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Itos formula for fractional Brownian motion

and fractional Bessel processes

Joao Guerra (Departamento de Matematica, ISEG, UTL)

ENSPM, June 23, 2006

Its formula for fBm and fractional Bessel processes

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Fractional Brownian motion

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fractional Brownian motion (fBm) is a centered Gaussian process BH =

BHt t0 with covarianceRH(s, t) =

1

2

t2H+ s2H |t s|

2H

, (1)

H (0, 1).

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1

2

t2H+ s2H |t s|

2H

, (1)

H (0, 1).

1. BH has Holder continuous paths with < H.

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1

2

t2H+ s2H |t s|

2H

, (1)

H (0, 1).


2. IfH = 1/2 then BH has dependent increments.

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1

2

t2H+ s2H |t s|

2H

, (1)

H (0, 1).


2. IfH = 1/2 then BH has dependent increments.

Its formula for fBm and fractional Bessel processes 1

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3. BH is self-similar.

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4. BH exhibits long range dependence if H > 12.

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5. IfH = 1/2 then BH is not a semimartingale =There is no Ito integralfor BH.

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5. IfH = 1/2 then BH is not a semimartingale =There is no Ito integralfor BH.

H = 0.2


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H = 0.5

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H = 0.5


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H = 0.7

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H = 0.7

p-variation properties:

n1k=0

BHtnk+1

BHtnk

p L1()

if p < 1HTE

BH1 1/H if p = 1H0 if p > 1H


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Malliavin calculus for fBm

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H 12, 1

. Hilbert space H:

1[0,t],

1[0,s]

H = RH(t, s).

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H 12, 1

. Hilbert space H:

1[0,t],

1[0,s]

H = RH(t, s).

The map 1[0,t] BHt can be extended to an isometry between H and

the Gaussian space G:

E

BHt BHs

=1[0,t],1[0,s]

H

. (2)

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H 12, 1

. Hilbert space H:

1[0,t],

1[0,s]

H = RH(t, s).

The map 1[0,t] BHt can be extended to an isometry between H and

the Gaussian space G:

E

BHt BHs

=1[0,t],1[0,s]

H

. (2)


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Wiener integral ( H): BH() = T

0dBH

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0dBH

|H|-Function space:

2|H| := H

T

0 T

0

|r| |u| |r u|2H2 drdu < .

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0dBH

|H|-Function space:

2|H| := H

T

0 T

0

|r| |u| |r u|2H2 drdu < .

S: random variables of the form

F = f

BH(1), . . . , BH(n)

, n 1, f Cb (Rn) ei H.

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0dBH

|H|-Function space:

2|H| := H

T

0 T

0

|r| |u| |r u|2H2 drdu < .

S: random variables of the form

F = f

BH(1), . . . , BH(n)

, n 1, f Cb (Rn) ei H.


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Malliavin derivative in S (with values in H)

DF :=ni=1

f

xi

BH(1), . . . , B

H(n)

i. (3)

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DF :=ni=1

f

xi

BH(1), . . . , B

H(n)

i. (3)

D1,2

: closure of S w.r.t. the norm

F21,2 := E

|F|2

+ E

DF2H

.

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DF :=ni=1

f

xi

BH(1), . . . , B

H(n)

i. (3)

D1,2

: closure of S w.r.t. the norm

F21,2 := E

|F|2

+ E

DF2H

.

Divergence operator is the adjoint operator of D:

E [F (u)] = E [DF,uH] , u L2 (; H) (4)


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Dom() is the set of random variables (processes). u L2 (; H) such

that |E

DF,uH| c F2, F S.

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that |E

DF,uH| c F2, F S.

D1,2(H) Dom() e

E

(u)2

= E

u2H

+ E

Du, (Du)HH

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that |E

DF,uH| c F2, F S.

D1,2(H) Dom() e

E

(u)2

= E

u2H

+ E

Du, (Du)HH

Integration by parts formula: F D1,2 and u Dom():

(F u) = F (u) DF,uH


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Relationship between and the pathwise integral:

T0

utdBHt = (u) + H

T0

T0

Dsut |t s|2H2 dsdt (5)

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T0

utdBHt = (u) + H

T0

T0


L1,pH : processes u D1,p(|H| ) such that

upL1,pH

:= E

up

L1/H([0,T])+ E

Dup

L1/H([0,T]2)

< .

Estimate: (u)pp Cp up

L1,pH

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T0

utdBHt = (u) + H

T0

T0


L1,pH : processes u D1,p(|H| ) such that

upL1,pH

:= E

up

L1/H([0,T])+ E

Dup

L1/H([0,T]2)

< .

Estimate: (u)pp Cp up

L1,pH


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Let u L1,pH , with p >1H and

T0

|E [us]|p ds +

T0

E

T0

|Dsur|1/Hds

pHdr < .

Then0 usBHs is -Holder continuous if < H 1/p.

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Let u L1,pH , with p >1H and

T0

|E [us]|p ds +

T0

E

T0

|Dsur|1/Hds

pHdr < .

Then0 usBHs is -Holder continuous if < H 1/p.

Theorem 1. Let u L1,1/HH . Then

n1i=0

tni+1

tni

usBHspL1()

n CHT0

|us|1/Hds,


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Itos formula

F of class C2 and H > 12 =t0

F(BHs )ds (Riemann-Stieltjes) existsand

F

BHt

= F(0) +t0

F(BHs )dBHs . (6)

F

BHt

t[0,T] D1,2 (|H|) and

t

0

F(BHs )dBHs =

t

0

F(BHs )BHs + H

t

0

F(BHs )s2H1ds (7)

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Itos formula

F of class C2 and H > 12 =t0

F(BHs )ds (Riemann-Stieltjes) existsand

F

BHt

= F(0) +t0

F(BHs )dBHs . (6)

F

BHt

t[0,T] D1,2 (|H|) and

t

0

F(BHs )dBHs =

t

0

F(BHs )BHs + H

t

0

F(BHs )s2H1ds (7)


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Theorem 2. If F is of class C2 and maxxR

{|F(x)| , |F(x)| , |F(x)|}

c exp

x2

then (Itos formula):

F(BHt ) = F(0) +

t0

F(BHs )BHs + H

t0

F(BHs )s2H1ds (8)

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Theorem 2. If F is of class C2 and maxxR

{|F(x)| , |F(x)| , |F(x)|}

c exp

x2

then (Itos formula):

F(BHt ) = F(0) +

t0

F(BHs )BHs + H

t0

F(BHs )s2H1ds (8)

Multidimensional Itos formula (BH-d-dimensional fBm ):

FBHt = F (0)+d

i=1

t

0

F

xi(BHs )B

(i)s +H

d

i=1

t

0

2F

x2i

(BHs )s2H1ds.

(9)

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Fractional Bessel processes

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d 2, H (1/2, 1), d-dimensional fBm:

BH =

BHtt[0,T]

=

B(1)t , . . . , B

(d)t

t[0,T]

Fractional Bessel processes:

Rt :=BHt = B(1)t 2 + . . . + B(d)t 2, (10)

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d 2, H (1/2, 1), d-dimensional fBm:

BH =

BHtt[0,T]

=

B(1)t , . . . , B

(d)t

t[0,T]

Fractional Bessel processes:

Rt :=BHt = B(1)t 2 + . . . + B(d)t 2, (10)


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Theorem 3. B(i)sRs

L1,1/HH,i for each i = 1, . . . , d and we have the repre-

sentation

Rt =di=1

t0

B(i)s

RsB(i)s + H(d 1)

t0

s2H1

Rsds. (11)

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Theorem 3. B(i)sRs


sentation

Rt =di=1

t0

B(i)s

RsB(i)s + H(d 1)

t0

s2H1

Rsds. (11)

Analogous representation for standard Brownian motion:

RWt =di=1

t0

W(i)s

RWsdW(i)s +

1

2(d 1)

t0

1

Rsds (12)

and the process Wt :=

di=1

t0W

(i)sRs

dW(i)s

t[0,T]

is a one dimensional

standard Brownian motion

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Theorem 3. B(i)sRs


sentation

Rt =di=1

t0

B(i)s

RsB(i)s + H(d 1)

t0

s2H1

Rsds. (11)

Analogous representation for standard Brownian motion:

RWt =di=1

t0

W(i)s

RWsdW(i)s +

1

2(d 1)

t0

1

Rsds (12)

and the process Wt :=

di=1

t0W

(i)sRs

dW(i)s

t[0,T]

is a one dimensional

standard Brownian motion


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Is Ht := di=1 t

0B(i)sRs

B(i)s a fBm?

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Is Ht := di=1

t

0B(i)sRs

B(i)s a fBm?

1. Ht is self-similar2. Ht has finite 1/H-variation as fBm.

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Is Ht := di=1

t

0B(i)sRs

B(i)s a fBm?


Partial results:

1. If 12 < H 12, then Ht is not a B-adapted fBm.

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Is Ht := di=1

t

0B(i)sRs

B(i)s a fBm?


Partial results:

1. If 12 < H 12, then Ht is not a B-adapted fBm.


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References

[1] E. Alos and D. Nualart, Stochastic integration with respect to the frac-tional Brownian motion, Stochastics and Stoch. Reports, 75 (2003),129-152.

[2] J. M. E. Guerra and D. Nualart, The 1/H-variation of the divergenceintegral with respect to the fractional Brownian motion for H>1/2and fractional Bessel processes, Stochastic Processes and their Appli-cations, 115 (2005), 91-115.

[3] Y. Hu and D. Nualart, Some processes associated with fractional Bessel

Processes, J. Theoretical Probability 18 (2005), 377-397.


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