Potential Analysis 13: 249–268, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

249

Anticipating Integral Equations JORGE A. LEÓN1∗ and DAVID NUALART2† 1 Department of Mathematics, CINVESTAV–IPN, Apartado Postal 14-740, 07000 México, D.F.,

Mexico 2 Department of Mathematics, University of Barcelona, Gran Via De Les Corts Catalanes 585,

08007 Barcelona, Spain. (e-mail: [email protected]) (Received: 7 October 1997; accepted: 23 February 1999) Abstract. In this paper we deduce some estimates of the Lp ()-norm of the Skorohod and the forward integrals. These estimates allow us to study the existence of a unique solution to anticipating Volterra equations of the Skorohod and forward type. The coefficients Fi (t, s, x), t > s, are F t measurable and satisfy some differentiability conditions (in the sense of the stochastic calculus of variations). Mathematics Subject Classifications (2000): 60H07, 60H20. Key words: Anticipating stochastic calculus, Skorohod integral, stochastic Volterra equations.

1. Introduction Consider the stochastic Volterra equation of the form Z t Z t Xt = Yt + F1 (t, s, Xs ) ds + F2 (t, s, Xs ) dWs , 0

t ∈ [0, T ].

(1.1)

0

Here W = {Wt , t ∈ [0, T ]} is a real-valued Wiener process, the coefficients F1 (t, s, x) and F2 (t, s, x) are Ft -measurable, where {Ft } is the filtration generated by W , and Y is a progressively measurable process. Observe that although the process X is adapted, the stochastic integral in (1.1) has to be anticipative. The case where the coefficients Fi (t, s, x), i = 1, 2, are Fs -measurable and the stochastic integral in (1.1) is the Itô integral has been investigated among others in [3], [4] and [9]. Berger and Mizel [2] have considered the linear case under the assumption that F2 (t, s, x) is Ft -measurable. They use a forward integral to construct the solution of Equation (1.1). Pardoux and Protter [8] have studied Equation (1.1) of the Skorohod type (i.e., the stochastic integral is the Skorohod one) under the assumption that F2 (t, s, x) = G(t, s, Ht , x), where H is an adapted process and G(t, s, h, x) is Fs -measurable. Alòs and Nualart [1] have also studied Equation (1.1) of the Skorohod type assuming that the coefficient F2 (t, s, x) is infinitely differentiable (in the sense of ∗ Partially supported by CONACyT grant 3050P-E9607. † Supported by the DGICYT grant no. PB93-0052.

250

´ AND DAVID NUALART JORGE A. LEON

the stochastic calculus of variations) and all its derivatives are Lipschitz in the variable x. In the case where W is a cylindrical Wiener process defined on a Hilbert space and F2 (t, s, x) = G(t, s)H (s, x), where G(t, s) is an Ft -measurable stochastic evolution system and H (s, x) is adapted, León and Nualart [5] have considered Equation (1.1) of both Skorohod and forward type assuming that G(t, s) is only twice-differentiable in the Malliavin calculus. They use the semigroup property of the random system G(t, s) to prove a maximal inequality of the Lp ()-norm of both the Skorohod and forward integrals. These inequalities are their main tool to study Equation (1.1). A related maximal inequality in probability has been deduced by Tudor [12], when G is a deterministic family of bounded operators, to study delay stochastic evolution equations. In this paper we analize Equation (1.1) of both Skorohod and forward type when F2 (t, s, x) = G(t, s)H (s, x), where H (s, x) is a progressively measurable process and G(t, s) is Ft -measurable and infinitely differentiable in the Malliavin calculus sense. As in [5], Lp ()-norm estimates for the Skorohod and forward integrals are our main tool. We observe that we have to assume that G(t, s) is infinitely differentiable because it does not have the semigroup property. The paper is organized as follows. Section 2 contains some elements of the Malliavin calculus that will be used later. The basic estimate for the Lp ()-norm of the Skorohod integral is obtained in Section 3. The forward integral is studied in Section 4. Finally, in Section 5 we show the existence and uniqueness results. 2. Preliminaries Let {Wt , t ∈ [0, T ]} be a real-valued Wiener process defined on a complete probability space (, F , P ). Let Ft be the σ -field generated by {Ws , s 6 t} and the P -null sets. S is the set of all smooth random variables of the form F = f (Wt1 , . . . , Wtn ),

(2.1)

with t1 , . . . , tn ∈ [0, T ] and f ∈ Cb∞ (R n) (i.e., f and all its partial derivatives are bounded). Given a random variable F of the form (2.1) we introduce its derivative as the stochastic process n X ∂f Ds F = (Wt1 , . . . , Wtn )1[0,tj ] (s), s ∈ [0, T ]. ∂xj j =1

More generally, the iterated derivative operator of the random variable F (given by (2.1)) is defined by Dsn1 ,...,sn F = Ds1 · · · Dsn F.

251

ANTICIPATING INTEGRAL EQUATIONS

These operators are closeable (see Nualart [7]) and we denote by Dn,2 the closure of S with respect to the norm ||F ||2n,2 = ||F ||2L2 () +

n X

||D i F ||2L2 (×[0,T ]i ) .

i=1

The adjoint of the derivative operator D is the Skorohod integral δ. That is, the domain of δ, denoted by Dom δ, is the set of processes u ∈ L2 ( × [0, T ]) such that there is δ(u) ∈ L2 () verifying the duality relationship Z T E(δ(u)F ) = E (Ds F )us ds, for any F ∈ S. 0

It is well-known that Ln,2 :=R L2 ([0, T ]; Dn,2 ), n > 1, is contained in Dom δ. T Sometimes we denote δ(u) by 0 us dWs . A process φ = {φt , t ∈ [0, T ]} is called a smooth step process if there is a partition 0 = t0 < t1 < . . . < tn = T and smooth random variables Fi ∈ S, i ∈ {0, 1, . . . , n − 1}, such that φs =

n−1 X

Fi 1]ti ,ti+1 ] (s),

s ∈ [0, T ].

i=0

Ln,2,f is the family of stochastic processes, introduced by Alòs and Nualart [1], that are differentiable in the future. That is, Ln,2,f is the closure of the smooth step processes with respect to the norm Z n X 2 2 ||u||n,2,f = ||u||L2 (×[0,T ]) + E |Ds1 ...sj us |2 ds1 . . . dsj ds, j =1

1Tj

where 1Tj = {(s1 , . . . , sj , s) ∈ [0, T ]j +1 , s1 > . . . > sj > s}. We set L∞,2,f = ∩n> 1 Ln,2,f . LEMMA 2.1. Fix v ∈ [0, T ]. Let u ∈ L1,2,f and φ ∈ L2 ( × [0, T ]) be such that 1[v,T ] (·)φ· ∈ Dom δ. Then  Z T  Z T  E ut φs dWs = E (Ds ut )φs ds for a.a. t ∈ [0, v]. v

v

Proof. The definition of the space L1,2,f implies that there is a sequence {un , n > 1} of smooth step processes such that Z T n 2 E|ut − ut | + E |Ds (ut − unt )|2 ds → 0 as n → ∞ t

for a.a. t ∈ [0, T ].

(2.2)

252

´ AND DAVID NUALART JORGE A. LEON

Let t ∈ [0, v] be such that (2.2) holds. Then, for any n,  Z E unt



T

Z =E

φs dWs v



T

(Ds unt )φs

ds ,

v

2

and taking the limit as n tends to infinity yields the result.

LEMMA 2.2. Fix v ∈ [0, T ]. Let F ∈ L2 ( × [0, T ]) and u ∈ L1,2,f be such that: (i) (ii) (iii) (iv)

1[v,T ] (·)F· ∈ Dom δ. ut F· ∈ L2 ( × [0, T ]) for a.a. t ∈ [0, T ]. RT ut v Fs dWs ∈ L2 () for a.a. t ∈ [0, v]. RT 2 v Fs (Ds ut ) ds ∈ L () for a.a. t ∈ [0, v].

Then, for a.a. t ∈ [0, v], 1[v,T ] (·)ut F· ∈ Dom δ and Z

T

Z

T

ut Fs dWs = ut

v

Z

T

Fs dWs −

v

Fs (Ds ut ) ds. v

Proof. Let G ∈ S and t ∈ [0, v] be such that Assumptions (i)–(iv) and (2.2) hold. Then [1, Proposition 2.5] and Lemma 2.1 imply Z

T

E

Z

T

(Ds G)ut Fs ds = E

v

[Ds (Gut ) − G(Ds ut )]Fs ds

v

 Z = E Gut



T

Fs dWs v

 Z −E G



T

(Ds ut )Fs ds , v

and therefore the definition of δ yields the result.

2

1,2,f

LC denotes the class of processes u ∈ L1,2,f for which there exists a version of Du such that: (i) the mapping s 7 → Dt us is continuous from [0, t] into L2 (), uniformly with respect to t; (ii) sup06 s
For u ∈ LC

the following limit exists in L2 (), uniformly in t ∈ [0, T ]

Dt− ut = lim Dt ut −ε . ε↓0

253

ANTICIPATING INTEGRAL EQUATIONS

3. Some Estimates of the Lp ( )-norm of the Skorohod Integral p In this section of R t we obtain some inequalities for the L ()-norm of processes the form { 0 us φs dWs }, where φ is an {Ft }-adapted process belonging to Lp ( × [0, T ]). We start with a technical proposition.

PROPOSITION 3.1. Let u be a process belonging to L2,2 and let ϕ: R → R be a twice continuously differentiable function such that ϕ(0) = ϕ 0 (0) = 0 and ϕ 00 > 0. Then Z t  Eϕ us dWs 0

Z 6

1 E 2

t

u2s ϕ 00

Z

ur dWr

0

Z +E



s

ds

0 t

|us |ϕ

00

Z

0

s 0

 Z s ur dWr Ds ur dWr ds,

t ∈ [0, T ].

(3.1)

0

REMARK. Observe that the assumptions of the proposition imply that ϕ is a nonnegative function. Thus the left-hand side of (3.1) is well-defined and it may be equal to +∞. Proof. For t ∈ [0, T ], x ∈ R and N > 0, we denote Z t Z xZ y Xt = us dWs and ϕN (x) = (ϕ 00 (r) ∧ N) dr dy. 0

0

0

Now we decompose the proof in two steps. Step 1. Here we assume that u is a smooth step process. Therefore u ∈ L2,p , for any p > 2. Thus the Itô formula for the Skorohod integral (see [6]) applied to the twice continuously differentiable function ϕN yields Z t Z 1 t 00 ϕN (Xt ) = ϕN0 (Xs )us dWs + (ϕ (Xs ) ∧ N)u2s ds 2 0 Z t Z s0 + (ϕ 00 (Xs ) ∧ N)us Ds ur dWr ds. (3.2) 0

0 1,p

By [11, Theorem 2.1] we have that X ∈ L , p > 2. Therefore [6, Proposition 4.8] implies that ϕN0 (X)u belongs to L1,2 . So the first term in the right-hand side of (3.2) has zero expectation. Hence, taking expectation in equality (3.2) gives Z t 1 EϕN (Xt ) 6 2 E (ϕ 00 (Xs ) ∧ N)u2s ds 0

Z +E 0

t

Z s (ϕ (Xs ) ∧ N)|us | Ds ur dWr ds. 00

0

(3.3)

254

´ AND DAVID NUALART JORGE A. LEON

m Step 2. There is a sequence {um }∞ m=1 of smooth step processes such that u converges to u in L2,2 . Hence, by step 1, inequality (3.3) still holds for u ∈ L2,2 . Thus (3.1) follows from the monotone convergence theorem. 2

For every p > 2 we set γp = (2T )(p−2)/2(p − 1)p/2 . PROPOSITION 3.2. Let p > 2, u ∈ L2,p and let φ be an adapted smooth step process. Then for every positive constant K and t ∈ [0, T ], we have p   Z t E us φs dWs 0

 Z t 6 γp (1 + 2p−2 K) E(|us φs |p ) ds 0

p   Z t  Z s −1 +K (3.4) E (Ds ur )φr dWr ds . 0 0 Rt Proof. Set Xt = 0 us φs dWs , t ∈ [0, T ]. Observe that X is well-defined because our hypotheses imply that uφ ∈ L2,p . By Proposition 3.1, with ϕ(x) = |x|p , and Hölder’s inequality, we obtain E(|Xt |p ) 6 2−1 p(p − 1)

Z

t

E(|Xs |p−2 (us φs )2 ) ds

0

Z



t

+ p(p − 1)

E |Xs | 0

6 2−1 p(p − 1)

p−2

 Z s |us φs | (Ds ur )φr dWr ds 0

Z

t

(E(|Xs |p ))(p−2)/p (E(|us φs |p ))2/p ds

0

Z

t

+ p(p − 1)

(E(|Xs |p ))(p−2)/p

0

p/2 !!2/p Z s E us φs (Ds ur )φr dWr ds. 0

Hence, the Lemma of Zakai [13, page 171] implies   Z t p E(|Xt | ) 6  (p − 1)(E(|us φs |p ))2/p  0  p/2 p/2 !!2/p  Z s ds  +2(p − 1) E us φs (Ds ur ) φr dWr  0

255

ANTICIPATING INTEGRAL EQUATIONS

6 (p − 1)p/2 t (p/2)−1 Z 0

6 (p − 1)p/2 t (p/2)−1 Z

t

+2p−1 0



Z

t

2(p/2)−1

E(|us φs |p ) ds

0

 p/2 !  Z s ds E us φs (Ds ur )φr dWr  0

t

+2p−1

 

  

Z 2(p/2)−1

t

E(|us φs |p ) ds

0

 p 1/2    Z s (E(|us φs |p ))1/2 E (Ds ur )φr dWr ds .  0

This implies that for any constant K > 0, E(|Xt |p )  Z t Z t p (p/2)−1 (p/2)−1 6 γp E(|us φs | ) ds + 2 γp +2 K E(|us φs |p ) ds 0

+2

1−(p/2)

0

K

−1

Z 0

t

p    Z s E (Ds ur )φr dWr ds , 0

which implies that (3.4) holds. Thus the proof is finished.

2

Henceforth {St , t > 0} will denote the Ornstein–Uhlenbeck semigroup and 1tn will denote the set {(s1 , . . . , sn+1 ) ∈ [0, t]n+1 , s1 > . . . > sn+1P }. That is, for each t > 0, 2 −nt St is a contraction operator on L () defined by St = ∞ Jn , where Jn is n=0 e the orthogonal projection on the nth Wiener chaos. PROPOSITION 3.3. Let φ be an adapted smooth step process, M a positive integer and u a process belonging to LM,2,f ∩ Lp ( × [0, T ]), for some p > 2, such that Dsn . . . Ds1 (u· 1[0,sn ] (·)) ∈ Lp ( × [0, T ]),

(3.5)

for a.a. (s1 , . . . , sn ) ∈ 1Tn−1 and n ∈ {1, . . . , M}. Then for any sequence KM = {Kn , n ∈ {1, . . . , M + 1}} of positive constants and r > 0, we have p   Z t E (Sr us )φs dWs 0

6

M X γpn+1 (1 + 2p−2 Kn+1 ) n=0

K1 . . . Kn

256

´ AND DAVID NUALART JORGE A. LEON

Z ×

+

1tn

E(|(Dsn . . . Ds1 Sr usn+1 )φsn+1 |p ) dsn+1 . . . ds1

γpM+1 K1 . . . KM+1

Z 1tM

 Z E

sM+1

(DsM+1 . . . Ds1 (Sr usM+2 )) 0

p  ×φsM+2 dWsM+2 dsM+1 . . . ds1 .

(3.6)

Proof. Fix r > 0. Since u ∈ LM,2,f , by the properties of the Ornstein– Uhlenbeck semigroup, we have Dsn . . . Ds1 (Sr usn+1 ) = e−nr Sr (Dsn . . . Ds1 usn+1 ), for n ∈ {1, . . . , M} and (s1 , . . . , sn+1 ) ∈ 1Tn . Hence (3.5) and [7, Exercise 1.5.7] imply that Sr u and Dsn . . . Ds1 (Sr (u· 1[0,sn ] (·))) belong to L2,p for n ∈ {1, . . . , M} and a.a. (s1 , . . . , sn ) ∈ 1Tn−1 . Therefore (3.6) follows from Proposition 3.2 by an iteration procedure. 2 DEFINITION 3.4. Given p > 2, a sequence K = {Kn , n > 1} of positive constants and u ∈ L∞,2,f we define ||u||K,p,T =

∞ X n=0

sup ω∈ 06s6T

γpn+1 (1 + 2p−2 Kn+1 ) K1 . . . Kn

Z ×

{s 6 sn 6 ···6 s1 6 T }

|Dsn . . . Ds1 us |p dsn . . . ds1 .

PROPOSITION 3.5. Let φ be an adapted smooth step process, p > 2 and u ∈ L∞,2,f ∩ Lp ( × [0, T ]) such that (i) For any positive integer n, Dsn . . . Ds1 (u· 1[0,sn ] (·)) ∈ Lp ( × [0, T ]), for a.a. (s1 , . . . , sn ) ∈ 1Tn−1 . (ii) There exists a sequence K = {Kn , n > 1} of positive constants such that ||u||K,p,T < ∞. Then, for every r > 0 we have Z t p E (Sr us )φs dWs 0

257

ANTICIPATING INTEGRAL EQUATIONS

6

∞ X γpn+1 (1 + 2p−2 Kn+1 )

K1 . . . Kn

n=0

Z ×

1tn

E|(Dsn . . . Ds1 Sr usn+1 )φsn+1 |p dsn+1 . . . ds1 .

(3.7)

Proof. Fix r > 0. By [7, Theorem 3.2.1], there is a constant Cp > 0 such that, for s1 > · · · > sM+1 , Z sM+1 p E (DsM+1 . . . Ds1 (Sr usM+2 ))φsM+2 dWsM+2 0

(Z

p/2

sM+1

6 Cp

2

(E(DsM+1 . . . Ds1 (Sr usM+2 )φsM+2 )) dsM+2 0

Z

sM+1

+E

Z

0

p/2 )

T

|Dv (DsM+1 . . . Ds1 (Sr usM+2 )φsM+2 )|2 dv dsM+2

.

0

 6 C1 (p, T ) Z +||φ||p∞

Z ||φ||p∞

sM+1

Z

sM+1

E|DsM+1 . . . Ds1 usM+2 |p dsM+2

0 T

p/2 |Dv Sr (DsM+1 . . . Ds1 usM+2 )| dv 2

E 0

dsM+2

0

Z +||Dφ||p∞

sM+1

 E|DsM+1 . . . Ds1 usM+2 | dsM+2 . p

0

Hence [7, Exercise 1.5.7] implies that there exists a constant Mp > 0 such that Z sM+1 p E (DsM+1 . . . Ds1 (Sr usM+2 ))φsM+2 dWsM+2 0

 6 C1 (p, T ) Z

sM+1

×

||φ||p∞

  p   e−r p 1 + Mp √ + ||Dφ||∞ 1 − e−2r

E|DsM+1 . . . Ds1 usM+2 |p dsM+2 .

(3.8)

0

Then (3.8) gives that there is a constant C2 = C2 (p, T , kφk∞ , kDφk∞ , r) such that p Z sM+1 Z γpM+1 E (DsM+1 . . . Ds1 (Sr usM+2 ))φsM+2 dWsM+2 t K ...K 1

M+1

1M

0

258

´ AND DAVID NUALART JORGE A. LEON

×dsM+1 . . . ds1 6 6

Z

γpM+1 C2 K1 . . . KM+1

E|DsM+1 . . . Ds1 usM+2 |p dsM+2 . . . ds1

1tM+1

Z

γpM+1 C2 T K1 . . . KM+1

T

Z

sM

...

sup ω∈ 06s6T

Z

s1

s

s

|DsM+1 . . . Ds1 us |p dsM+1 . . . ds1

s

 γpM+2 (1 + 2p−2 KM+2 ) T C2  sup 6 γp K1 . . . KM+1 ω∈ 06s


Z ×

{s 6 sM+1 6 ···6 s1 6 T }

| DsM+1 . . . Ds1 us |p dsM+1 . . . ds1  .

(3.9)

Finally, Proposition 3.3, Hypothesis (ii) and (3.9) yield the result. PROPOSITION 3.6. Under the assumptions of Proposition 3.5 we have that, Rt for any t ∈ [0, T ], 1[0,t ] uφ belongs to Dom δ and 0 (Sr us )φs dWs converges to Rt p as r ↓ 0. 0 us φs dWs in L () RT Proof. Since E 0 |(Sr us − us )φs |2 ds converges to zero as r ↓ 0, it is enough to prove that Z t p Z t E (Sr us )φs dWs − (Sr 0 us )φs dWs → 0 as r, r 0 ↓ 0. (3.10) 0

0

Observe that for every positive integer n and (s1 , . . . , sn+1 ) ∈ 1tn we have E|Dsn . . . Ds1 (Sr usn+1 − Sr 0 usn+1 )|p 6 E|Dsn . . . Ds1 (Sr∨r 0 −r∧r 0 usn+1 − usn+1 )|p 0

0

= E|(e−n(r∨r −r∧r ) Sr∨r 0 −r∧r 0 − 1)Dsn . . . Ds1 usn+1 |p → 0 as r, r 0 ↓ 0.

(3.11)

Hence, using that Sr is a linear contraction operator on Lp () for every p > 1, we have Z E|(Dsn . . . Ds1 (Sr usn+1 − Sr 0 usn+1 ))φsn+1 |p dsn+1 . . . ds1 → 0 1tn

as r, r 0 ↓ 0, and

Z 1tn

E|(Dsn . . . Ds1 (Sr usn+1 − Sr 0 usn+1 ))φsn+1 |p dsn+1 . . . ds1

(3.12)

259

ANTICIPATING INTEGRAL EQUATIONS

Z 6 2p−1 ||φ||p∞ 62

p−1

1tn

||φ||p∞ T

E|Dsn . . . Ds1 usn+1 |p dsn+1 . . . ds1 Z sup

{s 6 sn 6 ···6 s1 6 T }

ω∈ 06s
|Dsn . . . Ds1 us |p dsn . . . ds1 . (3.13)

Finally (3.7), (3.12) and (3.13) give that (3.10) holds. Thus the proof is complete. 2 PROPOSITION 3.7. Let φ, p and u be as in Proposition 3.5. Then we have Z t p E us φs dWs 0

∞ X γpn+1 (1 + 2p−2 Kn+1 )

6

K1 . . . Kn

n=0

Z

×

1tn

E|(Dsn . . . Ds1 usn+1 )φsn+1 |p dsn+1 . . . ds1 .

(3.14)

Proof. By Propositions 3.5 and 3.6 we only need to show that the right-hand side of (3.7) converges to the right-hand side of (3.14) as r ↓ 0. But (3.14) follows from (3.11–3.12) and (3.13) with r 0 = 0. 2 PROPOSITION 3.8. Suppose that φ, p and u satisfy the assumptions of Proposition 3.5. Then we have Z t p Z t E us φs dWs 6 ||u||K,p,T E|φs |p ds. (3.15) 0

0

Proof. For any positive integer n we have Z E|(Dsn . . . Ds1 usn+1 )φsn+1 |p dsn+1 . . . ds1 1tn

Z tZ

Z

t sn+1

0

Z =E

t

|φsn+1 |

6  sup ω∈ 06s6T

sn−1

···

sn+1

Z

|Dsn . . . Ds1 usn+1 |p |φsn+1 |p dsn . . . ds1 dsn+1

sn+1 t

Z

s1

p sn+1

0



Z

s1

=E

sn+1

Z

Z ···

sn−1

 |Dsn · · · Ds1 usn+1 | dsn . . . ds1 dsn+1 p

sn+1



|Dsn . . . Ds1 us | dsn . . . ds1 

Z

{s 6 sn 6 ···6 s1 6 T }

Thus (3.15) follows from (3.14).

t

E|φr |p dr.

p

0

2

260

´ AND DAVID NUALART JORGE A. LEON

PROPOSITION 3.9. Let p and u be as in Proposition 3.5. Then for any adapted process φ in Lp ( × [0, T ]) we have that 1[0,t ] uφ belongs to Dom δ, for each t ∈ [0, T ], and Z t p Z t E us φs dWs 6 ||u||K,p,T E|φs |p ds. (3.16) 0

0

Proof. Let {φ , n > 1} be a sequence of adapted smooth step processes that converges to φ in Lp ( × [0, T ]) as n → ∞. Since ||u||K,p,T < ∞, then we obtain Z t E |us (φs − φsn )|2 ds n

0

Z 6

||u||2∞ E

T

|φs − φsn |2 ds → 0 as n → ∞

(3.17)

0

and, by Proposition 3.8, Z t p Z t n m E us φs dWs − us φs dWs 0

Z

6 ||u||K,p,T

0

t

E|φsn − φsm |p ds → 0 as n, m → ∞.

(3.18)

0

Finally, Proposition 3.8, (3.17) and (3.18) yield that (3.16) holds. Thus the proof is complete. 2 4. An Estimate of the Lp ()-Norm of some Forward Integrals In this section we obtain an estimate similar to (3.16) for the forward integral. For the convenience of the reader we recall the following definition: DEFINITION 4.1. Let u be a measurable process with integrable paths. We say that u ∈ Dom δ − if 1 ε

Z

T

us (W(s+ε)∧T − Ws ) ds 0

converges in probability as ε ↓ 0.This limit is denoted by the forward integral of u with respect to W .

RT 0

us dWs− and is called

REMARK. The forward integral has been studied, among other authors, by Russo and Vallois [10]. In order to state the results of this section we need the following definition:

261

ANTICIPATING INTEGRAL EQUATIONS

DEFINITION 4.2. Let K = {Kn , n > 1} be a sequence of positive integers, p > 2 and u ∈ L∞,2,f . We define |||u|||K,p,T =

∞ X γpn+1 (1 + 2p−2 Kn+1 )

K1 . . . Kn

n=0

Z

×

{06 sn 6 ···6 s1 6 T }

sup |Dsn . . . Ds1 us |p dsn . . . ds1 . ω∈ s6sn

REMARK 4.3. Note that ||u||K,p,T 6 |||u|||K,p,T , where ||·||K,p,T was introduced in Definition 3.4. LEMMA 4.4. Let p > 2 and u ∈ L∞,2,f ∩ Lp ( × [0, T ]) be such that there is a sequence K = {Kn , n > 1} of positive constants such that |||u|||K,p,T < ∞. Also assume that φ is an adapted process in Lp ( × [0, T ]). Then, for each t ∈ [0, T ],  Z Z r 1 T 1[0,t ] (s)φs (us − ur )ds dWr → 0 (4.1) ε 0 (r−ε)∨0 in Lp () as ε → 0. REMARK 4.5. In the proof of the lemma we show that (r 7 → ur )ds) ∈ Dom δ for each t ∈ [0, T ]. Proof. Fix t ∈ [0, T ]. For ε > 0 define Z 1 r vrε = 1[0,t ] (s)φs (us − ur ) ds. ε (r−ε)∨0

Rr

(r−ε)∨0 1[0,t ] (s)φs (us −

Now we divide the proof into three steps. Step 1. Here we assume that φ is an adapted smooth step process. Fix ε > 0. Let n > 1, then Z |Dsn . . . Ds1 vrε |p dsn . . . ds1 {r 6 sn 6 ···6 s1 6 T }

Z

=

{r 6 sn 6 ···6 s1 6 T }

Z r p 1 × 1[0,t ] (s)φs Dsn . . . Ds1 (us − ur ) ds dsn . . . ds1 ε (r−ε)∨0 Z p p 6 2 ||φ||∞ sup |Dsn . . . Ds1 us |p dsn . . . ds1 , {06 sn 6 ···6 s1 6 T }

ω∈ s6sn

262

´ AND DAVID NUALART JORGE A. LEON

which implies, together with the fact that |||u|||K,p,T < ∞, ||v ε ||K,p,T < ∞.

(4.2)

Step 2. Now we prove that v ε ∈ Dom δ and Z T p ε E vr dWr 0

Z ∞ X γpn+1 (1 + 2p−2 Kn+1 )

6

K1 . . . Kn

n=0

1Tn

E|Dsn . . . Ds1 vsεn+1 |p dsn+1 . . . ds1

(4.3)

for any ε > 0. There exists a sequence {φ n , n > 1} of adapted smooth step processes converging to φ in Lp ( × [0, T ]). Fix ε > 0 and define Z 1 r v(n, r) = 1[0,t ] (s)φsn (us − ur ) ds. ε (r−ε)∨0 By Step 1 (see (4.2)) and Proposition 3.7 we have that v(n, ·) ∈ Dom δ and Z T p E (v(k, r) − v(m, r)) dWr 0

6

∞ X γpn+1 (1 + 2p−2 Kn+1 )

K1 . . . Kn

n=0

Z

×

1Tn

 6 2p 

E|Dsn . . . Ds1 (v(k, sn+1 ) − v(m, sn+1 ))|p dsn+1 . . . ds1

K1 . . . Kn

n=0

Z

T

× 0



Z ∞ X γpn+1 (1 + 2p−2 Kn+1 ) 1Tn−1

sup |Dsn . . . Ds1 us |p dsn . . . ds1  ω∈ s6sn

 Z r p 1 k m E |φ − φs |ds dr → 0 as m, k → ∞. ε (r−ε)∨0 s

On the other hand by the assumptions of the lemma we obtain Z T E|v(n, r) − vrε |2 dr 0

Z 6

4||u||2∞

T 0

Z 6

4||u||2∞

 Z r 2 1 n E |φ(s) − φ (s)|ds dr ε (r−ε)∨0

T

E|φ(s) − φ n (s)|2 ds → 0 as n → ∞. 0

(4.4)

263

ANTICIPATING INTEGRAL EQUATIONS

Thus (4.4) and the dominated convergence theorem imply that v ε ∈ Dom δ and that (4.3) holds. Step 3. Finally we show that (4.1) holds. We have that for a.a. (ω, (s1 , . . . , sn+1 )) ∈  × 1Tn , Dsn . . . Ds1 vsεn+1 Z 1 sn+1 = 1[0,t ] (s)φs Dsn . . . Ds1 (us − usn+1 ) ds → 0 as ε → 0, ε (sn+1 −ε)∨0 and 



|Dsn . . . Ds1 vsεn+1 |p 6 2p  sup |Dsn . . . Ds1 us |p  ω∈ s
 Z sn+1  1 |φs |p ds . ε (sn+1 −ε)∨0

Hence, using (4.3) and the dominated convergence theorem we obtain that (4.1) holds. 2 PROPOSITION 4.6. Let p, φ and u be as in Lemma 4.4. Also assume that u ∈ 1,2,f LC . Then the process {1[0,t ] (s)us φs , s ∈ [0, T ]} belongs to Dom δ − and Z 0

t

us φs dWs−

Z = 0

t

Z

t

us φs dWs +

(Ds− us )φs ds.

(4.5)

0

REMARK. Observe that Remark 4.3 and Proposition 3.9 imply that the process {1[0,t ] (s)us φs , s ∈ [0, T ]} belongs to Dom δ. Proof. Fix t ∈ [0, T ]. R tBy the definition of the forward integral we only need to show that 3ε := 1ε 0 us φs (W(s+ε)∧T − Ws ) ds converges in probability to the right-hand side of (4.5) as ε ↓ 0. By Lemma 2.2 and the fact that ||u||K,p,T < ∞ we have Z (s+ε)∧T  Z 1 t 3ε = us φs dWr ds ε 0 s Z Z Z Z 1 t (s+ε)∧T 1 t (s+ε)∧T = us φs dWr ds + (Dr us )φs dr ds. ε 0 s ε 0 s Hence, Fubini’s theorem yields  Z Z r 1 T 3ε = 1[0,t ] (s)us φs ds dWr ε 0 (r−ε)∨0 Z Z 1 T r + 1[0,t ] (s)(Dr us )φs ds dr := I1ε + I2ε . ε 0 (r−ε)∨0

(4.6)

264

´ AND DAVID NUALART JORGE A. LEON

Using the definition of I1ε we obtain Z t ε I1 − us φs dWs 0

1 = ε

Z

T

Z



r

1[0,t ] (s)φs (us − ur ) ds dWr (r−ε)∨0

0

Z

T

+ 0

 Z r  1 ur 1[0,t ] (s)φs ds − 1[0,t ] (r)φr dWr ε (r−ε)∨0

:= J1ε + J2ε .

(4.7)

From Proposition 3.9 we have p Z T Z r 1 ε p E|J2 | 6 ||u||K,p,T E 1[0,t ] (s)φs ds − 1[0,t ] (r)φr dr → 0 ε (r−ε)∨0 0 as ε ↓ 0. Hence Lemma 4.4 and (4.7) yield p Z t ε E I1 − us φs dWs → 0

as ε ↓ 0.

(4.8)

0

1,2,f

On the other hand, using that u ∈ LC Z t ε − E I2 − (Dr ur )φr dr

we have

0

Z

T

0

1 ε

Z

T

6E +E 0

Z

r

|φr |1[0,t ] (r)|Dr− ur − 1[(r−ε)∨0,r](s)Dr us |ds dr

r−ε

1 ε

Z

r

|Dr us ||φr 1[0,t ] (r) − φs 1[0,t ] (s)|ds dr (r−ε)∨0

 Z r 1 6 (E|φr | ) (E|Dr− ur − 1[(r−ε)∨0,r](s)Dr us |2 )1/2 ds dr ε 0 r−ε  1/2 Z T 1 + sup E|Dr us |2 s
T

2 1/2

(r−ε)∨0

Finally (4.5) follows from (4.6) and (4.8)–(4.9).

2

265

ANTICIPATING INTEGRAL EQUATIONS

PROPOSITION 4.7. Let p, φ and u be as in Proposition 4.6. Also assume that ||D − u||∞ := supt ∈[0,T ],ω∈ |Dt− ut | < ∞. Then, for each t ∈ T Z t p Z t − p−1 p−1 − p E us φs dWs 6 2 (||u||K,p,T + T ||D u||∞ ) E|φs |p ds. 0

0

Proof. The result is an immediate consequence of Propositions 3.9 and 4.6. 2 5. Existence and Uniqueness of Solutions for Anticipating Integral Equations 5.1.

VOLTERRA EQUATIONS OF THE SKOROHOD TYPE

In this section we study the existence and uniqueness of solutions for anticipating Volterra equations of the form Z t Xt = Yt + F (t, s, Xs ) ds Z +

0 t

G(t, s)H (s, Xs ) dWs ,

t ∈ [0, T ],

(5.1)

0

where Y is a progressively measurable process belonging to Lp ( × [0, T ]) for p some p > 2 (Y ∈ Lprog( × [0, T ])) for short. The coefficients F :  × 1T1 × R → R, G:  × 1T1 → R and H :  × [0, T ] × R → R are measurable functions that satisfy the following assumptions: (H.1) For each (t, x) ∈ [0, T ] × R we have: (i) F (t, ·, x) is F ⊗B([0, t])-measurable and, for every s ∈ [0, t], F (t, s, x) is Ft -measurable. (ii) H (·, x) is progressively measurable. (H.2) For each t ∈ [0, T ]: (i) G(t, s) is Ft -measurable for every s ∈ [0, t]. (ii) G(t, ·) ∈ L∞,2,f ∩Lp (×[0, t]) and, for any positive integer n, Dsn . . . Ds1 (G(t, ·)1[0,sn ] (·)) ∈ Lp ( × [0, t]) for a.a. (s1 , . . . , sn ) ∈ 1tn−1 . (iii) There exists M ∈ L1 ([0, T ]) and a sequence Kt = {Knt , n > 1} of positive integers such that ||G(t, ·)||Kt,p,t 6 Mt . (H.3) Lipschitz property: For all (t, s) ∈ 1T1 and x, y ∈ R we have: (i) There is f ∈ Lp ([0, T ]) such that |F (t, s, x) − F (t, s, y)| 6 ft |x − y|. (ii) There exists a constant M > 0 such that |H (s, x) − H (s, y)| 6 M|x − y|.

266

´ AND DAVID NUALART JORGE A. LEON

(H.4) Linear growth condition: For any (s, t) ∈ 1T1 we have: |F (t, s, 0)| 6 ft

|H (s, 0)| 6 M,

and

where f and M are given in (H.3). It is well-known that every adapted process has a progressively measurable version. Henceforth we will always deal with such a version. p

THEOREM 5.1. Let p > 2 and Y ∈ Lprog ( × [0, T ]). Then Hypotheses (H.1)– p (H.4) imply that (5.1) has a unique solution in Lprog ( × [0, T ]). p

Proof of uniqueness. Let X, Z ∈ Lprog ( × [0, T ]) be two solutions of (5.1). Hence, Proposition 3.9 and Hypothesis (H.3) give Z t E|Xs − Zs |p ds 0

Z 62

p−1

Z

t

(T

p−1

fsp

+ M Ms )

s

p

 E|Xu − Zu | du ds, p

0

t ∈ [0, T ],

0

thus Gronwall’s lemma implies that X = Z in Lp ( × [0, T ]). Proof of existence. We begin an iteration procedure with X (0) ≡ Y and, for n > 1 and t ∈ [0, T ], Z

t

Xt(n) = Yt +

Z

t

F (t, s, Xs(n−1) ) ds +

0

G(t, s)H (s, Xs(n−1) ) dWs .

(5.2)

0

It is not difficult to see (using induction on n) that Proposition 3.9 and Hypothep ses (H.1)–(H.4) imply that {X (n) } is a sequence in Lprog ( × [0, T ]). By (5.2), Proposition 3.9 and (H.2)–(H.4) we obtain E|Xt(n) − Xt(n−1) |p Z

n−1

t

6 (1/(n − 1)!)Kh(t)

h(s) ds

,

t ∈ [0, T ],

(5.3)

0

with Z K=

T

E(1 + |Ys |)p ds

and

p

h(t) = 2p−1 (T p−1 ft + M p Mt ).

0 p

Note that (5.3) yields that {X (n) , n > 0} is a Cauchy sequence in Lprog( × [0, T ]). p Let X be such that X (n) converges to X in Lprog( × [0, T ]). Finally, using Propop sition 3.9, (H.2) and (H.3) again, we have that X is a solution of (5.1) in Lprog ( × [0, T ]). 2

267

ANTICIPATING INTEGRAL EQUATIONS

5.2.

VOLTERRA EQUATIONS OF THE FORWARD TYPE

Here we establish an existence and uniqueness result for anticipating integral equations of the form Z t Xt = Yt + F (t, s, Xs ) ds Z +

0 t

G(t, s)H (s, Xs ) dWs− ,

t ∈ [0, T ].

(5.4)

0

Here Y , F and H are as in Section 5.1 and G satisfies: (H.5) For each t ∈ [0, T ]: (i) G(t, ·) satisfies (H.2). 1,2,f (ii) G(t, ·) ∈ LC . (iii) There exists M ∈ L1 ([0, T ]) and a sequence Kt = {Knt , n > 1} of positive integers such that |||G(t, ·)|||Kt ,p,t < ∞ and p ||G(t, ·)||Kt ,p,t + ||(D·− G(t, ·))I[0,t ] (·)||∞ 6 Mt . With these assumptions we can state the following result p

THEOREM 5.2. Let p > 2 and Y ∈ Lprog ( × [0, T ]). Then (H.1)–(H.5) implies p that there is a unique solution in Lprog( × [0, T ]) for Equation (5.4). Proof. We only need to proceed as in the proof of Theorem 5.1 using Proposition 4.7 instead of Proposition 3.9. Acknowledgement This work was done while J. A. León was visiting the Universitat de Barcelona. He is thankful for its hospitality. References 1. 2. 3. 4. 5. 6. 7.

Alòs, E. and Nualart, D.: ‘Anticipating stochastic Volterra equations’, Stoch. Proc. Appl. 72 (1997), 73–95. Berger, M. A. and Mizel, V.: ‘An extension of the stochastic integral’, Ann. Probab. 10 (1982), 435–450. Berger, M. A. and Mizel, V.: ‘Volterra equations with Itô integrals – I, II’, J. Integral Equations 2 (1980), 187–245, 319–337. Kolodii, A. M.: ‘On the existence of solutions of stochastic Volterra equations’, Theory of Random Processes 11 (1983), 51–57. (In Russian). León, J. A. and Nualart, D.: ‘Stochastic evolution equations with random generators’, Ann. of Probab. 26 (1998), 149–186. Nualart, D. and Pardoux, E.: ‘Stochastic calculus with anticipating integrands’, Probab. Theory Rel. Fields 78 (1988), 535–581. Nualart, D.: ‘The Malliavin Calculus and Related Topics’, Springer-Verlag, 1995.

268 8. 9. 10. 11. 12. 13.

´ AND DAVID NUALART JORGE A. LEON

Pardoux, E. and Protter, P.: ‘Stochastic Volterra equations with anticipating coefficients’, Ann. Probab. 18 (1990), 1635–1655. Protter, P.: ‘Volterra equations driven by semimartingales’, Ann. Probab. 13 (1985), 519–530. Russo, F. and Vallois, P.: ‘Forward, backward and symmetric stochastic integration’, Probab. Theory Rel. Fields 97 (1993), 403–421. Sugita, H.: ‘On a characterization of the Sobolev spaces over an abstract Wiener space’, J. Math. Kyoto University 25 (1985), 717–725. Tudor. C.: ‘Some properties of mild solutions of delay stochastic evolution equations’, Stochastics 17 (1986), 1–18. Zakai, M.: ‘Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations’, Israel J. Math. 5 (1967), 170–176.