Equations differentials stochastics involving fractional brownian motion two parameter

Chapter 3

Existence and Uniqueness of the

Solutions of SDE with

Two-Parameter Fractional

Brownian Motion

Next for K > 0 we define the closed sets

C[_a,b],H(K) = {? ? C[a,b],H : I?1[a,b],H = K},

and for ?i ? C[ai,bi],ái,

{CT,á1,á2,00(K, ?1, ?2) = x ? CT,á1,á2,00 : x(a1, .) = ?1, x(., a2) = ?2, 1x1T,á1,á2 = K,

sup

a1<t1<b1

< }

|x(., t2 ) | [a1 ,b1], K

_al - .

|x(t1, .)|[a2,b2],á2 = K, sup

a2<t2<b2

By using the Hölder spaces of functions we obtain the following local contraction property of an integral operator between such spaces, which is useful in the next existence and uniqueness result.

Proposition 3.1. Let â1, â2 ? (1/2, 1] and á1, á2 be such that âi > ái > 1 - âi.Let g ? CR²,â1,â2 and b, ó : R ? R be such that b is bounded and Lipschitz and ó ? C²_b(R) with ó^" Lipschitz. Then for every K > 0 and ai, bi ? R,ai < bi, i = 1, 2, there exists å0 > 0 independent of ai, bi, such that

,ái(K) the operator

for every ?i ? C[ai,ai+å0]

F : C[a1,a1+å0]x[a2,a2+å0],á1,á2,00(2K, ?1, ?2) ? C[a1,a1+å0]x[a2,a2+å0],á1,á2,00(2K,?1,?2)

defined by

Z s_J t s_f t

(F x)st = ?1(s) + ?2(t) + b(x_u,v)dudv + ó(x _u,v)dg(u, v),

a2 a1a2

Existence and Uniqueness of the Solutions of SDE with 26 Two-Parameter Fractional Brownian Motion

is a contraction.

Proof. Clearly we have

~ ~ Z .ja2b(xu,v)dudv T,á1,á2,8 = kb1₈(b1 - a1)^1-á1(b2 - a2)^1-á2 (3.1)

× [(b1 - a1)^á1(b2 - a2)^á2 + 1] .

By using (2.18) it follows

~ Z . ~Z. ~~ a1 a2

~ ~ ó(xu,v)dg(u, v) = Iló(x)11T,á1,á2,811g1IT,â1,â2 T,á1,á2

×(b1 - a1)^â1?á1(b2 - a2)^â2?á2 [(b1 - a1)^á1(b2 - a2)^á2 + 1] . (3.2)

Next

ó(x) ([s1, t1] × [s2, t2]) = (xt1,t2 - xt1,s2)

_- -(xs1 ,t2 xs1,s2)

(ëxt1,t2 + (1 - ë)xt1,s2) dë + (1 - ë)xs1,s2) dë

1

J0

1

Z0

Then

1

1

ó (x) ([s1,t1] × [s2, t2]) = (xt1,t2 - xt1,s2 - x _s1,t2 + xs1,s2) I ó' (ëxt1,t2 + (1 - ë)xt1,s2dë)

^+(x:(¹ëx

,t2 _s2 s1,s2) [ó^f _(ëxt1,t2 + (1 - ë)xt1,s2)

-ó

+ (1 - ë)xs1,s2)] dë.

(3.3)

Then (3.3) implies

{

|ó(x)([s1,t1] × [s2,t2])| 1,

110110011x1IT,aá2 + Iló^'IlL11x(s1, .)1[a2,b2],á2

1

×

o

[ë t2) II [a1,b1],á1 + (1 - ë) Mx(., s2)1[a1,b1],á1 ] dë} (t1 -s1)^á1(t2 -

and hence, if x ? CT,á1,á2,8(K, ?1, ?2), then

Ió(x)1T,á1,á2 = K (Mól8 + 1ó¹1_L) . (3.4)

Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion 27

From (3.2),(3.2) and (3.4) it follows that

Fx E C[a1,b1]x[a1,b1],á1,á2,00

if x E _{C[a1,b1]x[a1,b1],á1,á2,00} and also for å1 > 0 enough small,

Fx E C[a1,a1+å1]x[a2,a2+å1],á1,á2,00(2K, p1, p2)
if x E C[a1,a1+å1]x[a2,a2+å1],á1,á2,00(2K, p1, p2).

Next we have

[ó(x) - ó(y)] ([s1, t1] x [s2, t2])

Existence and Uniqueness of the Solutions of SDE with 28 Two-Parameter Fractional Brownian Motion

1

0₁

,

-(xt1,s2 - yt1,s2) I ó^' (ëxt1 s2 + (1 - ë)yt1,s2) dë

I 1

- (xs1,t2 - ys1,t2) a

I (ë - ë)ys1,t2) dë

o 1 _ol _(Ax_x,_: t_{2 82} + (¹

+(xs1,s2 - ys1,s2) j + (1 - ë)ys1,s2) dë

1

= (x - y) ([s1,t1] × [s2, t2]) I ó^f (ëxt1,t2 + (1 - ë)yt1,t2) dë

o

= (xt1 ,t2 - yt1 ,t2 ) ₁ ó^f (ëxt1,t2 + (1 - ë)yt1,t2) dë

1

+(xt1,s2

[ó^' (ëxt1 ,t2

- yt1,s2) I

- ó^' (ëx t1,s2 + (1 - ë)yt1 ,t2)

+ (1 - ë[ó) y, t(1ë, sx2d

t⁾1^] , t2 ë

1

(1 - ë)yt1,t2)

+(xs1,t2 - ys1,t2) I

+ (1 - ë_[ )_uy_,s₍1 _A, t _x2 )], , dë

t 2+

- ó^' (ëxs1,t2 0

1

- (xs1,s2 - ys1,s2) i + (1 - ë)yt1,t2)

0

- ó' (ëxs1,s2 + (1 - ë)ys1,s2)] dë

1

= (x - y) ([s1, t1] × [s2, t2]) j ^óf (ëxt1,t2 + (1 - ë)yt1,t2) dë

+ [(xs1,t2 - ys1,t2) - (xs1,s2 - ys1,s2)]

1

0

× 10 [a'(ëxt1,t2 + (1 .-, (ë) Y :2₂ ) - a' (ëxs1,t2 + (1 - ë)ys1,t2)] dë

+ +(óx:(1ë,sx2 s1,sy2 s1,s2) 11

[0 ëx _ti, + (1 - ë)ys1,s2) - ó^' (ëxt1,s2 + (1 - ë)yt1,s2)] dë

+ (1 - ë)yt1,t2) - ^ó' (ëxs1,t2 + (1 - ë)ys1,t2)

1

= (x - y) ([s1,t1] × [s2, t2]) I a' (ëxt1,t2 + (1 - ë)yt1,t2) dë

0

+ [(xs1,t2 - ys1,t2) - (xs1,s2 - ys1,s2)] 1

× jo [ó^f _(ëxt1,t2 + (1 - ë)yt1,t2) - ó^f (ëxs1,t2 + (1 - ë)ys1,t2)] dë

1

+(xs1,s2 - ys1,s2) [A(xt1,t2 - X81,t2 ) I + (1 - ë)(yt1,t2 - ys1,t2)]

1

0

× /

0 ó00 (u (ëxt1 ,t2 + (1 - ë)yt1,t2 ) + (1 - u) (ëxs1,t2 + (1 - ë)ys1,t2))dudë

1

- (xs1,s2 [A(xt1,82 - x81,82) Y81,82) I + (1 - A)(yt1,s2 - ys1,s2)]

1

o

×

+ (1 - ë)ys1,s2)) dudë

1

o ó00 (u _(ëxt1,s2 + (1 - ë)yt1,s2) + (1 - u) (ëxs1,s2

Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion 29

Therefore

[ó(x) - ó(y)] ([s1,t1] × [s2, t2])

1

= (x - y) ([s1, t1] × [s2, t2]) f ó^f (ëxt1,t2 (1 - ë)yt1,t2) dë

+ [(xs1,t2 - ys1,t2) - (xs1,s2 - ys1,s2)]

Z 1

×

[ó^f (ëxt1,t2 + (1 - ë)yt1,t2) - ó0 (ëxs1,t2 + (1 - ë)ys1,t2)] dë

0 1

+(xs1,s2 - ys1,s2) [ëx ([s1,t1]× [s2, t2]) + (1 - ë)y ([s1, t1] × [s2, t2])]

₁ 0

× ó⁰⁰ (u (ëxt1,t2 + (1 - ë)yt1,t2) + (¹ - u)(ëxs1,t2 + (1 - ë)ys1,t2))dudë

0 1

+ (xs1 ,s2 ys1,s2 ) f ,s2 xs1,s2) ( 1 - (yt1,s2 ys1,s2)]

0

1

× 0 [ó^" (u _(ëxt1,t2 + (1 - ë)yt1 ,t2 ) + (1 - u) (ëxs1,t2 + (1 - ë)ys1,t2))

-ó^" (u (ëxt1,s2 + (1 - ë)yt1,s2) + (1 - u) (ëxs1,s2 + (1 - ë)ys1,s2))]dudë.

(3.5)

If x, y ? C[a1,a1+å1]×[a2,a2+å1],á1,á2,8(K, ?1, ?2), then (3.5) yields

|ó(x) - ó(y)|T,á1,á2 = C (K, 1ó¹1₈, mó^tkL, ,ó^ffIL) Ilx - yIT,á1,á2. (3.6)

From (3.1), (3.2) and (3.6) it follows that there exists å2 > 0 enough small, independent of ai, bi, such that

1Fx - Fyk[a1,a1+å2]×[a2,a2+å2],á1,á2,8 = 11x - y1[a1,a1+å2]×[a2,a2+å2],á1,á2,8, (3.7)

for some 0 < d < 1, and hence, denoting å0 = min(å1, å2), we obtain that

F : C[a1,a1+å0]×[a2,a2+å0],á1,á2,8(2K, ?1, ?2) ? C[a1,a1+å0]×[a2,a2+å0],á1,á2,8(2K, ?1, ?2)

is a contraction.

An existence and uniqueness result for ordinary differential equations with Hölder continuous forcing is obtained in [12]. The global solution is constructed, first in small time interval, when the contraction principle can be applied, by using estimates in terms of Hölder norms. For the two-parameter case we have the following result.

Theorem 3.1. Let â1, â2 ? (1/2, 1] and á1, á2 be such that âi > ái > 1- âi. Let g ? CR²,â1,â2 and b, ó : R ? R be such that b is bounded and Lipschitz and ó ? C2 b (R) with ó⁰⁰ Lipschitz. Then for every a1 < b1, a2 < b2 and

Existence and Uniqueness of the Solutions of SDE with 30 Two-Parameter Fractional Brownian Motion

? ? C[ai,bi],ái with ?1(a1) = ?2(a2), the equation

Z ^sxs,t = ?1(s) + ?2(t) - ?1 (a1) + it

b(x_u,v)dudv

a2

(3.8)

s _ft

+ ó(x_u,v)dg(u, v), (s, t) ? T,

a1 a1

has a unique solution in CT,á1,á2,00.

Proof. Let K > 0 be such that ?i ? C[ai,bi],ái(K). Then from Proposition 3.1 we obtain the existence of the solution x of (3.8) on the rectangle [a1, a1 + å0]×[a2, a2+å0], å0 independent of ai, bi (but dependent on K). If a1+å0 < b1, let n0 be the biggest integer such that n0å < b1. Then x ? CT,á1,á2,00(2K) and inductively we obtain the existence of the solution on

[a1 + å0, a1 + 2å0] × [a2, a2 + å0], ..., [a1 + n0å0, b1] × [a2, a2 + å0], and then on

[a1, a1 + å0] × [a2 + å0, a2 + 2å0], ..., [a1 + n0å0, b1] × [a2 + å0, a2 + 2å0],

and continuing again by induction we obtain the existence on T . Let now x1, x2 be two solutions of (3.8). In particular, there is K > 0 such that x1, x2 ? CT,á1,á2,00(K). From (3.7) we deduce the existence of å0 > 0 (which does not depend on ai, bi) and 0 < d < 1 such that

1x1 -x21[a1,a1+å0]x[a2,a2+å0] = dlx1 - x21[a1,a1+å0]x[a2,a2+å0],

and therefore x1 = x2 on [a1, a1 + å0] × [a2, a2 + å0].Inductively (see the existence part) we obtain that x1 = x2 on T.