Stochastic differential equations involving the two- parameter fractional brownian motion

1.1.5 Fractional Integral and Fractional Derivative of Function

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 left and Right sided fractional integrals on R are defined as

J x

1

(I^á ₊f)(x) := 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.

The Riemann-Liouville left and Right sided fractional derivatives of f of order á on IR are defined by

(I^-á + f)(x) = (D^á+f)(x) :=			Z x 1 d f(t)(x - t)^-ádt, (1 - á) dx -8
and	(I^-á -f)(x) = (D^á-f)(x) :=	-1		Z 8 d f(t)(t - x)^-ádt, dx x
		(1 - á)

respectively.

For f ? I^á#177;(L_p(118)) with p > 1 the Riemann-Liouville left and Right sided derivatives coincide with the Marchaud fractional derivatives

( 15á+f)(x) := (1 1 á) /L#177; (f(x) - f(x- y))y^-á-1dy,

and

( fi^áf)(x) := (1 1 á) L_#177; (f(x) - f(x+ respectively.

Proposition 1.1.5.1. [4] 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

j

every 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, (1.2) a

^~~Z^~~_~ a b

~

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

~

i

(1.3)