University of Saïda Faculty of Sciences and
Technology Department of Mathematics & Computer machine Probability
& Applications
Memory of Master
Equations Differentials Stochastics
Involving Fractional Brownian Motion
Twoparameter
HAMADA.I
Si tu veux courir, cours un kilomètre, si tu
veux changer ta vie, cours un marathon Emil Zatopek1.
Contents
1 Element of Fractional Brownian Motion 5
1.1 Fractional Brownian Motion 5
1.1.1 Selfsimilarity 5
1.1.2 Hölder Continuity 6
1.1.3 Path Differentiability 7
1.1.4 The Fractional Brownian Motion is not a
SemimartingaleforH=61 _{2 } 7 1.1.5 Fractional Integrals and
Fractional Derivatives of Func
tions 8
1.2 Twoparameter Fractional Brownian Motion 9
1.2.1 The Main Definition 9
1.2.2 Fractional Integrals and Fractional Derivatives of Two
parameter Functions 10
1.2.3 Hölder Properties of Twoparameter fbm 12
2 Stochastic Integration with Respect to Twoparameter Fractional
Brownian Motion 15
2.1 Pathwise Integration in Twoparameter Besov Spaces 15
2.2 Some Additional Properties 23
3 Existence and Uniqueness of the Solutions of SDE with
TwoParameter Fractional Brownian Motion 25
4 CONTENTS
Chapter 1
Element of Fractional Brownian
Motion
1.1 Fractional Brownian Motion
Definition 1.1.0.1. The (twosided, normalized) fractional
Brownian motion (fBm) with Hurst index H E (0, 1) is a Gaussian process
B^{H} = {BH _{t} , t E R} on (Ù, F, P), having the
properties:
1. BH 0 = 0,
2. EBH t = 0; t E R,
1 (t^{2H} + s^{2H}  t  s^{2H}) ;
t, s E R,
3. EBH t BH s = 2
1.1.1 Selfsimilarity
Definition 1.1.1.1. We say that an R^{d}valued random
process X = (Xt)t=0 is selfsimilar or satisfies the property of
selfsimilarity if for every a > 0 there exist b > 0 such that:
law (Xat, t = 0) = law (bXt, t = 0) (1.1)
Note that (1.1) means that two process _{Xat} and bXt
have the same finitedimensional distribution functions, i.e., for every choice
t1, ..., t_{n} E R,
P (Xat0 = x0, ..., Xat_{n} = x_{n}) =
P(bXt_{0} = x0, ..., bXt_{n} = x_{n}) For every x0,
..., x_{n} E R.
Definition 1.1.1. A stochastic process X = {Xt, t E R} is called
bselfsimilar if
{X_{a}t,t E R} ^{d} ={a^{b}Xt,t E R} in
the sense of finitedimensional distributions.
1.1.2 Hölder Continuity
We recall that according to the Kolmogorov criterion [3], 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}] = kt 
s^{1+â}
for all s,t E R.
Theorem 1.1.2.1. Let H E (0, 1). The fractional brownian
motion B^{H} 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) = Mt  s^{á},
For every s, t E R. For any á > 0 we have
E ~BH t  BH _{t} 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 [4] we have
lim
t+0+

sup

~_{~}BH ~ (t)

= CH

t^{H}s/log log t1

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.
