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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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Rn
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Rn
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D(A)
Gmd,1
Zt
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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
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t [0, T]
L2
E
t0
V(Xxs) +
di=1
Vi(Xxs)2
ds
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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
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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]
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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
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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
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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
=.
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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].
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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,
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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).
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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.
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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
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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
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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).
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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
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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
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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
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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
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d
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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
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I A \ {}
f
fI(t) =f(i1,...,ik)(t) =
0
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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
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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
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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
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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
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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.
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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 {
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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
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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
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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
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(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,
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(, 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)
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(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)
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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,
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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
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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.
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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
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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
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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]
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0 =t0< t1
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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
,
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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 .
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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
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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),
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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)
,
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63/191
Dki
2(y1 yk) =y1 yk
kr=1
(r, r; ei)+ 2
r
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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
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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} .
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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
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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
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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 .
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{ [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.
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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,
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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
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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
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Ptf(x) =E(f(Xxt))
f : H R
x H
(t, x) Ptf(x)
HhH A Ah
Hh
Xxt()
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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
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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)
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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
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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
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7/24/2019 Christian Bayer Thesis
81/191
T > 0
m 1
r(m)
x D(Ar(m))
m
0 = t0 < t1
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7/24/2019 Christian Bayer Thesis
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
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7/24/2019 Christian Bayer Thesis
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))
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7/24/2019 Christian Bayer Thesis
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)|
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7/24/2019 Christian Bayer Thesis
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)
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7/24/2019 Christian Bayer Thesis
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
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7/24/2019 Christian Bayer Thesis
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]
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7/24/2019 Christian Bayer Thesis
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.
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7/24/2019 Christian Bayer Thesis
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
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7/24/2019 Christian Bayer Thesis
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 ))
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7/24/2019 Christian Bayer Thesis
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.
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7/24/2019 Christian Bayer Thesis
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
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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,
-
7/24/2019 Christian Bayer Thesis
94/191
-
7/24/2019 Christian Bayer Thesis
95/191
B = (Bt)t[0,[ (, F, (Ft)t[0,[, P)
D
int D
D
D
-
7/24/2019 Christian Bayer Thesis
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
-
7/24/2019 Christian Bayer Thesis
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
-
7/24/2019 Christian Bayer Thesis
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)
-
7/24/2019 Christian Bayer Thesis
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
-
7/24/2019 Christian Bayer Thesis
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
-
7/24/2019 Christian Bayer Thesis
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
-
7/24/2019 Christian Bayer Thesis
102/191
1
h0
X
Ni Z
Ni i= 0, . . . , N X
x
Zx
0 = t0
-
7/24/2019 Christian Bayer Thesis
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))
-
7/24/2019 Christian Bayer Thesis
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
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7/24/2019 Christian Bayer Thesis
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
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7/24/2019 Christian Bayer Thesis
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