TIME FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION WITH

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TIME FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION WITHRANDOM JUMPSPEIXUE WU, ZHIWEI YANG, HONG WANG, AND RENMING SONGAbstract. We study a class of variable-order time fractional stochastic differential equationswith jumps. We prove the well-posedness of this class of stochastic differential equations andstudy the regularity of their solutions. For the existence and uniqueness, we do not assumeany condition on the initial distribution and the integrability of the large jump term eventhough the solution is non-Markovian. To get moment estimates, some extra assumptionsare needed. As an application of moment estimates, we prove the Hölder regularity of thesolutions.Keywords: variable-order time fractional stochastic differential equation, stochastic Volterraequation, Lévy noise, well-posedness, moment estimates, regularity.1. Introduction1.1. Background and outline of the paper. Stochastic differential equations (abbreviatedas SDE) are used to describe phenomena such as particle movements with a random forcing,often modeled by Brownian motion noise (continuous)[13, 18, 26] or Lévy noise (jump type)[2, 30].The classical Langevin equation describes the random motion of a particle immersed in aliquid due to the interaction with the surrounding liquid molecules. Let m be the mass of theparticle, and x(t) and u(t) be the instantaneous position and velocity of the particle. ThenNewton’s equation of motion for the Brownian particle is given by the Langevin equationdu γu ξ(t).(1) LanEqdtHere γ is the friction coefficient per unit mass, and ξ(t) is the random force that accounts forthe effect of background noise and is usually described by a white noise, i.e., the correlationfunction satisfieshξ(t)i 0, hξ(t1 ), ξ(t2 )i 2Dδ(t1 t2 ).mIt is well known that (1) is equivalent to the following SDE:γ1u(t)dt dWt .mmwhere Wt is Brownian motion, and which generalizes to the general SDE:du du f (t, u(t))dt g(t, u(t))dWt .(2) ?LanEq: S(3) ?LanEq: SHowever, when the particle is immersed in viscoelastic liquids, the temporal moments of therandom force has memory effect and is often a power function of time thη(t)i 0, hη(0), η(t)i Γ(α)t α ,leading to the generalized Langevin equation [19]Z tdu K(t s)u(s)ds η(t)dt01(4) fLanEq

2PEIXUE WU, ZHIWEI YANG, HONG WANG, AND RENMING SONGwith K being the kernel, which is equivalent to the following stochastic Volterra equationdriven by fractional Brownian motion:Z tκ(t, s)u(s)ds dW H (t)(5) fLanEq: fdu 0Hwhere W (t) is a fractional Brownian motion with Hurst index H. We refer the reader to[17] for the modeling, and [9] for the well-posedness of (5).Different fractional Langevin equations were derived via the Laplace transform based on thegeneralized Langevin equation (4). For example, in [7, 22], the fractional Langevin equationof the formdu γDtα u η(t), 0 α 1.(6) ?fLanEq:edtwas derived. In [33], they similarly derived a fractional Langevin equation that assumesdifferent forms at different time scales. Namely,dudu1/2 u η(t), 0 t τ γDt u η(t), t τ(7) ?fLanEq:edtdtwhere τ is the mean collision time of the liquid molecules with the particle.Moreover, the surrounding medium of the particle may change with time, which leads to thechange of the fractional dimension of the media that in turns leads to the change of the orderof the fractional Langevin equation via the Hurst index [11, 23] and yields a variable-orderfractional Langevin equation of the formα(t)du γDtudt dW H (t)(8) ?fLanEq:eAll of the above examples focus on fractional Langevin equation with (fractional) Browniannoise. However, when the surrounding medium of the particle exhibits strong heterogeneity,the particle may experience large jumps and lead to an additional Lévy driven noise [3]. Inthis paper, we will focus on pure jump noise. Our model can be expressed as the followingvariable-order time fractional SDE driven by a multiplicative Lévy noise which also includesa large jump term:Z R α(t)e (dt, dz)du(t) λ · 0 Dt u(t) f (t, u(t)) dt g t, u(t ), z N z 1Z(9) model h t, u(t ), z N (dt, dz), u(0) u0 ,t [0, T ], z 1α(t)where T 0, λ R, Ru(t) represents friction with memory effect, f (t, u(t)) is the0 Dtexternal force and the random terms represent the jump type noise. Here the variable-orderRiemann-Liouville derivative is given byZd tu(s)R α(t)u(t) : ds,(10) fraction0 Dtdt 0 Γ(1 α(t))(t s)α(t)where α, f, g, h are functions in proper classes which make the integrals meaningful. One cansee similar definitions of fractional derivatives from [14, 15, 23, 24, 28, 32, 35, 39].As we will see in (12), our equation can be seen as a special form of stochastic Volterraequation, with no memory effect on the random noise. For the properties of general stochasticVolterra equation, with memory effect on the random noise, we refer the reader to [6, 27, 38]for white noise driven stochastic Volterra equation and [1, 4, 8, 12, 29] for Lévy driven noise.The reason why we do not consider memory effect on the random noise is that it can betreated similarly once we apply the moment inequalities, see Lemma 3.7. Also, the memoryeffect on the drift term already makes the equation essentially different from the usual SDE

TIME FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION WITH RANDOM JUMPS3because the solution will be non-Markovian. Therefore, it is not easy to argue that the equation has a unique solution with any initial distribution which is used to add the large jumpnoise.In our paper, we aim to prove that our equation has a unique solution with any initialdistribution, even though we do not have Markov property. Using this fact, with extra analysis, we show that without any condition on the large jump term, existence and uniquenessof (9) holds. According to the authors’ knowledge, this is the first time showing that fact innon-Markovian setting.Moreover, since the kernel function is given by fractional derivative, which is special inthat it has some scaling properties, we discover that the decaying speed in the approximatingprocedure of the solution is slower than the usual SDE, and it can be given explicitly by thefractional order of the derivative, see Theorem 2.2. Also, the explicit relation between the regularity of the solution and the regularity of the fractional order is also derived, see Theorem 2.5.Outline of the paper: The rest of this paper is organized as follows: In the remaining ofSection 1, we will talk about an illustrative example. Also, some preliminaries and notationswill be mentioned. In Section 2, we will formulate the assumptions and main theorems.In Section 3, we prove the well-posedness of (9). Indeed, we will prove a stronger resultProposition 3.9 which implies the well-posedness. In Section 4, we prove the moment estimatesof the solution and as an application, we prove the Hölder regularity of the solution. In Section5, we discuss some further questions, including some questions which are interesting but areoutside the scope of this paper.1.2. An illustrative example. Before we formulate our main problem, we consider thefollowing illustrative example. α(t)du(t) λ · R0 D0 u(t) f (t, u(t)) dt h(t, u(t ))δt0 (t), t [0, T ], u(0) u0 R,where t0 (0, T ) is a fixed time and δt0 is the Dirac measure supported at {t0 }. Then theabove equation can be understood as the following Volterra equation with a jump at time t0 :Zu(t) u0 λ0tu(s)ds Γ(1 α(t))(t s)α(t)ZtZ0th(s, u(s ))δt0 (ds).f (s, u(s))ds 0α(t)Due to the memory of the term Ru(t), solving the above deterministic equation is not0 Dtas easy as the memoryless case. The reason is that after the jump, the solution not onlydepends on the behavior at the jump time, but also depends on the past. Thus we need afiner piecewise analysis as below:If t t0 , the equation reduces to the Volterra equation with no jump:Zu(t) u0 λ0tu(s)ds Γ(1 α(t))(t s)α(t)Ztf (s, u(s))ds,0and the solution is denoted as v0 (t), t t0 .If t t0 , the solution is given byZu(t0 ) u0 λ0t0v0 (s)ds Γ(1 α(t))(t0 s)α(t0 )Zt0f (s, v0 (s))ds h(t0 , v0 (t0 )),0

4PEIXUE WU, ZHIWEI YANG, HONG WANG, AND RENMING SONGIf t t0 , the solution is given by the following equationZ t0Z t0v0 (s)f (s, v0 (s))ds h(t0 , v0 (t0 ))u(t) u0 λds Γ(1 α(t))(t s)α(t)00Z tZ tu(s) λf (s, u(s))ds.ds α(t)t0 Γ(1 α(t))(t s)t0If we definet0Zk(t) : u0 λ0v0 (s)ds Γ(1 α(t))(t s)α(t)Zt0f (s, v0 (s))ds h(t0 , v0 (t0 )),0then the solution for t t0 is given by the following Volterra equation:Z tZ tu(s)u(t) k(t) λf (s, u(s))ds.ds α(t)t0 Γ(1 α(t))(t s)t0(11) detexamplUnder some conditions on k(t), which means we need some conditions on α, f, h, the aboveequation (11) has a unique solution denoted as v1 (t), t t0 . Thus the global solution on [0, T ]can be given by t t0 v0 (t),u(t) v0 (t0 ) h(t0 , v0 (t0 )), t t0 v (t),t0 t T1In order to show the differences between ordinary differential equation and the variable-orderfractional differential equation, we plot 30.40.50.60.70.80.91tFigure 1. Plots of solutions: ordinary differential equation solutions (‘bluecolor’ or ‘left one’) and the variable-order fractional differential equation (‘redcolor’ or ‘right one’) with a same jump at t t0 0.5. As we can see, after thejump, the solution of the variable-order fractional differential equation dependson the past.?hplot ui?In this paper, we actually deal with a random analog of the above example. We will usea Poisson random measure to model jumps. There are two types of jumps, the first type arethe “small jumps” and there are infinitely many of them in any finite time interval. We dealwith the small jumps by compensating the jumps and the procedure is similar to the whitenoise case. For large jumps, we use the interlacing procedure, see chapter 2 of [2]. However,due to the memory term, we need a finer analysis as the above example.

TIME FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION WITH RANDOM JUMPS5 1.3. Preliminaries and notation. Let Ω, F, {Ft } 0 , P be a filtered probability space andN (dt, dz) be an Ft -adapted Poisson random measure on R Rd \{0} with N (0, ·) 0 andwithintensity measure dtν(dx), where ν is a Lévy measure which means ν{0} 0 andR(1 x 2 )ν(dx) . The compensated Poisson random measure is defined asRde (dt, dz) : N (dt, dz) ν(dz)dtNwhich is a martingale measure, see chapter 4 of [2] for the definition of martingale measure.Without loss of generality, for any t 0, Ft can be chosen as the completion and augmentationof the filtration generated by the Poisson random measure.The argument of this paper also works for Rd -valued SDEs. Since we are mainly interestedin the interplay of the memory term and the random noise, we concentrate on scalar SDEs.Denote R : [0, ). We call the mappings f : R R R, g : R R Rd \{0} R, h :R R Rd \{0} R the drift, small jump, large jump coefficients, which are measurablefunctions. To ensure the predictability of the jump coefficients, we always have the followingassumption:Assumption (A0): For fixed t 0, z Rd \{0},x 7 g(t, x, z) and x 7 h(t, x, z),are continuous functions.The above assumption ensures the predictability of the jump coefficients. For any càdlàgprocess {U (t)}t 0 , the following classes of functions will make the stochastic integral meaningful:RtR(1) L : {ϕ : R R Rd \{0} R : t 0, 0 Rd \{0} ϕ(s, U (s ), z) N (ds, dz) , a.s.}RtR(2) Lp,loc : {ϕ : R R Rd \{0} R : t 0, 0 Rd \{0} ϕ(s, U (s ), z) p ν(dz)ds , a.s.}, where 1 p 2.See chapter 4 of [2] for a complete definition of the stochastic integral when the jump coefficients are in L and Li,loc , i 1, 2. For the case 1 p 2, we use the standard truncatingtechnique ϕ ϕ1 ϕ 1 ϕ1 ϕ 1 to define the integral.For the variable-order time fractional integral operator defined by (10), if we integrate bothsides of (9), we have the following integral form:Z tZ t u(t) u0 κ(t, s)u(s)ds f s, u(s) ds00(12) integralZ tZZ tZ eg s, u(s ), z N (ds, dz) h s, u(s ), z N (ds, dz),0 z 10 z 1whereλ.Γ(1 α(t))(t s)α(t)Thus we have the following definition of (strong) solution to (9):κ(t, s) : (13) kappaDefinition 1.1. We say an Ft adapted càdlàg process {u(t)}t 0 is a strong solution of (9)if for each t 0,Z tZ tZ tZ e (ds, dz), κ(t, s)u(s) ds, f (s, u(s)) ds,g s, u(s ), z N000 z 1Z tZ h(s, u(s ), z) N (ds, dz).0 z 1are well-defined and finite P-almost surely, and (12) holds P a.s.

6PEIXUE WU, ZHIWEI YANG, HONG WANG, AND RENMING SONGWe say the strong solution of (9) is (pathwise) unique if two solutions ũ(t), u(t) with u(0) ũ(0) a.s. satisfyP(ũ(t) u(t), t 0) 0.In this paper, we always deal with strong solutions in the sense above. For simplicity, wewill only speak of solutions from now on.Notational convention: Throughout the paper, we use capital letters C1 (·), C2 (·), · · · todenote different constants in the statement of the results, the arguments inside the bracketsare the parameters the constant depends on. The lowercase letters c1 (·), c2 (·), · · · will denoteconstants used in the proof, and we do not care about their values and they may be differentfrom one appearance to another. They will start anew in every proof. The dependence ofthe integral coefficients α, λ, f, g, h will not be mentioned. Indeed, only the dependence onthe power p and time T 0 will be mentioned explicitly. g is the small jump coefficientand is defined on R R {z : 0 z 1}, we understand g : R R Rd \{0} Ras g(s, x, z)1{z:0 z 1} (z). h is the large jump coefficient and we understand it in a similarRtRmanner. The integral with respect to time 0 will always be understood as (0,t] in this paper.2. Assumptions and statement of the theoremBefore we give the formal assumptions, we introduce the definition of Lp Lipschitz continuityand Lp linear growth condition for the jump coefficients.Definition 2.1. Let p 0. We say that the small jump coefficient g : R R {z Rd :0 z 1} R is Lp Lipschitz continuous if there is a locally bounded function L(·) definedon [0, ) such that for any t 0, u R,Z g(t, u, z) g(t, ũ, z) p ν(dz) L(t) u ũ p . z 1We say that g satisfies the Lp linear growth condition if there is a locally bounded functionL(·) defined on [0, ) such that for any t 0, u R,Z g(t, u, z) p ν(dz) L(t)(1 u p ). z 12We remark that L -Lipschitz continuity and L2 -linear growth condition are the same as theusual sense. Our assumptions on the coefficients are:Assumption (A1) (Fractional order condition): The fractional order function α : R [0, 1)is a continuous function. Thus we can define its maximal functionα (t) : sup α(s) [0, 1), t 0.(14) ?def:alph0 s tAssumption (A2) (Lipschitz condition):For some p [1, 2], the small jump coefficient g is Lp -Lipschitz continuous, i.e., there is alocally bounded function L(·) defined on [0, ) such thatZ g(t, u, z) g(t, ũ, z) p ν(dz) L(t) u ũ p , t 0, u, ũ R. z 1The drift coefficient f is also Lipschitz continuous, i.e., for the same locally bounded functionL(·) above, it holds that f (t, u) f (t, ũ) L(t) u ũ ,t 0, u, ũ R.Assumption (A3) (Linear growth condition):For the same p [1, 2] as in (A2), the small jump coefficient g satisfies Lp -linear growth

TIME FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION WITH RANDOM JUMPS7condition, i.e., there is a locally bounded function L(·) defined on [0, ) such that for anyt 0, u R,Z g(t, u, z) p ν(dz) L(t)(1 u p ). z 1The drift coefficient f satisfies the linear growth condition, i.e., for the same locally boundedfunction L(·) above, it holds that f (t, u) L(t)(1 u ),t 0, u R.Under the above assumptions, the well-posedness of the solution can be established: Underthe above assumptions, we can prove the well-posedness of (9).hmain:1i Theorem2.2. If Assumptions (A1)–(A3) hold, then there exists a unique solution to (9)for any given initial distribution u0 F0 .Remark 2.3. (A2) can be weakened to non-Lipschitz coefficients as in [34]. However, the trickis also based on the classical approximating procedure, which is standard so we only considerthe Lipschitz case. The non-trivial part is due to the presence of the memory term. Unlike in[12], we do not need any condition on the large jump term and the initial distribution.If the initial distribution u0 does not have finite moment, then we can not expect the solutionto have finite moment. Thus to get Lp moment estimates for the solution to (9), first we needto assume E( u0 p ) . Apart from that, we also need some assumptions on the large jumpcoefficients. The moment estimates of the solution to (9) are as follows:hmain:2i Theorem2.4. Let u(t) be a solution to (9).Case 1: Suppose 1 p 2, u0 Lp (i.e., u0 pp : E u0 p ), the drift coefficientf satisfies the linear growth condition and the jump coefficients g, h satisfy Lp linear growthcondition, i.e., there is a locally bounded function L(·) defined on [0, ) such that for anyt 0 and u R, f (t, u) L(t)(1 u )Z g(t, u, z) p ν(dz) L(t)(1 u p ) z 1Z h(t, u, z) p ν(dz) L(t)(1 u p ) z 1Then for any T 0, we have E sup u(t) p C1 (p, T, u0 p ) ,(15) moment co0 t TwhereC1 (p, T, u0 p ) : c(p, T, u0 p )E1 α (T ),1 (c(p, T )Γ(1 α (T ))T 1 α (T )) ,and Ep,q is the Mittag-Leffler function defined in Lemma 3.5.Case 2: Suppose that p 2, and that, in addition to the assumptions in Case 1, thesmall jump coefficient g satisfies L2 linear growth condition. Then (15) holds.As an application of the moment estimates, we can establish the Hölder regularity of thesolution u(t) to (9). Hölder regularity means that for any two times t1 t2 , usually t2 t1 issmall, we have for some p 0, β 0,E u(t2 ) u(t1 ) p c(p) t2 t1 β .To establish the Hölder regularity of the solution to (9), we first need a more restrictivecondition on the fractional order α(t):

8PEIXUE WU, ZHIWEI YANG, HONG WANG, AND RENMING SONGAssumption (A1’): The fractional order function α : R (0, 1) is locally Hölder continuous with order γ 0, i.e., for any T 0, there exists C2 (T ), such that for any 0 t1 t2 T , α(t2 ) α(t1 ) C2 (T ) t2 t1 γ .hmain:3i Theorem2.5. Suppose u(t) is a solution to (9) and that for some p 1 E sup u(t) p C1 : C1 (p, T, u0 p ) .0 t TWe further assume that (A1’) holds, f satisfies the linear growth condition and g, h satisfythe Lp linear growth condition. If p 2, we also assume that g satisfies the L2 linear growthcondition. Then for any T 0, there exists C3 : C3 (p, T ) 0, C4 : C4 (p, T ) 0 such thatE u(t2 ) u(t1 ) p C3 t2 t1 C4 , 0 t1 t2 T.(16) {?}3. Existence and uniqueness of the solution3.1. Some lemmas. Before we prove the existence and uniqueness, we summarize somefrequently used results which are either easy to prove or already known. We will either givea quick proof or the reference to interested readers.Lemma 3.1. (Discrete Jesens’s inequality [21]) For any ai R and p 0,mmXXpp 1ai max{m , 1} ai p .i 1(17) ?Jesen?i 1We denote by D[0, ) the space of all càdlàg functions defined on [0, ) with values inR. The natural topology is the Skorohod topology. We will not dig into that topology andonly need the following simple lemma:?hconvi? Lemma3.2. Let f be a function on [0, ). If fn (·) is a sequence of càdlàg functions suchthat for all T 0, we havelim sup fn (t) f (t) 0,n 0 t Tthen f (·) D[0, ).Proof. Without loss of generality, we only prove the case for t 0. We would like to showthatlim f (s) f (t) and lim f (s) exists.s t,s ts t,s tWe will only prove that lims t,s t f (s) exists, because the proof of right continuity is similar.Indeed, for any ε 0 and fixed t 0, we have N : N (ε, t) 0, s

Keywords: variable-order time fractional stochastic di erential equation, stochastic Volterra equation, L evy noise, well-posedness, moment estimates, regularity. 1. Introduction 1.1. Background and outline of the paper. Stochastic di erential equations (abbreviated as SDE) are used to describe phenomena such as particle movements with a random .

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