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Stochastic Volterra Equation Driven by Wiener Process and Fractional Brownian Motion
For a mixed stochastic Volterra equation driven by Wiener process and fractional Brownian motion with Hurst parameter , we prove an existence and uniqueness result for this equation under suitable assumptions.
In recent years, there has been considerable interest in studying fractional Brownian motion (fBm) due to some of its compact properties such as self-similarity, stationary increments, Hölder's continuity, and long-range dependence (when ), and also due to its applications in various scientific areas including telecommunications, turbulence, image processing, finance, and other fields. It is a suitable generalization of standard Wiener process. Some surveys and complete literatures could be found in Biagini et al. , Hu , Mishura , Nualart , and Zähle . The so-called fBm with index is a zero mean Gaussian process with the covariance:
Since is neither a semimartingale nor a Markov process unless , the classical Itô theory is not available when dealing with it. There are several approaches to define an integral with respect to fBm. One possibility is Skorokhod or divergence integral introduced in the fractional Brownian setting in Decreusefond and Üstünel . We will use a pathwise integral, defined first in Zähle , for fBm with as a Young integral. The aim of this paper is to study the following mixed stochastic Volterra equation on : where , , , is an -dimensional standard Wiener process, and is an -dimensional fBm with . We assume that the processes and are independent, the integral with respect to is Itô type, and the integral with respect to is a pathwise Riemann-Stieltjes integral in the sense of Zähle .
The mixed stochastic differential equation: was first considered in Kubilius , where unique solvability was proved in one-dimensional case for time-independent coefficients and zero drift; that is, . Later, in Zähle , existence of a solution to (3) was proved under less restrictive assumptions, but only locally, that is, up to a random time. In Guerra and Nualart , global existence and uniqueness of solution to (3) was established under the assumption that and are independent. The latter result was obtained in Mishura and Shevchenko [10, 11] without the independence assumptions, and it was also shown in Mishura and Shevchenko  that all moments of the solution are finite for . The motivation to consider such equations comes from some financial applications, where Wiener process as a model is inappropriate because of the lack of memory, and fBm with is too smooth. A model driven by both processes is free of such drawbacks. Stochastic Volterra equations are also studied by many authors (see, e.g., [12–16]). In this paper, we consider the mixed stochastic Volterra equations driven by both Wiener process and fBm, and also give an existence and uniqueness result for this equation under suitable assumptions.
This paper is organized as follows. Section 2 contains some preliminaries on function spaces and lists our assumptions on the coefficients of (2). It also states the main results in this section. In Section 3, we will give some estimates of fractional and Itô integrals, based on the fractional calculus. Finally, in Section 4, we will prove Theorem 1 using the classical Yamada and Watanabe theorem [17, 18] to check the pathwise uniqueness property and the existence of weak solutions of (2).
2. Preliminaries and Main Results
Let be a complete probability space equipped with a filtration satisfying standard assumptions. Denote an -dimensional fBm with and an -dimensional standard Wiener process, independent of . is a -dimensional random variable independent of (). For each , we denote by the -field generated by the random variables and the -null sets. denotes the bigger filtrations such that is right-continuous and contains the -null sets, , are -measurable, and is a -Wiener process.
Let us now introduce some function spaces for later use. Leting and , for any measurable function , we introduce the following notation: Denote by the space of measurable function such that For any , let be space of -Hölder continuous functions such that where . Given any , we have the following inclusions: Notice that both the fBm with and the standard Wiener process have their trajectories in .
In what follows, we will assume the following standard hypotheses. Throughout the paper, the symbol will denote a generic constant, whose value is not significant and can change from one line to another. To emphasize its dependence on some parameters, we will put them into subscripts consider the following. is a measurable function such that there exists with , and for all there exists such that for all and . is a measurable function and there exists a constant such that for all and . is a measurable function. Moreover, there exist the derivatives , , and constants , and such that the following properties hold: for all , , .
Denote by the conditional expectation given , that is, given and . A strong solution of (2) is a -dimensional -adapted stochastic process such that a.s. the trajectories of belong to and , which satisfies (2) a.s. The main result of our paper is the following theorem on the existence and uniqueness of a solution for (2).
Theorem 1. Suppose that the coefficients , and satisfy the assumptions , , and with , , respectively. If and , then (2) has a unique strong solution .
Notice that in all our results we can replace the fBm with an arbitrary stochastic process with Hölder continuous trajectories of order .
3. Integral Estimates
In this section, we will first briefly recall some basic definitions of fractional integrals and derivatives and define the integral with respect to fBm as a generalized Stieltjes integral (see, e.g., Nualart and Răşcanu  and Zähle ). We will also give some estimates of this integral.
Let and let . Consider the left-sided and right-sided fractional Riemann-Liouville integrals of of order : if the integrals exist for almost all respectively, where denotes the gamma function defined by Let (resp., ) be the image of , by the operator (resp., ). If (resp., ) then the Weyl derivatives of : are defined for almost all (the convergence of the integrals at the singularity holds pointwise for almost all if and moreover in -sense if ), respectively.
Let and let (supposing that the limits exist and are finite) and define Assuming that , , exists, and for some , , , the generalized Stieltjes Integral is defined as The following properties hold. (i) If , under the preceding assumptions we have and we can write (ii) If and with , then the generalized Stieltjes integral exists; it is given by (16) and coincides with the Riemann-Stieltjes integral.
The linear spaces are Banach spaces with respect to the norms: and the same is true for . If then the norms on and are equivalent and if , then Fix a parameter ; denote by the space of measurable functions such that Clearly, for all . Moreover, if belongs to , we define We also denote by the space of measurable functions such that Then, if and , the integral exists for all and we have Furthermore, the following estimate holds: Given such that for any , , we can also consider the integral: and the estimate:
Now we will derive some useful estimates for the integrals involved in (2). Fix a parameter . Consider first the ordinary Lesbesgue integral. Given a measurable function we define
Proposition 2. Assume that satisfies with and . Let be a measurable function. If , then , for all , one has
Proof. It follows from Proposition 2.1 in Besalú and Rovira  and the growth assumption in that as required.
Proposition 3. Assume that satisfies with and . Let be measurable functions. If , then for all , one has
Proof. By Proposition 2.1 in Besalú and Rovira  and the growth assumption in , one obtains
Given a measurable function , let us define Notice that if , one has .
Proposition 4. Assume that satisfies with , , then for all and we have
Proof. It follows from Proposition 3.1 in Besalú and Rovira  and the growth assumption in that
Proposition 5. Assume that satisfies with , , then for all one has where
Proof. Following Proposition 3.1 in Besalú and Rovira  and the assumptions in , we get that
By Lemma A.1 of Besalú and Rovira , the hypothesis imply that for all , , , one has Thus, we obtain that This completes the proof.
Finally, we will consider the estimate of Itô integral based on the Itô calculus. Let be a measurable function and define
Proposition 6. Let be a -adapted stochastic process. If and a.s., then for all a.e., one has
Proof. It follows from Lemma 3.7 in Guerra and Nualart  and the growth assumption in .
Proposition 7. Let be -adapted stochastic processes. If and , a.s., then for all a.e., one has
Proof. By Lemma 3.7 in Guerra and Nualart  and the Lipschitz assumption in , one has which completes the proof.
4. Proof of the Theorem 1
To prove Theorem 1, we will use the classical Yamada and Watanabe theorem [17, 18], which asserts that pathwise uniqueness and existence of weak solutions imply the existence of a strong solution. A weak solution of (2) in this paper is a triple , , such that:(i) is a complete probability space, is right-continuous, and contains the P-null sets, (ii) is a fBm which is -measurable, (iii) is a --dimensional Wiener process, (iv) is -adapted and has the trajectories belong to and a.s, (v) satisfies (2) a.s.
The pathwise uniqueness property holds for (2) means that if and are two weak solutions, defined on the same probability space with the same filtration and a.s., then .
Let . By Propositions 2.2 and 3.2 in Besalú and Rovira , the sample paths of the integral processes and are continuously differentiable and -Hölder continuous of order , respectively. Therefore, if is a weak solution of (2), then the trajectories of are -Hölder continuous for all . Consequently, the pathwise uniqueness property holds for (2). It is exactly the same result as that of Theorem 4.4 given in Guerra and Nualart . So it remains to prove that the existence of weak solutions for (2).
Let us now introduce the Euler approximation for (2). Take a sequence of partitions of the interval such that as . Defined as and for all , where . We first prove the tightness of the law of the sequence in the space of -Hölder continuous functions, with , such that Using the criterion given by Hamadouche , is sufficient to prove the following lemma.
Lemma 8. For any , there exists a random variable , depending on and , such that for all and .
Proof. Firstly, we will show that there exists a random variable such that
for all and . In fact, from (46), we have
Using the estimate (28) and Hölder inequality, we get that
By Burkhölder-Davis-Gundy and Hölder inequalities, we obtain
Similarly, we get
Now applying Fubini’s theorem and using the growth assumption in , we have
As a consequence,
Further, applying the estimate (33) and Hölder inequality, we get
Combining these estimates, we have
Therefore, by the generalized Gronwall lemma (Lemma 7.6 in Nualart and Răşcanu ), the estimate (49) is satisfied.
So it remains to prove that the estimate (48) is satisfied. Indeed, from (46), we have It follows from the growth assumption in (), the estimate (49), and Hölder inequality that By Burkhölder-Davis-Gundy and Hölder inequalities and using the estimate (49), we obtain Finally, using the estimates (24) and (49), we get Summing up, we deduce the desired result.
Let , , be the sequence of probability measure induced by on , then the sequence is tight (a similar result to Proposition 5.2 in Guerra and Nualart ).
Theorem 9 (existence of weak solution). Suppose that the coefficients , , and satisfy the assumptions , , and with , , , respectively. If and , then (2) has a unique weak solution.
The authors would like to thank the referees for valuable comments and suggestions on this paper. The works was supported by the NSFC (11171062, 11201062, 40901241), the Innovation Program of Shanghai Municipal Education Commission (12ZZ063), the Research Project of Education of Zhejiang Province (Y201326507), the Natural Science Foundation of Zhejiang Province (Y5090377), the Key Natural Science Foundation of Anhui Educational Committee (KJ2013A133), Natural Science Foundation of Anhui Province (1308085QA14), and the Fundamental Research Funds for the Central Universities.
F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2006.
D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, Germany, 2nd edition, 2006.View at: MathSciNet
Y. S. Mishura and G. M. Shevchenko, “Existence and uniqueness of the solution of stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index ,” Communications in Statistics, vol. 40, no. 19-20, pp. 3492–3508, 2011.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Mishura and G. Shevchenko, “Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3217–3227, 2012.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet