Stochastic differential equations involving the two- parameter fractional brownian motion

Chapter 2

Stochastic Integration with

Respect to Two-parameter

Fractional Brownian Motion

2.1 Pathwise Integration in Two-parameter Besov Spaces

The next result gives an estimate of the Stieltjes integral for smooth functions in terms of Hölder norms and represents the essential step for extending the Stieltjes integral to Hölder functions of two variables.

Proposition 2.1.1. Let ái + âi > 1, ái, âi ? (0, 1], f, g ? C²(T) and let 0 < åi < ái + âi - 1. Then

_iab1 b2
¹

a

2

~~

f(t1, t2)dg(t1, t2) ~~

~~~~

= C(ái, âi)1gkT,â1,â2 {1f1T,á1,á2(b1 - a1)^á1+â1(b2 - a2)^á2+â2 +kf(., a2)1[a1,b1],á1(b1 - a1)^1+å1(b2 - a2)^â2 +1f(a1, .)1[a2,b2],á2(b1 - a1)^â1(b2 - a2)^1+å2 + |f(a1, a2)| (b1 - a1)^â1(b2 - a2)^â2~

(2.1)

Moreover, for every partition A = (si, _tj)i,j, a1 = s1 < ... < sn1 = b1, a2 = t1 < ... < t_n2 = b2, 1A1 = max(si+1 - si) + max(tj+1 - tj) , we have

i j

X 1

j

~~Z _b1 Z b2 n1-1_X n2-

f(si, tj)g ([si, si+1] × [tj, tj+1])

~_~f(u1, u2)dg(u1, u2) -

~ a1 a2 i=1 =1

Z _b1 Z b2 ~~~~ = (b1 - a1)^å1(b2 - a2)^å2

f(t1, t2)dg(t1, t2) å1å2(å1)(å2)

a1 a2

~~~~

kf1 * h1k8

(b1 - a1)^1+å1(b2 - a2)^1+å2

=

å1å2(å1)(å2) kf1k8kh1k8.

(2.2)

+ C(ái,âi)11fIIT,á1,á2 [11g1IT,â1,â211Äll^{á1+â1+á2+â2} (11g11T,â1,â2 .)1[a2,b2],â2) 1ÄM^á1

+ (1g1T,â1,â2 + kg(., _{a2)k[a1,b1],â1)} 1Ä1^á2] .

Proof. Assume first that f = 0 on ?1T and define

h(t1, t2) = g(b1 - t1, b2 - t2) - g(b1 - a1, b2 - t2) (2.3)

-g(b1 - t1, b2 - a2) + g(b1 - a1, b2 - a2).

Then

Z _b1 Z b2 f(t1, _t2)?²g(t1, t2) dt1dt2 = ?²(f * h)(b1, b2) .
?t1?t2 ?t1?t2

a1 a2

Choose åi > 0, 0 < á⁰_i < ái, 0 < â⁰_i < âi withá0 _i + â0 i = 1 +åi. By proposition 1.2.3.1 the function f1 = D_a^á+^' f, h1 = D_a^â+^' h are in L₈ and satisfy

I

á⁰ _Iâ⁰
_a+f1 = f, _a+h1 = h. (2.4)
Then by proposition 1.2.3.1, (2.3) and (2.4) we have

Z _b1 Z b2 Z _b1 Z b2 f(t1, _t2)?²g(t1, t2)

f(t1, t2)dg(t1, t2) = dt1dt2

?t1?t2

a1 a2 a1 a2

?²(f * h)(b1,b2)

=

?2

?t1?t2

_{h i}

_Iá⁰

=

?t1?t2
?2

=

_a+f1 * Iâ0

_a+h1 (b¹, b2)

_{h i}

_Iá⁰+â⁰

_a+ (f1 * h1) (b1, b2)

?t1?t2

?2 ~I1

= _a+I^å a+(f1 * h1) (b1, b2)

?t1?t2

= Iå _a+(f1 * h1)(b1, b2), such that I¹_a+ = I(1,1) _a+, I^å_a+ = Iå1,å2

_a+

and then

Next, the integration by parts for functions of two variables (see[14]) yields

(1 - á0(1 - á^'₂) _a2

(x1, x2) =

x1 1 r2 df ([t1,x1] × [t2, x2])

(x1 - t1)^{á 4} (x2 - t2)^{á 4}

1 f ([t1,x1] × [t2, x2]) lim (1 - á0(1 - á^'₂) t%?x% (x1 - t1)^á4 (x2 - t2)^á4

lirn

x1 f ([t1, x1] [t2, )

dt

t2?x2 x21,

- j_{a1 (x1 toá,} (x2 _t2)^j ál 2 1

- lim

f ([t1,x1]×[t2, x2]) dt2 j_a2 (x1 - t1)^á4 (x2 - t2)^á4

+áV1á^02

x1 r2 f ([t1,x1] × [t2, x2]) dt1dt2 a1 } Ja2 (x1 - t1)^á4+1(x2 - t2)^á2+1

r2 f ([t1,x1] × [t2, x2]) dt dt

(1 - áo(1 - _a1 J_a2 (x1 - t1)^á1+1(x2 - t2)^á2+1

i 2,

so that

₂₎kfkT,á1,á2

(1 - á⁰ ₁)(1 - á0

á0 1á0 2

x1 Ix2

Ja1 a

2

(2.6)

× (x1 - t1)^á1?á01-1(x2 - t2)^á2?á02-1dt1dt2

= ckfkT,á1,á2(b1- a1)^á1?á01(b2 - a2)^á2?á02.

Similary

kh1k8 = c1kgkT,â1,â2(b1- a1)^â1?â01(b2 - a2)^â2?â02. (2.7)

By using (2.6) and (2.7) in (2.5) we obtain (2.1) if f = 0 on ?1T. If f is not necessarily null on ?1T then we define

f(t1, t2) = f ([a1, t1] × [a2, t2]) .

Then f = 0 on ?1T and f, f have the same increments. Then we have

Z _b1 Z b2 f(t1, _t2)?²g(t1, t2) dt1dt2

?t1?t2

a1 a2

Z _b1 Z b2 Z _b1 Z b2

f(t1, _t2)?²g(t1, t2) f(a1, _t2)?²g(t1, t2)

= dt1dt2 + dt1dt2

?t1?t2 ?t1?t2

a1 a2 a1 a2

2

a

Z _b1 Z b

+

a1 2

=

Z _b1 Z b2

a1 a2

f(t1, _a2)?²g(t1, t2) dt1dt2 + f(a1, a2)g ([a1, b1] × [a2, b2]) ?t1?t2

Z b2 ~

f(t1, _t2)?²g(t1, t2) ~?g(b1, t2) - ?g(a1, t2)

dt1dt2 + f(a1, t2) dt2

?t1?t2 ?t2 ?t2

a2

Z b1 ~?g(t1, b2) ~

- ?g(t1, a2)

+ f(t1, a2) dt1 + f(a1, a2)g ([a1, b1] × [a2, b2])

?t1 ?t1

a1
4

=

X k=1

Jk.

(2.8)

From the previous reasoning we have

~~Z _b1 Z b2 ~~~~

~f(t1, _t2)?²g(t1, t2)

~ dt1dt2

?t1?t2

a1 a2

= C(ái, âi)kfkT,á1,á2kgkT,â1,â2(b1 - a1)^á1+â1(b2 - a2)^á2+â2. (2.9)

Next by using (1.2) we have

|J2| =

~~Z b2 ~~ a2

r?g(b1, t2) ?g(a1, t2) 1 dt2 [f (a1, t2) - f(a1, a2)] ?t2 - ?t2

+ |f(a1, a2)g ([a1, b1] × [a2, b2])|

= C(ái, âi) kf(a1, _{.)1[a2,b2],á2} kg(b1, .) - g(a1, .)k[a2,b2],â2 (b2 - a2)^1+å2

+ |f(a1, a2)| 1g1T,â1,â2(b1 - a1)^â1(b2 - a2)â2,

so that

11J211ái,âi)11g1IT,â1,â2 {11f(a1,.)1[a2,b2],á2 (b1 -a1)^â1(b2 -a2)^1+å2 + |f (a1, a2)| (b1 - a1)^â1(b2 - a2)^â2}.

(2.10)

Similarly

11J311 = C(ái, âi)11g11T,â1,â2{11f(., _{a2)k[a1,b1],á1} (b1 - a1)^1+å1(b2 - a2)^â2 +|f(a1, a2)|(b1 - a1)^â1(b2 - a2)^â2}.

(2.11)

Replacing (2.10) and (2.11) in (2.8) we obtain (2.2). Next we have

Z _b1 Z b2 _X

IÄ = f(u1, u2)dg(u1, u2) - f(si, tj)g ([si, si+1] × [tj, tj+1])

a1 a2 i,j

X=

[f(u1, u2) - f(si,tj)] dg(u1,u2)

Z _si+1 Z tj+1

i,j

X=

si tj

Z _si+1 Z tj+1

si tj

[f(u1, u2) - f(u1, tj) - f(si, u2) + f(si,tj)] dg(u1,u2)

i,j

+E

Z _si+1 Z tj+1

i,j

+E

si tj [f(u1, tj) - f(si,tj)] dg(u1, u2)

Z _si+1 Z tj+1

i,j

= I¹ _Ä+ I²_Ä + I³ _Ä.

si tj [f(si, u2) - f(si,tj)] dg(u1, u2)

(2.12)

From (2.1) it follows that

|I^1Ä| = ClIfIlT,á1,á211g1IT,â1,â2E i,j

(si+1 - si)^á1+â1(tj+1 - tj)^á2+â2

(2.13)

=C1kfkT,á1,á2kgkT,â1,â2kÄM^{á1+â1+á2+â2-2}.

Next define

Then (1.2),(1.3) imply

= E

= E

Z b1

^i,j X ~?g(u1, tj+1)?g(u1,tj)] = f1(u1, tj)du1

?u1 ?u1

a1j

Z _si+1 Z tj+1 [f(u1, tj) - f(si, tj)] ?²g(u1, u2) du1du2

?u1?u2

i,j si tj

[f (u1 , tj) - f (si, tj)] ?u1

? g (u1 , _tj+1) ?g(u1 1 tj)1 du1

?u

Z si+1

si

~~I2 ~~ = C Z b1 kf1(u1, _{.)k[a2,b2],á2} kg(u1, _{.)k[a2,b2],â2} du1. (2.14)

Ä

a1

Since u1 ? [si, si+1) we have

kf1(u1, .) [a2,b2],á2 = 1f1T,á1,á2 (u1 - si)^á1 = IfkT,á1,á2 kÄká1

and

Mg(u1,.)1[a2,b2],â2 = (b1 - a1)â1 ,gkT,â1,â2 + 1g(a1,.)1[a2,b2],â2 (2.15)

It follows by replacing in (2.14) that

~|I² Ä| = C1kfkT,á1,á2kÄká1 ~ kgkT,â1,â2 + kg(a1, _{.)k[a2,b2],â2} . (2.16)

Similarly

~|I³ Ä| = C1kfkT,á1,á2kÄká2 ~ kgkT,â1,â2 + kg(., _{a2)k[a1,b1],â1} . (2.17)

Finally using (2.13),(2.16) and (2.17) in (2.12) we (2.2).

Next we define CT,á1,á2,8 the space _CT,á1,á2 endowed with the norm

kxkT,á1,á2,8 = 1x1₈+ sup

a1=t1=b1

kx(t1, .) _[a2,b2],á2+ sup

a2=t2=b2

1x(., t2)1[a1,b1],á1+1xIT,á1,á2.

The space (CT,á1,á2,8, k.kT,á1,á2,8) is a Banach space.

The convergence of Riemann-Stieltjes sums to the integral for Hölder functions of one variable in shown in [[4],[15],[16]]. The corresponding result for functions of two variables is given in the next theorem.

Theorem 2.1.1. Let T0 = [a1 - å0, b1 + å0] × [a2 - å0, b2 + å0], å0 > 0, and let

á1, á2, â1, â2 ? (0, 1] be such that ái + âi > 1. If f ?CT0,á1,á2, g ? CT0,â1,â2,

Z _b1 Z b2

a1 a2

every sequence of partitions Äⁿ = (sⁿ_i ,tⁿ_j ), a1 = s0 < ... < sk(n) = b1, a2 = t0 < ... < tk(_n) = b2, with 1Äⁿ1 ? 0, the Riemann-Stieltjes sums

then there exists a unique real number f(u, v)dg(u, v) such that for

Sog= E i

X~ × ~tn ~~ , f(sn i , tn j )g ~~sn i , sn j , tn i+1 j+1 j

Z _b1 Z b2

converge to f(u, v)dg(u, v). Moreover, the following estimate holds:

a1 a2

_/b1 f

_a f (u, v)dg(u, v) f - a1)^â1(b2 - a2)^â2.

1 a2

(2.18)

Proof. It is enough to prove that for every ä > 0 there exist ç > 0 such that for every two partitions (Äi)i=1,2, ai = uⁱ₀ < ... < uim(i) = bi with kÄik < ç we have

S _,^f4 - = ä. (2.19)

Let J ? C^°°(R²) be such that J = 0, J(x) = 0 if 114 = 1 and J(x)dx = 1

_R2

and define J_å(x) = å^-2 J (^x). Consider the regularizations of f_å, g_å of f,g. å

Recall that

f_å(x) =_R J_å(x - y)f(y)dy = f (x - åy)J(y)dy,

²

and for g_å similarly (as usual f,g are extended as 0 on R² \ T0). It is well known that f_å ? f,g_å ? g uniformly on T. Also it is easily seen that

få ? CT,á1,á2, g_å ? CT,â1,â2.

Next we show that if 0 < á⁰_i < ái, 0 < â0 i < âi, then

f_å ? f in CT,á⁰ (2.20)

1,á0 ₂, g_å ? g in CT,â⁰ 1,â0 2,

f_å(a1, .) ? f(a1, .) in C[a2,b2],á⁰₂,

g_å(a1, .) ? g(a1, .) in C[a2,b2],â0 ₂, (2.21)

f_å(., a2) ? f(., a2) in C[a1,b1],á^'₁,

g_å(.,a2) ? g(., a2) in C[a1,b1],â⁰₁. (2.22)

We have

(f_å - f) ([s1,t1] × [s2, t2]) = J(u,v){f([s1 - åu, t1 - åu] × [s2 - åv, t2 - åv])

(0,1)

and then for every å, ä > 0,

sup

si6=ti

|(f_å - f) × [s2, t2])|

|s1 - t1|^á01|s2 - t2|^á02

= sup

{ | (få - f)([s1,t1]× [s2,t2])| , |si - ti| > ä, i = 1,2} - t1|^á4|s2 - t2|^á4

+ sup {|(f_å - f)([s1,t1] × [s2,t])|

₂ , |s1 - t1| > ä or |s2 - t2| > ä

|s1 - t1|^{á0 1}|s2 - t2|^á0

1

= sup {|f(u1, v1) - f (u2, v2)| , |ui- vi|< å, ? T0, i = 1, 2}

äá1+á2

- f ([s1, t1] × [s2, t2])} dudv,

? 0 as å ? 0,ä ? 0.

+CU max(äá1-á'1, äá2-á2)

Similarly one prove(2.21),(2.22).

Next we choose 0 < á⁰_i < ái, 0 < â0 i < âi with á⁰_i + â⁰ _i> 1. Then from (2.20),(2.22) and (2.12) we obtain

~~~~~ - ₌f ^,gå +

Ä1 Ä2 ^,g - S^feS^f,g - Sfå,gå

Ä2 Ä2

+Saf-å;gå- _fb1 b2

dg_å Skf^å - fådgå

d

b1 b2

a,

1

a

2

fa

1

a

2

=- S;^t6,.^å;^gå + - S kf^å

+C (II fåIIT,á4,á4 + IlgålITAA) {(1 + 11Ä21e^+â1+á4+%-2

+ + 11Ä21)^á4 + _(llÄ1ll + 11Ä21e}

=

f,g- S;^t6_,.^å;^gå + - gå

st

+C1 {(11Ä1ll + 11Ä211)^{á4+â1+á4+â?2} + _(11Ä1ll + 11Ä211)^á4

+ (kÄ1k + kÄ21)^á02o ?0,

as å ? 0 and then IÄiI ? 0. The previous computation also shows that

lim b1b2 b1b2

fådgå =

f dg,

å?0

la

¹

12

²

1

¹

12

2

and this fact and(1.2) imply (2.18).

28 2.1 Pathwise Integration in Two-parameter Besov Spaces