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

• Ke Hu , Niels Jacob and Chenggui Yuan

## 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

(1.1)0Dtαu(x,t)+λu(x,t)=g(x,t),for0<α<1,

for example under the initial data

(1.2)0Dtα1u(x,0)=0.

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

(2.1)0Dtαu(x,t)+υ(Dx)u(x,t)+p(x,Dx)u(x,t)0tf(u(x,θ))dB(θ)=g(x,t),

with the initial data

(2.2)

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

(2.3)p(x,Dx)υ(x)=(2π)1/2eixξp(x,ξ)υ^(ξ)dξ

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

(2.4)p(x,Dx)u(,t)L2μu(,t)L2,u(,t)L2(),

where μ is independent of t, and

(2.5)u(,t)L22=|u(x,t)|2dx.

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

(2.6)L2(Ω×;L([0,T])):={u:Ω×L([0,T])|(Esup0tT|u(x,t)|2dx)1/2<}

is a Banach space equipped with norm

(2.7)uL2(Ω×;L([0,T]))=(Esup0tT|u(x,t)|2dx)1/2⋅.

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

(2.8)u(x,t)=(2π)1/20t(tτ)α1g(xy,τ)(F1(Eα,α(υ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α1(0τf(u(xy,θ))dB(θ))×(F1(Eα,α(υ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α1(p(x,Dx)u(xy,τ))×(F1(Eα,α(υ()(tτ)α)))(y)dτdy,

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

(2.9)Eα,β(z):=n=0znΓ(αn+β)

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

(2.10)ψ(ξ)=\{0}(1cosyξ)ν(dy)

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,

|dlυ(ξ)dξl|{υ(ξ),l=0;C1υ1/2(ξ),l=1;C2,l=2.

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

(2.11)υ(ξ)Cυ(1+|ξ|2)γ/2

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

(3.1)u(x,t)=(2π)1/20t(tτ)α1g(xy,τ)(F1(Eα,α(υ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α1(0τf(u(xy,θ))dB(θ))×(F1(Eα,α(υ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α1(p(x,Dx)u(xy,τ))×(F1(Eα,α(υ()(tτ)α)))(y)dτdy,P-a.e.

Let us introduce the operator

(3.2)Ltu(x,t)=(2π)1/20t(tτ)α1g(xy,τ)(F1(Eα,α(υ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α1(0τf(u(xy,θ))dB(θ))×(F1(Eα,α(υ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α1(p(x,Dx)u(xy,τ))×(F1(Eα,α(υ()(tτ)α)))(y)dτdy,P-a.e.

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.

(3.3)Ltu=u.
THEOREM 3.1

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

(3.4)f(u)f(v)L2(Ω×;L)[0,T]))K1uvL2(Ω×;L)[0,T])),
and the linear growth condition
(3.5)f(u)L2(Ω×;L)[0,T]))K2(1+uL2(Ω×;L)[0,T]))),
whereK1and K2are two positive constants. ForR > 0 consider
(3.6)B(g,R):={vL2(Ω×;L([0,T]))|vgL2(Ω×;L([0,T]))R}
and assume
(3.7)gL2(Ω×;L([0,T]))R4.

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

(3.8)=(2π)1/20t(tτ)α10τ(f(u(xy,θ))f(v(xy,θ)))dB(θ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α1(p(x,Dx)(u(xy,τ)v(xy,τ)))×(F1(Eα,α(ψ()(tτ)α)))(y)dτdyL2(Ω×;L([0,T]))(2π)1/20t(tτ)α10τ(f(u(xy,θ))f(v(xy,θ)))dB(θ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdyL2(Ω×;L([0,T]))+(2π)1/20t(tτ)α1(p(x,Dx)(u(xy,τ)v(xy,τ)))×(F1(Eα,α(ψ()(tτ)α)))(y)dτdyL2(Ω×;L([0,T]))(2π)1/20t(tτ)α1(0τ(f(u(,θ))f(v(,θ)))dB(θ))*(F1(Eα,α(ψ()(tτ)α)))(x)dτL2(Ω×;L([0,T]))+(2π)1/20t(tτ)α1(p(x,Dx)(u(,τ)v(,τ)))[4pt]*(F1(Eα,α(ψ()(tτ)α)))(x)dτL2(Ω×;L([0,T])),

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

(3.9)0t(tτ)α1(0τ(f(u(,θ))f(v(,θ))dB(θ))*(F1(Eα,α(ψ()(tτ)α)))(x)dτdyL2(Ω×;L([0,T]))[4pt]={Esup0tT(0t(tτ)α1(0τ(f(u(,θ))f(v(,θ)))dB(θ))*(F1(Eα,α(ψ()(tτ)α)))(x)dτ)2dx}1/2{E[(sup0tT0t(tτ)α1(0τ(f(u(,θ))f(v(,θ)))dB(θ))*(F1(Eα,α(ψ()(tτ)α)))(x)dτ)2dx}1/2={E[(sup0tT0t(tτ)α1(0τ(f(u(,θ))f(v(,θ)))dB(θ))*(F1(Eα,α(ψ()(tτ)α)))(x)dτ)2dx)1/2]2}1/2{E[(sup0tT0t(tτ)α10τ(f(u(,θ))f(v(,θ)))dB(θ)L2×(F1(Eα,α(ψ()(tτ)α)))(x)dτL1)2}1/2={E(sup0tT0t(tτ)α1h(t,τ)×0τ(f(u(,θ))f(v(,θ)))dB(θ)L2dτ)2}1/2

Here we have for α > 0

h(t,τ):=F1(Eα,α(ψ()(tτ)α))L1.

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

dsup0tT00t(tτ)α1h(t,τ)dτk<1.

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

(3.10)0t(tτ)α1(0τ(f(u(,θ))f(v(,θ)))dB(θ))*(F1(Eα,α(ψ()(tτ)α)))(x)dτdyL2(Ω×;L([0,T])){E(sup0tT0t(tτ)α1h(t,τ)×sup0tT0τ(f(u(,θ))f(v(,θ)))dB(θ)L2dτ)2}1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)×{E(sup0τT0τ(f(u(,θ))f(v(,θ)))dB(θ)L2)2}1/2=(κ1α(γ1)Tα(γ1)/γ+κ2αTα)×{Esup0τT(0τ(f(u(,θ))f(v(,θ)))dB(θ))2dx}1/24(κ1α(γ1)Tα(γ1)/γ+κ2αTα)×{E0τ(f(u(,θ))f(v(,θ)))2dθdx}1/24T1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)×{Esup0tT(f(u(,))f(v(,)))2dx}1/24T1/2K1(κ1α(γ1)Tα(γ1)/γ+κ2αTα)uvL2(Ω×;L([0,T])).

For the second term in (3.8), we have

(3.11)0t(tτ)α1(p(x,Dx)(u(,τ)v(,τ)))*(F1(Eα,α(ψ()(tτ)α)))(x)dτL2(Ω×;L([0,T]))={Esup0tT(0t(tτ)α1(p(x,Dx)(u(,τ)v(,τ)))*(F1(Eα,α(ψ()(tτ)α)))(x)dτ)2dx}1/2{Esup0tT(0t(tτ)α1(p(x,Dx)(u(,τ)v(,τ)))*(F1(Eα,α(ψ()(tτ)α)))(x)dτ)2dx}1/2{E(sup0tT0t(tτ)α1h(t,τ)p(x,Dx)(u(,τ)v(,τ))L2dτ)2}1/2,

where

h(t,τ):=F1(Eα,α(ψ()(tτ)α))L1.

Using (2.4), we find

p(x,Dx)(u(,τ)v(,τ))L2μu(,τ)v(,τ)L2

which implies that

(3.12)0t(tτ)α1(p(x,Dx)(u(,τ)v(,τ)))*(F1(Eα,α(ψ()(tτ)α)))(x)dτL2(Ω×;L([0,T])){E(sup0tT0t(tτ)α1h(t,τ)×sup0tTp(x,Dx)(u(,t)v(,t))L2dτ)2}1/2μ(κ1α(γ1)Tα(γ1)/γ+κ2αTα)uvL2(Ω×;L([0,T])).

Combining (3.10) and (3.12), we have

(3.13)LtuLtvL2(Ω×;L([0,T]))(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)×(4K1T1/2+μ)uvL2(Ω×;L([0,T])).

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

(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)(4K1T1/2+μ)

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

Now, we note

(3.14)LtuL2(Ω×;L([0,T]))=(2π)1/20t(tτ)α1g(xy,τ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdy+(2π)1/20t(tτ)α10τf(u(xy,θ))dB(θ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdy++(2π)1/20t(tτ)α1p(x,Dx)u(xy,τ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdyL2(Ω×;L([0,T]))(2π)1/20t(tτ)α1g(xy,τ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdyL2(Ω×;L([0,T]))+(2π)1/20t(tτ)α10τf(u(xy,θ))dB(θ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdyL2(Ω×;L([0,T]))+(2π)1/20t(tτ)α1p(x,Dx)u(xy,τ)×(F1(Eα,α(ψ()(tτ)α)))(y)dτdyL2(Ω×;L([0,T])),

and further,

(3.15)LtuL2(Ω×;L([0,T]))=(2π)1/20t(tτ)α1g(,τ)*(F1(Eα,α(ψ()(tτ)α)))(x)dτdyL2(Ω×;L([0,T]))+(2π)1/20t(tτ)α1(0τf(u(,θ))dB(θ))*(F1(Eα,α(ψ()(tτ)α)))(x)dτdyL2(Ω×;L([0,T]))+(2π)1/20t(tτ)α1p(x,Dx)u(,τ)*(F1(Eα,α(ψ()(tτ)α)))(x)dτdyL2(Ω×;L([0,T]))4(2π)1/2T1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)f(u(,t))L2(Ω×;L([0,T]))+μ(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)uL2(Ω×;L([0,T]))+(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)gL2(Ω×;L([0,T])).

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

(3.16)LtuL2(Ω×;L([0,T]))4(2π)1/2T1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)K2(1+uL2(Ω×;L([0,T])))+μ(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)uL2(Ω×;L([0,T]))+(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)gL2(Ω×;L([0,T]))=C1+C1uL2(Ω×;L([0,T]))+C2uL2(Ω×;L([0,T]))+C3gL2(Ω×;L([0,T])),

where

(3.17)C1:=4(2π)1/2T1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)K2,
(3.18)C2:=μ(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα),

and

(3.19)C3:=(2π)1/2(κ1α(γ1)Tα(γ1)/γ+κ2αTα)

Thus we obtain

(3.20)LtugL2(Ω×;L([0,T]))LtuL2(Ω×n;L([0,T]))+gL2(Ω×;L([0,T]))C1+C1uL2(Ω×;L([0,T]))+C2uL2(Ω×;L([0,T]))+C3gL2(Ω×;L([0,T]))+gL2(Ω×;L([0,T])).

In addition for uB(g, R) it follows

(3.21)uL2(Ω×;L([0,T]))gL2(Ω×;L([0,T]))ugL2(Ω×;L([0,T]))R,

or

(3.22)uL2(Ω×;L([0,T]))5R4.

Eventually, we arrive at

(3.23)LtugL2(Ω×;L([0,T]))C1+C15R4+C25R4+C3R4+R4.

We can make C1+C15R4+C25R4+C3R4+R4R, i.e. 4C135C15C2C3R and we find ‖LtugL2(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.

## References

[1] Awawdeh, A, Rawashdeh, EA, Jaradat, HM, Analytic solution of fractional integro-differential equations. Annals of the University of Craiova, Mathematics and Computer Science Series 38 2011 1–10.Search in Google Scholar

[2] El-Borai, MM, On some stochastic fractional integro-differential equations Adv Dyn Syst Appl 1 2006 49–57.Search in Google Scholar

[3] Gorenflo, R, Mainardi, F, Random walk models for space-fractional diffusion processes Fract Calc Appl Anal 1 1998 167–191.10.1016/S0378-4371(99)00082-5Search in Google Scholar

[4] Gorenflo, R, Vivoli, A, Mainardi, FDiscrete and continuous random walk models for space-time fractional diffusion Nonlinear Dyn 38 2004 101–106.10.1007/s11071-004-3749-5Search in Google Scholar

[5] Hilfer, R, Threefold introduction to fractional derivatives. In: Klages, R, Radons, G, I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008), 17–74.10.1002/9783527622979.ch2Search in Google Scholar

[6] Hu, K, On an Equation Being a Fractional Differential Equation with Respect to Time and a Pseudo-differential Equation with Respect to Space Related to Lévy-Processes. Ph.D. Thesis, Swansea University, 2012.10.2478/s13540-012-0009-0Search in Google Scholar

[7] Hu, K, Jacob, N, Yuan, C, On an equation being a fractional differential equation processes. FractCalcand ApplAnal15 (2012), 128–140; DOI: 10.2478/sl3540-Ol2-0009-0; http://www.degruyter.com/view/j/fca.2012.15.issue-1/issue-files/fca.2012.15. issue-l.xml.10.2478/sl3540-Ol2-0009-0Search in Google Scholar

[8] Jacob, N, Pseudo-differential Operators and Markov ProcessesVol. : Fourier Analysis and Semigroups. Imperial College Press, London (2001).10.1142/p245Search in Google Scholar

[9] Jacob, N, Pseudo-differential Operators and Markov ProcessesVolII: Generators and Their Potential Theory. Imperial College Press, London (2002).10.1142/p264Search in Google Scholar

[10] Jacob, N, Pseudo-differential Operators and Markov ProcessesVolIII: Markov Processes and Applications. Imperial College Press, London (2005).10.1142/p395Search in Google Scholar

[11] Kilbas, AA, Srivastava, HM, Trujillo, JJ, Theory and Applications of Fractional Differential Equations. North Holland Mathematics Studies 204, Elsevier, Amsterdam (2006).Search in Google Scholar

[12] Mao, X, Stochastic Differential Equations and Applications, 2nd Ed. Horwood Publishing Ltd. (2008).10.1533/9780857099402Search in Google Scholar

[13] Mainardi, F, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010).10.1142/p614Search in Google Scholar

[14] Meerschaert, MM, Nane, E, Vellaisamy, P, Fractional Cauchy problems on bounded domains. AnnProb37 (2009), 979–1007.10.1214/08-AOP426Search in Google Scholar

[15] Metzler, R, Klafter, J, The Random walk’s guide to anomalous diffusion: a fractional dynamical approach. PhysReports339 (2000), 1–77.10.1016/S0370-1573(00)00070-3Search in Google Scholar

[16] Podlubny, I, Fractional Differential Equations. Mathematics in Science and Engineering 198, Academic Press (1998).Search in Google Scholar

[17] Samko, SG, Kilbas, AA, Marichev, OI, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993).Search in Google Scholar

[18] Schilling, RL, Song, RM, Z. Vondraček, Bernstein Functions: Theory and Applications. De Gruyter Studies in Mathematics, De Gruyter, Berlin (2010).Search in Google Scholar

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