## Abstract

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

with the initial data

for *x* ∈ ℝ, 0 < *t* ≤ *T*, *T* < ∞. Here, 0 < *α* < 1, *ψ* : ℝ → ℝ is a continuous negative definite function of class *C*^{2}, 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*, *D _{x}*) 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 *ω* ↦ *u*(*ω*, *x*, *t*) be a random variable, which we often write just as *u*(*x*,*t*). The space *L*^{2} (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 *E*_{α, 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 *E*_{α, β}(*z*) are discussed. We now estimate the Mittag-Leffler function *E _{α,α}* (−

*ψ*(·)(

*t*−

*τ*)

^{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*|^{2}*ν*(*dy*) < ∞.We assume further that ψ is of class *C*^{2}, 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 *u* ∈ *L*^{2} (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 *u* ∈ *L*^{2} (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 *L _{t}*, i.e.

*Let u, v* ∈ *L*^{2} (Ω × ℝ; *L*^{∞}([0, *T*])) *and let f satisfy the Lipschitz conditions*

*and the linear growth condition*

*where*

*K*

_{1}

*and K*

_{2}

*are two positive constants. For*

*R*> 0

*consider*

*and assume*

*Then there exists T* > 0 *such that L _{t} is a contraction on L*

^{2}(Ω×ℝ;

*L*

^{∞}([0,

*T*]))

*and L*(

_{t}leaves B*g*,

*R*)

*invariant, i.e. u*∈

*B*(

*g*,

*R*)

*implies L*∈

_{t}u*B*(

*g*,

*R*).

*Consequently, L*(2.1)

_{t}has a unique fixed point which is a mild solution to the problem*and*(2.2).

Proof. For *u*, *v* ∈ *L*^{2} (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 *t*^{α − 1}*h*(*t*, *τ*) on every finite interval [0 *T*]. For *d* > 0 we can find *T*_{0} such that

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

For the second term in (3.8), we have

where

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, *L _{t}* is contractive.

Now, we note

and further,

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

where

and

Thus we obtain

In addition for *u* ∈ *B*(*g*, *R*) it follows

or

Eventually, we arrive at

We can make *L _{t}u* −

*g*‖

_{L2(Ω×ℝ; L∞([0, T]))}≤

*R*. □

## Acknowledgements

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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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

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