Equations differentials stochastics involving fractional brownian motion two parameter

1.1.3 Path Differentiability

By[5] we also obtain that the process B^H is not mean square differentiable and it does not have differentiable sample paths.

Proposition 1.1.3.1. Let H E (0, 1). The fractional brownian motion sample path B^H(.) is not differentiable. In fact, for every t0 E [0, 8)

~~~~

sup

lim

t--+t0

~~~~ = 8

B^H(t) - B^H(t0)

t - t0

With probability one.

1.1.4 The Fractional Brownian Motion is not a Semi-martingale for H =6 1 2

The fact that the fractional brownian motion is not a semimartingale for
H =61 ₂ has been proved by several authors. In order to verity B^H is not a
semimartingale for H =6 ¹ ₂, it is sufficient to compute the p-variation of B^H.

Definition 1.1.4.1. Let (X(t))tE[0,T ] be a stochastic process and consider a partition ð = {0 = t0 < t1 < ... < t_n = T}. Put

8_p(X,ð) := _Xn |X(ti) - X(ti_1)|^p

i=1

(i - 1

_~~B^H - BH

Ti Ti

Yn,p = TipH_1 _Xn

i=1

.

The p-variation of X over the interval [0, T] is defined as

Vp(X, [0, T]) := sup ð

8p(X, ð),

where ð is a partition of [0, T]. The index of p-variation of a process is defined as

I(X, [0, T]) := inf {p > 0; V_p(X, [0, T]) < 8}.

We claim that

I(B^H, [0, T ]) = 1H .

In fact, consider for p > 0,

81.1.5 Fractional Integrals and Fractional Derivatives of Functions

Since B^H has the self-similarity properity, the sequenceY_n,p, m ? N has the same distribution as

^eYn,p = m-1 _Xn ~_~B^H(i) - B^H(i - 1) ~~p

i=1

And by the Ergodic theorem [6] the sequence _eYn,p converges almost surely and in L¹ to E [~~BH(1) ^~~p] as n tends to infinity. It follows that

Vn,p =

Xn i=1

~( i "1 ~'\~~~~ ^p (i - 1 ~~B^H - BH m m

converges in probability respectly to 0 if pH > 1 and to infinity if pH < 1 as

1

n tends to infinity. Thus we can conclure that I(B^H, [0, T]) = H . Since for

every semimartingale X, the index I(X, [0, T]) must belong to [0, 1]?{2}, the

1

fractional brownian motion B^H can not be a semimartingale unless H = ₂.

1.1.5 Fractional Integrals and Fractional Derivatives of Functions

Let á > 0 (and in most cases below á < 1 though this is not obligatory). Define the Riemann-Liouville left- and right-sided fractional integrals on (a, b) of order á by

Z x

1

(I^á _a+f)(x) := f(t)(x - t)^á-1dt,

['(á) a

and

Z b

1

(I^á _b-f)(x) := f(t)(t - x)^á-1dt,

['(á) x

respectively.

We say that the function f ? D(Iá _a+(b-)) (the symbol D(.) denotes the domain of the corresponding operator), if the respective integrals converge for almost all (a.a.) x ? (a, b) (with respect to (w.r.t.) Lebesgue measure).

The Riemann-Liouville fractional integrals on R are defined as

(I^á +f)(x) :=

Z x 1 f(t)(x - t)^á-1dt, ['(á) -8

and

J 8

1

(I^á _-f)(x) := f(t)(t - x)^á-1dt,

['(á) x

respectively.

Z x

1 d

f(t)(x - t)^-ádt,

F(1 - á) dx -8

The Riemann-Liouville fractional derivatives of f of order á on R are defined by

(I+ - f)(x) = (D^á₊f)(x) :=

and

(I^?á -f)(x) = (D^á -f)(x) :=	-1	d 8 dx f (t)(t - x)^?á dt, x
	F(1 - á)

respectively.

For f ? I^á#177;(L_p(R)) with p > 1 the Riemann-Liouville derivatives coincide with the Marchaud fractional derivatives

_Z

( 15-7f)(x) := F(1 1 á) (f(x) - f(x - y)) R+

and

_Z

( ii^áf)(x) := F(1 1 á) (f(x) - f(x+ y)) R+

respectively.