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 ℱ0 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 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 ω ↦ u(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(·)(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|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 u ∈ L2 (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 ∈ L2 (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, v ∈ L2 (Ω × ℝ; L∞([0, T])) and let f satisfy the Lipschitz conditions
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 Ltu ∈ B(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, v ∈ L2 (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
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, Lt 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
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
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