Stochastic differential equations involving the two- parameter fractional brownian motion

1.1.2 Hölder Continuity

We recall that according to the Kolmogorov criterion [13], a process X = (Xt)t?R admits a continuous modification if there exist constants á = 1, â > 0 and k > 0 such that

E [|X(t) - X(s)|^á] = k|t - s|^1+â

for all s,t E R.

Theorem 1.1.2.1. Let H E (0, 1). The fractional Brownian motion BH admits a version whose sample paths are almost surely Hölder continuous of order strictly less than H.

Proof. We recall that a function f : R -? R is Hölder continuous of order á, 0 < á = 1 and write f E C^á(R), if there exists M > 0 such that

|f(t) - f(s)| = M|t - s|^á,

For every s, t E R. For any á > 0 we have

E ~|BH t - BH _s |á] = E [|BH 1 |á] |t - s|^áH;

Hence, by the Kolmogorov criterion we get that the sample paths of B^H are almost everywhere Hölder continuous of order strictly less than H. Moreover, by [1] we have

lim t-+0+	sup	~~BH ~~ t	= CH
		t^Hs/log log t-1

with probability one, where CH is a suitable constant. Hence B^H can not have sample paths with Hölder continuity's order greater than H.

1.1.3 Path Differentiability

By[9] 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

BH t - BH 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 =6¹ 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

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 finite 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,

Yn,p = npH_1

Xn i=1

~~'~~ ^p ~BH ( ^i n) - B^H . (i-1 n )

12

1.1.5 Fractional Integral and Fractional Derivative of Function

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

^eYn,p = m-1

Xn i=1

~'Ip . ~BH i - BH i-1

And by the Ergodic theorem [3] the sequence _eYn,p converges almost surely and in L¹ to E [~~BH ~~p] as n tends to infinity. It follows that

1

Vn,p =

Xn i=1

~~'~I ^p ~BH ( ^i n) - BH (i_1 n )

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.