Equations differentials stochastics involving fractional brownian motion two parameter

1.2 Two-parameter Fractional Brownian Motion

1.2.1 The Main Definition

For technical simplicity we consider two-parameter fbm (fbm field) {B^H_t , t ? R²₊}, where t = (t1, t2). We suppose that s = t if s = (s1, s2), t = (t1, t2) and si = ti, i = 1, 2.

Definition 1.2.1. The two-parameter process {B^H_t ,t ? R²₊} is called a (normalized) two-parameter fBm with Hurst index H = (H1, H2) ? (0, 1)², if it satisfies the assumptions:

(a) B^H is a Gaussian field, Bt = 0 for t ? ?R²₊;

(b) EBt^i = 0, EBti B^H = 1 11 ^s 4 i=1,2

(ti ^2Hi + s2Hi i - |ti -si |2Hi)

Evidently, such a process has the modification with continuous trajectoires, and we will always consider such a modification. Moreover, consider "two-parameter" increments:Ä_sB^H_t := B^H_t - BHs1t2 - BHt1s2 + B^H_s for s = t. Then they are stationary. Note, that for any fixed ti > 0 the process B^H

(ti,.)

will be the fbm with Hurst index Hj, i = 1, 2, j = 3 - i, evidently, nonnormalized.

1.2.2 Fractional Integrals and Fractional Derivatives of 10 Two-parameter Functions

1.2.2 Fractional Integrals and Fractional Derivatives of Two-parameter Functions

For á = (á1, á2) denote (á) =

(á1)¹(á2)

Definition 1.2.2. [2] Let f ? T := [a, b] := 11 [ai, bi], a = (a1, a2), b =

i=1,2

(b1, b2). Forward and backward Reimann-Liouville fractional integrals of orders 0 < ái < 1 are defined as

(I _a^á_r¹² f)(x) := (á) 1,, f (u) ,x] ?(x, u, 1 - á) du,

and

_Z (Ir² f)(x) := (á)

f

u () ?(x, u ^du, 1 -- a) du,

+ E i=1,2,j=3--i	ái	áj	xi f (x) - f (ui, x j) % du i i), ai (xi -- Ui)^l#177;a
	xj - aj

correspondingly, where [a, x] = 11 [ai, xi], [x, b] = 11 [xi, bi], du = du1du2,

i=1,2 i=1,2

?(u,x, á) =| u1 - x1 |^á1| u2 - x2 |^á2 ,u, x ? [a, b].

Definition 1.2.3. Forward and backward fractional Liouville derivatives of orders 0 < ái < 1 are defined as

2

(Da

á1+ a
^a2 f)(x) := 1^-1(1 a) \
du 8x1?x2 I [a ,] ?(x f (u) u, á₎

and

(Dbá1á2

l f

x

\

A )

:= (1 - á)

?x1?x2 f[x ,b] ? (x u, á) x ? [a, b]

f(u)

Definition 1.2.4. Forward fractional Marchaud derivatives of orders 0 < ái < 1 are defined as

15r

( ² f)(x) :=(1 - á) f (x) +

á1á2 Ä_u f (x)du

?(x, u,á)i[a,x] 40(x, u, 1 + á)

1.2.2 Fractional Integrals and Fractional Derivatives of Two-parameter Functions 11

Let 1 = p = 8, the classes I₊ ^á1á2(L_p(T)) := ~f|f = I_a^á_rⁱ²?, ? ? L_p(T)}, I_- ^á1á2(L_p(T)) := {f|f = I_bJ^á2?, ? ? L_p(T)}

Further we denote Dá1á2

_a+ :=I-(á1á2)

_a+ . Of course, we can introduce the notion

of fractional integrals and fractional derivatives on R²₊. For exemple, the Riemann-Liouville fractional integrals and derivatives on R²₊ are defined by the formulas

(Iá1á2 f)(x) := (á) L8,x] ?(x^f(u^t) á) dt,

Z

(I1^1á2 f)(x) := (á) f(ut) á) dt,
?(x [x,8) 2

(I_V^á1á2) f)(x) = (DT^á2 f)(x) := (1 - á) O x1Oxe (t) dt,

x2 _(-8,x] ?(x,t, á)

and

O2 f f(t)

(I^-(á1á2)f)(x) (D^á1á2f)(x)

:= (1 - á) Ox1Ox2 _i[x,8)?(x,t, á)dt,

0 < ái < 1. Evidently, all these operators can be expanded into the product of the form Iá1á2 += Iá1

+ ? Iá2

+ , and so on. In what follows we shall consider only the case Hi ? (1/2, 1). Define the operator

YM#177; 1 H2 f := i=1,2

4) Iá1á2 #177; f.

Definition 1.2.5. A random field {Xt,t ? R²₊} is a field with independent
increments if its increments {ÄsiXti, i = 1, n} for any family of disjoint

rectangles {(si, ti], i = 1, n} are independent.

Definition 1.2.6. The random field {Wt,t ? R²₊} is called the Wiener field if W = 0 on OR²₊. W is the field with the independent increments and

E(Ä_sWt)² = area((s,t]) = II (ti - si).

i=1,2

Let we have a probability space (Ù,F, P) with two-parameter filtration {Ft, t ? R²₊} on it. It means that F_s ? Ft ? F for s < t. Denote F^* _s:= ó{F_u, s = u}.

Definition 1.2.7. An adapted random field {Xt,Ft,t ? R²₊} is a strong martingale if X vanishes on ?R²₊, E|Xt| < 8 for all t ? R²₊ and for any s < t E(Ä_sXt|F^* _s) = 0.

Evidently, any random field with constant expectation and independent increments is a strong martingale, in particular, the Wiener field is a strong martingale.

Definition 1.2.8. Let

~ f ? L21H2 := f : R² -->R : J ((M5^1H2f)(t))²dt < 8}

R

2

Then we denote I f (t)dBr1 ^H2 as .1 (M- ^H1H2 f)(t)dWt for the underlying R2 R2

Wiener process W.