Stochastic differential equations involving the two- parameter fractional brownian motion

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) = {(P ? C[_a,b],H :1(PI[a,b],H = K}, and for (Pi ? C[ai,bi],ái,

(

CT,á1,á2,00(K, (P1, (P2) = x ? CT,á1,á2,00 : x(a1, .) = (P1, x(., a2) = (P2, Ix1T,á1,á2 = K,

sup a1<t1<b1

kx(t1, .)I[a2,b2],á2 = K, a2<sut2p<b2

}11x(., t2) 11 [a1 ,b1],á1 = K .

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 ? Rbe 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 for every (Pi ? C[ai,ai+å0],ái(K) the operator

F : C[a1,a1+å0]x[a2,a2+å0],á1,á2,00(2K, ^(P1, (P2) ? C[a1,a1+å0]x[a2,a2+å0],á1,á2,00(2K, (P1, (P2)

defined by

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

Z _s Z t Z _s Z t

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

a1 a2 a1 a2

is a contraction.

Proof.(We refer to read [8] for more detail.) Clearly we have

~ ~ ~ ~

Z . Z . a1 a2

~ ~ b(xu,v)dudv ~ = kb18(b1 - a1)^1-á1(b2 - a2)^1-á2 ~T,á1,á2,8 × [(b1 - a1)^á1(b2 - a2)^á2 + 1] .

(3.1)

By using (2.18) it follows

~ Z. ~Z. ~~ a1a2

~ ó(xu,v)dg(u, v) ~= (x)11T,á1,á2,811g11T,â1,â2 T,á1,á2,8

×(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)

ó0 (ëxt1,t2 + (1 - ë)xt1,s2) dë
ó0 (ëxs1,t2+ (1 - ë)xs1,s2) dë

1

_J

0 1

Z0

Z0 1 ó0 (ëxt1,t2 + (1 - ë)xt1,s2) dë

Then

ó(x) ([s1, t1] × [s2, t2]) = (xt1,t2 - xt1,s2 - xs1,t2 + xs1,s2)

Z 1

+(xs1,t2 - xs1,s2) [ó⁰ _(ëxt1,t2 + (1 - ë)xt1,s2)

0

-ó^' _(ëxs1,t2 + (1 - ë)xs1,s2)] dë.

(3.3)

Then (3.3) implies

{|ó(x) ([s1, t1] × [s2, t2])| = 1ó¹181x1T,á1,á2 + Ió^'IL1x(s1, .)1[a2,b2],á2

)

Z 1 ~ dë

× ~ëkx(., _{t2)k[a1,b1],á1} + (1 - ë)kx(., s2)k[a1,b1],á1 (t1-s1)^á1(t2-s2)^á1,

0

Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion 31

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

Ió(x)1T,á1,á2 = K (1ó¹1₈ + kó⁰k_L) . (3.4)

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

Fx ? C[a1,b1]×[a2,b2],á1,á2,8

a

if x ? C[a1,b1]×[a2,b2],á1,á2,8nd also for å1 > 0 enough small,
Fx ? C[a1,a1+å1]×[a2,a2+å1],á1,á2,8(2K, ?1, ?2)
if x ? C[a1,a1+å1]×[a2,a2+å1],á1,á2,8(2K, ?1, ?2).

Next we have

[ó(x) - ó(y)] ([s1, t1] × [s2, t2])

1

= (x - y)([s1,t1] × [s2, t2]) J ó0 (ëxt1,t2 + (1 - ë)yt1,t2) dë

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

1

0

Z× [ó⁰ _(ëxt1,t2 + (1 - ë)yt1,t2) - ó0 (ëxs1,t2 + (1 - ë)ys1,t2)] dë

0 Z

1

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

Z 1 0

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

+(

_Z

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

0

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

(3.5)

0 Z 1
xs1,s2 - ys1,s2) [ë(xt1,s2 - xs1,s2) + (1 - ë)(yt1,s2 - ys1,s2)]
0 1

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

kó(x) -ó(y)1T,á1,á2 = C (K, kól₈, Mó^tkL, kó^itkL)

Mx - ylT,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

11^F x-Fyl[a1,a1+å2]×[a2,a2+å2],á1,á2,8 = dlx-yl[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

Existence and Uniqueness of the Solutions of SDE with 32 Two-Parameter Fractional Brownian Motion

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)

is a contraction. 111

An existence and uniqueness result for ordinary differential equations with Hölder continuous forcing is obtained in [11]. 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 ó^" Lipschitz. Then for every a1 < b1, a2 < b2 and ?i ? C[ai,bi],ái with ?1(a1) = ?2(a2), the equation

t

x st=?1(s) ? 2(t) - ?1 (a1) + f i

b(x_u,v)dudv

l a

2

(3.8)

+ _Is ft

j_a1 ó(x_u,v)dg(u,v), (s, t) ? T,

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] = d1x1 - 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. 111

Existence and Uniqueness of the Solutions of SDE with Two-Parameter Fractional Brownian Motion 33

Theorem 3.2. Let (B^H_{t )tE[0,1]2} be a two-parameter fractional Brownian motion with Hi ? (1/2,1) and let ái,âi > 0 be such that 1/2 < âi < Hi,

âi > ái > 1-âi, i = 1, 2. Let b, ó : R ? R be such that b is bounded and Lipschitz and ó ? C²_b (R) with ó^" Lipschitz and let be the processes {?i(t)}tE[0,1] such that almost surely ?1(0) = ?2(0) and ?i ? C[0,1],á,. Then with probability one the stochastic equation

s t

Xs,t = ?1(s) + ?2(t) - ?1 (0) + b(X_u,v)dudv

f

0 0 (3.9)

+ ₁s _ft

0 ft (s,t)? [0,1]²,

has a unique solution _{{Xu,v}(u,v)E[0,1]2} with the paths in C[0,1]²,á1,á2,00.

Proof. From the Kolmogorov criterion (see [2, 7]) it follows that B^H has â-Hölder paths, i.e., there exists a random variable C such that for all ù ? Ù

2

(B^H ([t, s])) (ù) ~~ = C(ù)

II j=1

(sj - tj)^â' . (3.10)

s pt

Therefore almost surely we have by Theorem 2.1.1 and (3.10) that the Stieltjes integral

10 0 f (u, v)dBuH,v

is well defined for f ? C[0,1]²,á1,á2. Now the result is a consequence of Theorem 3.1 applied pointwise.

Existence and Uniqueness of the Solutions of SDE with 34 Two-Parameter Fractional Brownian Motion