1.2 Two-parameter Fractional Brownian Motion
1.2.1 The Main Definition
For technical simplicity we consider two-parameter fbm (fbm
field) {BHt , 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
{BHt ,t ? R2+} is called a
(normalized) two-parameter fBm with Hurst index H = (H1, H2) ? (0,
1)2, if it satisfies the assumptions:
(a) BH is a Gaussian field, Bt = 0 for t ?
?R2+;
(b) EBti = 0, EBti BH = 1 11 s
4
i=1,2
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(ti 2Hi + s2Hi i - |ti -si |2Hi)
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Evidently, such a process has the modification with continuous
trajectoires, and we will always consider such a modification. Moreover,
consider "two-parameter" increments:ÄsBHt
:= BHt - BHs1t2 - BHt1s2 + BHs for
s = t. Then they are stationary. Note, that for any fixed ti > 0
the process BH
(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 aá_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,
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+ E
i=1,2,j=3--i
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ái
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áj
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xi f (x) - f (ui, x j)
% du
i i),
ai (xi -- Ui)l#177;a
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xj - aj
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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 Äu 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(Lp(T)) := ~f|f =
Iaá_ri2?, ? ? Lp(T)},
I- á1á2(Lp(T)) := {f|f =
IbJá2?, ? ? Lp(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]
?(xf(ut) á) dt,
Z
(I11á2 f)(x) := (á) f(ut)
á) dt,
?(x [x,8) 2
(I_Vá1á2) f)(x) =
(DTá2 f)(x) := (1 - á) O x1Oxe (t) dt,
x2 (-8,x] ?(x,t, á)
and
O2 f f(t)
(I-(á1á2)f)(x)
(Dá1á2f)(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
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YM#177; 1 H2 f :=
i=1,2
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4) Iá1á2 #177; f.
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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(ÄsWt)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 Fs ? Ft ? F for s < t. Denote F* s:=
ó{Fu, 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(ÄsXt|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
((M51H2f)(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.
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