Equations differentials stochastics involving fractional brownian motion two parameter

1.2.3 Hölder Properties of Two-parameter fbm

We fix á = (á1, á2), ái ? (0,1] and let T = [a1, b1] × [a2, b2]. Let f the Riemann-Liouville fractional integral of order á i.e

1

,

(I^á_a+f)(x1, x2) = f(t1 ^t2)dt1dt2, x1x2) ? T

(á1)(á2) a1 a2 (x1 t1)^1-á1 (x2 - t2)^1-á2

x1

I

x2

The space Ëá,p = (I^á_a+)(L_p(T)) is called the Liouville space (or Besov space) and it becomes separable Banach space with respect to the norm IIáa+fIá,p = If1_p

Proposition 1.2.3.1. [7] For every á, â

Iá _a+I^â a+= Iá+â

_a+ ,

If f ? C²_b (T) and f = 0 on ?1T = ([a1,b1] × {b1}) ? ({a1} × [a2,b2])then the function

1 1x1 r2 _a²f (t1, t2) dt1dt2

+f (x1, x2) =

(1 - á1)(1 - á2) L₁ J_a2 ?t1?t2 (x1 - t1)^á1(x2 - t2)^á2

(1.2)

is the unique function from L₈(T) such that

I^á_a+D^á_a+f = f.

For a rectangle D = [s1, t1] × [s2, t2] ? T we define the increment on D of the function f : T ? IR by

f(D) = f(t1, t2) - f(t1, s2) - f(s1, t2) + ^f(s1, s2).

We denote by _C[ai,bi],ái the space of all ái-Hölder functions on [ai, bi] and

kfk[ai,bi],ái = sup u6=v,ai<u,v<bi

|f(u) - f(v)| (u - v)ái .

Also, we denote by _CT,á1,á2 the space of all (á1, á2)-Hölder functions on T, i.e., f ? CT,á1,á2 if f is continuous,

kf(a1, .)1[a2,b2],á2 < 8, 1f(., a2)I[a1,b1],á1 < 8

and

|f([u1, v1] × [u2, v2])|

< 8.

|u1 - v1|^á1|u2 - v2|^á2

1fkT,á1,á2 = sup

ui6=vi

Proposition 1.2.3.2. [8] Let 0 < â1 < á1,0 < â2 < á2 and p = 1. Then we have the continuous inclusions Ëá,p ? Ëâ,p,

Ëá,p ? Cá1_p^-1,á2_p^-1, Câ1,â2 ? Ëã,p if áip > 1, âi > ãi > 0

ici il faut tout d'abord définir la fonction généralisé

Proposition 1.2.3.3. [8] Assume that f, g are C¹([a, b])-function with f(a) = 0. Let á, â ? (0,1] be such that á+â > 1 and let ä := {a = t0 < ... < t_n = b} be a partition of [a, b] with the norm 1ä1 = max (tj+1 - tj). Then for every

j

0 < å < á + â - 1 the following estimates hold:

~~~~	fb f(t)dg(t)~~~~ = C(á, â)1fk[a,b],álgk[a,b],â(b - a)^1+å, (1.3) a

I^f b

~

f (t)dg(t) - E f(ti)[g(ti+1) - g(ti)]~= C(á, â)1f1[a,b],á1g1[a,b],â(b-a)^å.

~

i

(1.4)

14 1.2.3 Hölder Properties of Two-parameter fbm