VOLTERRA EQUATIONS WITH FRACTIONAL STOCHASTIC INTEGRALS

We assume that a probability space (Ω,η,P) is given, where Ω denotes the space C(R+, Rk) equipped with the topology of uniform convergence on compact sets, η the Borel σ-field of Ω, and P a probability measure on Ω. Let {Wt(ω)= ω(t), t ≥ 0} be a Wiener process. For any t ≥ 0, we define ηt = σ{ω(s); s < t}∨Z, where Z denotes the class of the elements in ηt which have zero P-measure. Pardoux and Protter discussed the existence and uniqueness of the solution of the stochastic integral equation of the form


Introduction
We assume that a probability space (Ω,η,P) is given, where Ω denotes the space C(R + , R k ) equipped with the topology of uniform convergence on compact sets, η the Borel σ-field of Ω, and P a probability measure on Ω.
Let {W t (ω) = ω(t), t ≥ 0} be a Wiener process.For any t ≥ 0, we define η t = σ{ω(s); s < t} ∨ Z, where Z denotes the class of the elements in η t which have zero P-measure.
Pardoux and Protter discussed the existence and uniqueness of the solution of the stochastic integral equation of the form whose solution {X t } should be R d -valued and η t adapted process; {H t } is an R p -valued (see [16]).It is supposed that F maps {t, s; In the present work, we study the existence, uniqueness, and continuity of the solution of the fractional stochastic integral equation of the form where 0 < β < 1, I β t F(t,s,X s ), the fractional integral of F(t,s,X s ), is defined by (see [22]) (1. 3) The fractional Wiener process W β t G i (H t ;t,s,X s ) of G i (H t ;t,s,X s ) is defined by (see [7]) (1.4) Stochastic Volterra equations have been studied in several papers (see [2,3,4,6,16,17,18]).In this work, we will use the Skorohod integral (see [9,10,11,12,13,14,15,19,20]) to interpret (1.2) as in [16] since the integrands in the stochastic integrals are not adapted; therefore we cannot use, as usual, the Ito integral to interpret the equation.In Section 2, we state some results concerning the Skorohod integral, which will be used later together with the precise interpretation of (1.2).In Sections 3 and 4, we prove the existence and uniqueness of a solution to (1.2) in two steps.In Section 5, we establish, under additional assumptions, the existence of almost surely (a.s.) continuous modification of the solution process.In Section 6, we show the continuity of the solution of (1.2).Equation (1.2) has many important financial applications.These systems arise if we consider the fractional analog of a portfolio (see [1]).The fractional Black-Scholes market consists of a bank account or a bond and a stock.The price process A t of the bond at time t is given by where r(s) ≥ 0, s ∈ [0,t], is the interest rate.A portfolio is a pair (u t ,v t ) of random variables for fixed t ∈ [0,T].The price X t of the stock could be governed by a fractional Volterra equation of the form (1.6) Here the drift µ ≥ 0 and volatility σ > 0 are continuous functions on [0, T].The numbers u t and v t are the bond and stock units, respectively (held by an investor).Hence, the corresponding value process is The process V t could be governed by the equation (1.8) Mahmoud M. El-Borai et al. 455

The Skorohod integral
We will now define the Skorohod integral.Most of this section is a review of some basic notations and a few results from [4,16].
For h ∈ L 2 (R + ;R k ), we denote the Wiener integral by

.2)
Let A denote the dense subset of L 2 (Ω,η,P) consisting of those classes of random variables of the form where n ∈ N (N denotes the set of nonnegative integers), b is the set of infinitely differentiable functions on [0, b] whose derivatives of any order are null at b.If F has the form (2.3), we define its derivative in the direction i as the process {D i t F; t ≥ 0} defined by (2.4) i be the closure of A with respect to the norm where is the mathematical expectation of X, and Similarly, the domain D 1,2 = k i=1 D 1,2 i is the closure of A with respect to the norm (see [10]) (2.7) We identify D i and D with their closed extensions (D 1,2 is the domain of D : We denote by D 1,2 i,loc the set of measurable F's which are such that there exists a sequence i with the two properties For F ∈ D 1,2 i,loc , we define without ambiguity loc is defined similarly. For i = 1,...,k, we define δ i , the Skorohod integral with respect to W i t , as the adjoint of D i , that is, Domδ i (the set of adapted processes for the Skorohod integral) is the set of u ∈ L 2 (Ω × R + ) which are such that there exists a constant c with i for any T > 0 and we can write The Skorohod integral is a local operation on i,loc denote the set of measurable processes u which are such that, for any T > 0, there exists a sequence For u ∈ L 1,2 i,loc , we can define its Skorohod integral with respect to W i t by i , and L 1,2 loc is defined similarly as L 1,2 i,loc .We now introduce the particular class of integrands which we will use below.Let u : R + × Ω × R P → R satisfy the following.
Let θ be a p-dimensional random vector such that We fix T > 0 and consider Define, moreover, v t = u(t,θ).Under conditions (i), (ii), (iii), and (iv), the following proposition holds.It is proved in [10] that The same relation is proved in [16] under slightly different conditions.Equation (2.15) is used to define the Skorohod integral T 0 u(t,θ)dW i t .Proposition 2.2.The random field {I i (x); x ∈ R p } defined above possesses an a.s.continuous modification so that the random variable I i (θ) can be defined, v ∈ Domδ i (see [10,16]).Condition (iv) can be replaced by (iv ): θ j ∈ D 1,2  i,loc , j = 1,..., p.Under conditions (i), (ii), (iii), and (iv ), v ∈ (Dom δ i ) loc in the sense that there exists a sequence Indeed, let {(Ω n ,θ n )} be a localizing sequence for θ in (D 1,2 i ) p , and let It is then natural to define, for every >0, the Skorohod integral t by formula (2.15) and the latter coincides with t exists in the mean by using the norm • 2 .

Statement of the problem: interpretation of (1.2)
Our aim is to study the equation We define D = {(t, s) ∈ R 2 + ; 0 ≤ s < t}.The coefficients F and G are given as follows: {H t } is a given progressively measurable p-dimensional process.It will follow from these hypotheses that we will be able to construct a progressively measurable solution {X t }.
Therefore, for each t, the process {G i (h;t,s,X s ); s ∈ [0,T]} is of the form v s = u(s,θ) with u(s,h) = G i (h;t,s,X s ) and θ = H t .We will impose below conditions on G, {H t }, and the solution {X t } so as to satisfy requirements (i), (ii), (iii), and (iv ) of Section 2.
In particular, we will consider only nonanticipating solutions.Therefore, the stochastic integrals in (3.1) will be interpreted according to (2.15), that is, In other words, we can rewrite (3.1) as where and the stochastic integrals are now the usual Itô integrals.We will show below that (3.3) makes sense for any progressively measurable process X which satisfies X ∈ t>0 L q (0,t) a.s., for some q > p.We will find such a solution to (3.3); it will then follow from (3.2) that it is a solution to (3.1).Similarly, uniqueness for (3.1) in the above class will follow from uniqueness for (3.3) in that class.

Existence and uniqueness under strong hypotheses
We formulate a set of further hypotheses (those stated in Section 3 are assumed to hold throughout the paper) under which we will establish a first result of the existence and uniqueness of a solution of (1.2).
It is easy to show, using, in particular, (H4) and Lebesgue's dominated convergence theorem, that the mapping Since q > p, we can infer from Sobolev's embedding theorem (see [13]) that It then follows from the Burkholder-Gundy inequality that where we have used (H4) and B is bounded.From (4.5) and (4.10), the proof is complete.
Mahmoud M. El-Borai et al. 461 Lemma 4.3.For any 0 < t ≤ T, there exists a constant c > 0 such that We are now in a position to prove the main result of this section.
Taking the limit as n → ∞, we find that (4.18) leads to γ t = 0.
To prove the existence, we define a sequence {X n t , 0 ≤ t ≤ T, n = 0,1,2,...} as follows: Using Lemma 4.2, we can see that It then follows from Lemma 4.3 that where By using a similar argument, we can write The last estimation implies that X n is a Cauchy sequence in L q prog(Ω × [0,T]).Then there exists X such that X n → X in t>0 L q prog(Ω × [0,T]), and again using Lemma 4.3, we can pass to the limit in (4.20), yielding that X solves (4.12).

An existence and uniqueness result under weaker assumptions
We formulate a new set of weaker hypotheses.
We assume that there exists an increasing progressively measurable process {U t , t ≥ 0} with values in R + such that (H3 Finally, we suppose that for any N > 0, there exists an increasing progressively measurable process Let, again, q be a fixed real number, with q > p, and set 1 − β = α/q, 0 < α < 1.We have the following theorem. Theorem 5.1.Equation (3.1) has a unique solution in the class of progressively measurable processes which satisfy X ∈ t>0 L q (0,t) a.s. ( Note that H t (ω) = H n t (ω) a.s. on Ω T n , for all t ∈ [0,T], where Ω T n ↑ Ω a.s. as n → ∞.We, moreover, define X n 0 = X 0 1 {|X0|≤n} , S n = inf{t; sup s<t |D s H n t | ∨ V n t ≥ n}.We consider the equation where (5.7) It is easy to see that theorem (4.15) applies to (5.6).Define (5.8) Proof.We need to show only that whenever X ∈ q>1 t>0 L q (0,t) a.s., {J t (X); t > 0} has an a.s.continuous modification.(a) We first show that t → t 0 F(t,s,X s )ds is a.s.continuous.Note that (H.6), (H.7), (H.8), and (H.9) imply that for all (s,x) ∈ R + × R d , t → F(t,s,x) is a.s.continuous on (s,+∞).

E 1 + 2 ( 6 . 9 )
X s q (r − s) −α/2 − (t − s) −α/2 ds ≤ cq |t − r| δq r 0 (r − s) −α/2 ds + r 0 (r − s) −α/2 − (t − s) −α/2 ds ≤ C q |t − r| δq r 1−α/2 + r 1−α/2 + (t − r) 1−α/2 − t 1−α/Mahmoud M. El-Borai et al. 467 which from the above estimate yields E I t (X,h) − I r (X,k) Proof.(a)We first see how (3.1) makes sense if X ∈ t>0 L q (0,t) a.s.That is, we have to show that for fixed t > 0, .s. continuous modification on {|h| ≤ N}.Since this is true for any n and N, and ∪ n {τ n ≥ t} = Ω a.s., the result follows.(b)Existence:we want to show existence on an arbitrary interval [0,T] (T will be fixed below).Let {H n ; n ∈ N} denote a progressively measurable localizing sequence for H in (L 1,2 ) p on [0, T].Since, from (H.3 ), sup t≤T |H t | is a.s.finite, we can and do assume, without loss of generality, that .2) Let first {t n ; n ∈ N} be a sequence such that t n < t for any n and t n → t as n → ∞; then and the latter tends a.s. to 0 as n → ∞. tn 0 F t,s,X s − F t n ,s,X s ds ≤ t 0 F t,s,X s − F t n ,s,X s ds(6.4)whichtends to 0 as n → ∞.A similar argument gives the same result when t n > t, t n → t.(b)We next show that t → I t (X,H t ) possesses an a.s.continuous modification.