)Q(tp) Q(t1)f(x)| p

r=1

Ptr f(x) Q(tr )Ptr1f(x)C

pr=1

supyStr (x)

mdeg(i0,...,ik)m+2

i0 ik Ptr1f(y) (tr) m+12C sup

yST(x), 0tTmdeg(i0,...,ik)m+2

|i0 ik Ptf(y)|p

r=1

(tr)m+12 ,

x

A


83/191

f

xPtf(x) D(An)

n 0

J0t(x)h Xxt

h D(An)

J0t(x) h=

=0

Xx+ht D(Ak).

J0t(x)h

dJ0

t(x)

h= A(J0t(x) h) + DXxt J0t(x) hdt

+d

i=1

Di

Xxt J0t(x) h dBit,

J00(x) h= h,

hH

xH

t0

Xx

Xxt, J0t(x) h)

H2

J0t(x) h

L2(,

F, P;

D(An))

=0

Ptf(x + h) =E

Df(Xxt) J0t(x) h

Df

D(An)

, 1, . . . , d

= R(, A)r(m), i=i R(, A)r(m),

i= 1, . . . , d

(A)

, 1, . . . , d D(Ak+r(m)) D(Ak+2r(m)) k N

, 1, . . . , d D(Ar(m))


84/191

f

f =g (R(, A)r(m))

C

g: H R

Xxt x D(Ar(m))

H

R >0

H

Xxt =Stx

R

R > 0

St R

> 0

[0, T]

f

sup0tT

supyH,yR

|i0 ik Ptf(y)|


85/191

Y

f(YyT) =g (R(, A)r(m)) (R(, A)r(m))(XxT) =g(XxT)

f

Y

g

X

E(g(XxT))

Xx

f :H R

E(f(Xxt)) =

n1,...,ne0

n11 nee

n1! ne! et1n1...tene tn1+...+ne

E(f(Xxt)|Njt =nj j = 1, . . . , e)

t0

j nj t

Xx

jn

Nj

n

(j1 , . . . , jk jk1, tjk )

Njt =k1 k tk Rk+1

m2n1. . . 2ne

j

zj= 0 Ljjk

=zj j = 1, . . . , e k1

j

E(f(Xxt)|Njt =nj j= 1, . . . , e)


86/191

m 2n1 . . . 2ne n1 + . . . + ne m+12

x D(Ar(m))

l

l > 0

t

nj

n1+ . . .+ne = n jq1

jq 1 q nj l,j,q nj = 0

m 1

Njt =nj j= 1, . . . , m n1 +. . .+ne =

n

l,j,q

m= m 2n1.

q

l1,...,ln

lr,j,q N

j

jq jq1

r(m)

0

E(f(Xxt)|Njt =nj)

Nl1,...,ln=1

l1. . . ln E(f(Xxt(l1,...,ln ))|Njt =nj)

Ct m

+12 max

(i1,...,ik)Adeg(i1,...,ik)m+2

supysupp(Xxs),

0st

|i1 ik E(f(Xyq,t(lq+1,...,ln )) | Njt =nj )|,

Xxt()

dXxt() =0(Xxt())dt +

di=1

i(Xxt()) dBit+

ej=1

j (Xxt)dL

jt ,

x j(x)

C

D(Ak)

[tnq, t[ yE(Xyq,t(lq+1q,...,ln )) 1 q n


87/191

D >0

E(f(Xxt)) 2(n1+...+ne)m

Nl1,...,ln=1

n11 nem

n1! ne! et1n1...tene

l1. . . ln E(f(Xxt(l1,...,ln ))|Njt =nj) Dt m+12 ,

m

m

m

tk

tk

d= 1

(T)1 (t) =

tT

, (T)2 =

tT

, t[0, T]

1 = 2 =

12 m = 3

[0, T]


88/191

p

2p

p= 10

p

E(f(XxT))

(j1,...,jp){1,...,N}pj1 jp f(XxT(j1,...,jp )).

j1 jp = 1

f(XxT()) {1, . . . , N }p

(j1, . . . , jp)

{1, . . . , N }p j1 jp

f(XxT(j1,...,jp ))

f(XxT())

Xxt

dXxt = X

xtdt + dBt

H = L2(]0, 1[)

]0, 1[ H D() =H10 (]0, 1[)H2(]0, 1[) Cc (]0, 1[)

C0 (St)t0 H H

Xxt =Stx +

t0

StsdBs

St x

x(u) = sin(u) u]0, 1[

Stx= e2tx

x

2

:H R

(y) =

10

y(u)du, yH.


89/191

E((Xx1 )) =E 1

0e

2sin(u)du +

10

10

S1s(u)dBsdu

=

10

e2

sin(u)du= 0.3293 104.

p

1 0.3601 1042 0.2192 1043

0.1226

104

4 0.0652 1045 0.0334 1046 0.0172 1047 0.0084 1048 0.0031 1049 0.0002 104

10 0.0013 104

p

]0, 1[

50

500

(u) = sin(u)

D()

(Xx1 )

sd((Xx1 )) =

2

4(1 e22) = 0.1433.

1012

101


90/191

2 4 6 8 10

1.0

0.8

0.6

0.4

0.2

0.0

Number of cubature steps

RelativeError

OUprocess

Example with Nemicky operator (regular data)

Example with Nemicky operator (irregular data)

x(u) = sin(u)

x

107

101m

107

m 1012

dXxt = X

xtdt + sin XxtdBt,

x(u) = sin(u)

Xx1 E((Xx1 ))


91/191

0 5 10 15 20 25 30

1.0

0.6

0.2

0.2

Number of cubature steps

RelativeError

Full cubature

Cubature with MC

Confidence interval

Xx

1

Xx1 =S1x +

10

S1ssin Xxs dBs

(Xx1 ) = (S1x) +

10

(S1ssin Xxs)dBs.

0

E((Xx1 )) = (S1x) = 0.3293 104

x(u) = sin(u)

dXxt =

Xxt

1

2cos Xxt sin Xxt

dt + sin XxtdBt.


92/191

l

1 0.2907 104

2 0.2163 1043 0.1467 1044 0.0961 1045 0.0622 1046 0.0385 1047 0.0228 1048 0.0142 1049 0.0086 104

10 0.0040 104

(sin Xxt)2 Xxt

t

50

100

l m

5 32 0.0567 104 0.1498 10410 1000 0.0325 104 0.0179 10415 1500 0.0184 104 0.0172 10420 2000 0.0128 104 0.0170 10425 2500 0.0179 104 0.0145 10430 3000 0.0596 104 0.0167 104

m


93/191

x(u) =

1

2

1 2 u 12 u 12 .

E((Xx1 )) = (S1x)

E((Xx1 )) = 0.3002104 x L2(]0, 1[)

x / D(A)

dXxt =

d

duXxtdt + sin XxtdBt,


94/191


95/191

B = (Bt)t[0,[ (, F, (Ft)t[0,[, P)

D

int D

D

D


96/191

X= (Xt)t[0,[ R a R X a

Lat (X) = lim0+

1

t01[a,a+[(Xs)d X , Xs ,

X , X

X

X

Lat (X) = lim0+1

2 t0 1]a,a+[(Xs)d X , Xs .

La = (Lat (X))t[0,[

{ t|Xt= a }

|Xt a|=|X0 a| +

t0

sign(Xs a)dXs+ Lat .

Lt = L

0t (B) 0

|Bt|= t0

sign(Bs)dBs+ Lt,

t=

t0

sign(Bs)dBs

B

|B|

0

Lt= sup

0st(s).

D Rd

x D = D\int D

Nx

Nx=r>0

Nx,r, Nx,r=

y Rd y= 1, B(x ry,r) D= ,

B(x, r)

r

x

D


97/191

r >0

Nx=Nx,r=

xD

B

d

V :DRd

V1, . . . , V d : DRd

dXxt =V(X

xt)dt +

di=1

Vi(Xxt)dB

it+ n(t)dZ

xt,

Xx0 = x D Zx0 = 0 R n(t)

n(t) NXxt XxtD

V, V1, . . . , V d

(Xx, Zx) = (Xxt, Zxt)t[0,[

D

[0, [

Xx0 =x Zx0 = 0 Z

x

Zxt =

t01D (X

xs)dZ

xs

t [0, [

n(t)

n(s) NXxs

Xxs D

n

D

n(t) = n(Xxt)

t

XxtD


98/191

Zx

{ t|XxtD }

Zx

Zxt

d= 1

V0 V11 D=]0, [

x= 0

Xt= Bt+ Zt,

n 1

W

t =

t0sign(Ws)dWs

|Wt|= t+ Lt

Lt Wt 0

Zt B

W

Bt=

t0

sign(Ws)dWs,

Zt= Lt(W) Xt=|Wt|

Xt |Bt| ZtLt(B).

L

tu(t, x) =Lu(t, x), (t, x)[0, T] D,u(0, x) =f(x), x

D,

nu(t, x) =h(x), xD,

f :DR h: DR n(x)

xD

nu(t, x) =u(t, x) , n(x) ,

u(t, x)

xD

uC1,2([0, T] D)


99/191

u

(t, x)[0, T] D

u(t, x) =E

f(Xxt) t

0h(Xxs)dZ

xs

.

T >0

u(T t, Xxt)

u(0, XxT) =u(T, x) + T

0 Lu(T t, x) (

t

u)(T

t, Xxt)dt

+n

i=1

T0

u(T t, Xxt) , Vi(Xxt) dBit

+

T0

u(T t, Xxt) , n(Xxt) dZxt.

dZxt {XxtD}

n u(Tt, Xxt) = h(Xxt)

Lu t u= 0

12

1

h0

#Nx = 1 x D

(x)D

x /D

T >0

xD


100/191

Xxti+1Xxti+ V

Xxti

ti+d

j=1

Vj

Xxti

Bji + n

Xxti

Zti .

Xi+1= Xxti+ V

Xxti

ti+d

j=1

Vj

Xxti

Bji ,

Xxti+1 Xi+1n

Xxti

Zi+1,

Zi+1 Xxti+1 Xi+1 .

0 = t0 < t1


101/191

u(T, x)EF

N

FN

u

u C3b ([0, T]D)

Bi

u(T, x) EFN CN1/2

,

Z

Ni =Z

Ni+1 ZNi i= 0, . . . , N 1

D

B

X

Nl Z

Nl

l = 0, . . . , i

XNi+1 Z

Ni+1 X

Ni

X

(XNi+1,

ZNi+1) ti+1

V

XNi

V1

XNi

, . . . , V n

X

Ni

D

X

XNi+1 D XNi+1 = XNi+1 Z

Ni+1= Z

Ni +

ZNi+1 XNi+1 D X

Ni+1= (

XNi+1)

u(T, x)

[ti, ti+1]

Z

Ni >0 X

Ni+1int D

h

XNi+1

h

XNi+1

12 1

(x) =V1(x) , n(x) V1(x) + + Vd(x) , n(x) Vn(x).

XNi+1

XNi+1 XNi+1 /D

XNi+1 D

X

Ni+1 int D D

XNi+1

XNi+1 D


102/191

1

h0

X

Ni Z

Ni i= 0, . . . , N X

x

Zx

0 = t0


103/191

v

D

c

XNi+1 /D

(II)

XNi+1 XNi = ti+1

ti

dXNs ,

v

ti+1, XNi+1

vti, XNi = ti+1ti

t

v

s, XNs

ds +

ti+1ti

LX

Ni

v

s, XNs

ds +

=

ti+1

ti L

XNi

L

v

s, X

Ns

ds + ,

t v=Lv

0

Ly y Rn

V(y), V1(y), . . . , V d(y)

Lyg(x) =d

j=1

Vj (y)

xjg(x) +

1

2

di,j=1

aij(y) 2

xixjg(x),

a(y) =(y)(y)T

(y) = (ij(y))

di,j=1 ij(y) =V

ij(y)

(II) =

ti+1ti

E

LX

Ni

Lvs, XNs ds.

(I)

XNi+1=

XNi+1+ n

XNi+1

Z

Ni .

(I) =vti+1, XNi+1 vti+1, X

Ni+1 nX

Ni+1Z

Ni+1=

nvti+1, XNi+1ZNi

1

0(1 )

2

n

XNi+1

2 vti+1, XNi+1 ZNi nXNi+1dZNi 2,

2

n

2

n(y)2g(x) =

2

2

=0

g(x + n(y))


104/191

x, y Rd n vti+1, XNi+1 = hX

Ni+1 ZNi = 0

(I)

(II)

v(0, x) =E

f

XNN

N1i=0

h

XNi+1

Z

Ni

v(0, x) v(0, x) =

T0

E

L

XNt

L

v

t, X

Nt

dt

EN1i=0

Z

Ni

2 10

(1) 2

n

XNi+1

2 vti+1, XNi+1ZNi nXNi+1d,

t = max { ti|i {0, . . . , N } , tit } t [0, T]

u(T, x) =v(0, x)

T0

E

LX

Nt

L

v

t, X

Nt

dt

D

E

N1i=0

Z

Ni

2 10

(1 ) 2

n

XNi+1

2 vti+1, XNi+1 ZNi nXNi+1d,

2

n2v

[0, T] D

CEN1

i=0

Z

Ni

2.

i

ZNi = 0

N

Z

Ni

ti Z

Ni


105/191

CEN1

i=0

Z

Ni

2 CN 1N

CN

,

45

d = 1

d > 1

n D

[n, V](x) = [n, Vi](x) = 0, xD, i= 1, . . . , d .

12 1

t

1

1/2

t

int D


106/191

A

v(0, x) v(0, x) = T

0E

LX

Nt

Lvt, XNt dt

12

E

N1i=0

Z

Ni

2 2n

XNi+1

2 vti+1, XNi+1

+1

2E

N1i=0

Z

Ni

3 1

Christian Bayer Thesis

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