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Publicly Available Published by De Gruyter March 9, 2016

Existence and Uniqueness for a Class of Stochastic Time Fractional Space Pseudo-Differential Equations

  • Ke Hu EMAIL logo , Niels Jacob and Chenggui Yuan


Some classes of stochastic fractional differential equations with respect to time and a pseudo-differential equation with respect to space are investigated. Using estimates for the Mittag-Leffler functions and a fixed point theorem, existence and uniqueness of mild solutions of the equations under consideration are established.

1. Introduction

The theory of the fractional calculus has been developed for more than 300 years, we highlight the contributions of Leibniz, Liouville, Riemann and Weyl, just to name a few. There exists an extensive literature in this area, we here only mention Awawdeh, Rawashdeh and Jaradat [1], Gorenflo and Mainardi [3], Hilfer [5], Mainardi [13], Metzler and Klafter [15] and Podlubny [16] among others. Moreover, many mathematicians (see [11], [13] and [16]) have paid attention to the Riemann-Liouville fractional derivative and fractional differential equations composed by Riemann-Liouville fractional derivatives, i.e. equations of a form such as


for example under the initial data


The main tool for solving such equations is to use the Laplace transform.

On the other hand, it is well-known that many phenomena in various branches of science and industry can be modelled by stochastic differential equations (SDEs). Recently, attempts were made to combine fractional derivatives and SDEs. For example, El-Borai [2] has investigated the existence and uniqueness of stochastic fractional integro-differential equations. In this paper, motivated by the previous references, we will study the existence and uniqueness for a class of stochastic fractional differential equations. Our results are different from [2], since we use the Riemann-Liouville fractional derivatives.

In Section 2 we deal with auxiliary results, and in Section 3 we prove the existence and uniqueness for a class of stochastic fractional differential equations.

2. Preliminaries

Throughout this paper, let (Ω, , P) be a complete probability space with a filtration ﹛t≥0 satisfying the usual conditions, i.e. it is right continuous and 0 contains all P-null sets. B(t) is a standard Brownian motion defined on (Ω, , P). We consider a stochastic fractional differential equation of the form


with the initial data


for x ∈ ℝ, 0 < tT, T < ∞. Here, 0 < α < 1, ψ : ℝ → ℝ is a continuous negative definite function of class C2, satisfying the growth condition (2.11), and


is a pseudo-differential operator with symbol p ℝ × ℝ → ℝ. For the details on Itô’s stochastic integral, we refer the reader to [12]. We will discuss later on certain properties which we have to impose on the symbol p(x, ξ). In this paragraph, however, the only property needed is that p(x, Dx) satisfies the estimate


where μ is independent of t, and


We refer to [7, Remark 3.2], where suitable conditions for (2.4) to hold were discussed.

Let ωu(ω, x, t) be a random variable, which we often write just as u(x,t). The space L2 (Ω × ℝ; L([0, T])) defined by


is a Banach space equipped with norm


Comparing [16, Example 4.3, p.140], (2.1) can be rewritten as:


and the equality holds P–almost everywhere for which we will write P–a.e. Here by Eα, β(z) we denote the (2-indices) Mittag-Leffler function defined by


This is an entire function and we refer to [11], [13] or [16] where basic properties of Eα, β(z) are discussed. We now estimate the Mittag-Leffler function Eα,α (−ψ(·)(tτ)α). For this we need the following theorem which can be found in[16, Section 1.2, p.16]. Let ψ : ℝ → ℝ be a continuous negative definite function. In fact we assume that ψ has the representation


with a Lévy measure ν satisfying ℝ\﹛0﹜ 1 ∧ |y|2ν(dy) < ∞.We assume further that ψ is of class C2, note that ψ(ξ) ≥ 0, then for l ≤ 2,


Moreover, we require a lower bound for ψ, i.e.


for some γ ∈ (1, 2) and all ξ ∈ ℝ. We refer to [8]-[10] where examples of continuous negative definite functions are given. Using the table of Bernstein functions provided in [18], we can easily construct many examples of type f(|ξ|2 where f is suitable Bernstein function.

3. Existence and Uniqueness

We start with a definition. A random function uL2 (Ω × ℝ; L([0, T])) is called a mild solution of the problem (2.1) and (2.2), if


Let us introduce the operator


It follows that uL2 (Ω × ℝ; L([0, T])) is a mild solution to (2.1) and (2.2) if and only if u is a fixed point of Lt, i.e.


Let u, vL2 (Ω × ℝ; L([0, T])) and let f satisfy the Lipschitz conditions

and the linear growth condition
whereK1and K2are two positive constants. ForR > 0 consider
and assume

Then there exists T > 0 such that Lt is a contraction on L2 (Ω×ℝ; L([0, T])) and Lt leaves B(g, R) invariant, i.e. u B(g, R) implies LtuB(g, R). Consequently, Lt has a unique fixed point which is a mild solution to the problem(2.1)and(2.2).

Proof. For u, vL2 (Ω × ℝ; L([0, T])) we have


where the * means convolution. In (3.8), we estimate the first term


Here we have for α > 0


By Lemma 2.2 and Corollary 2.2 in [7], compare also with [6], we have already proven the integrability properties of h(t, τ) and th˜(t,τ):=tα1h(t,τ)tα − 1h(t, τ) on every finite interval [0 T]. For d > 0 we can find T0 such that


Now, by using Burkholder-Davis-Gundy inequality, compare [12], we arrive at


For the second term in (3.8), we have




Using (2.4), we find


which implies that


Combining (3.10) and (3.12), we have


In (3.13), let T be sufficiently small such that


is less than or equal to 1. Under this condition, Lt is contractive.

Now, we note


and further,


By using the linear growth condition (3.5), we have






Thus we obtain


In addition for uB(g, R) it follows




Eventually, we arrive at


We can make C1+C15R4+C25R4+C3R4+R4R, i.e. 4C135C15C2C3R and we find ‖LtugL2(Ω×ℝ; L([0, T]))R. □


The authors would like to thank the editors and the referees for detailed comments and valuable suggestions that helped to improve the paper significantly.


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  1. Please cite to this paper as published in:

    Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 56–68, DOI: 10.1515/fca-20l6-0004

Received: 2015-4-12
Published Online: 2016-3-9
Published in Print: 2016-2-1

© 2016 Diogenes Co., Sofia

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