Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.     

Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

Abstract

We consider stochastic optimal control problems in Banach spaces, related to nonlinear controlled equations with dissipative non linearities: on the nonlinear term we do not impose any growth condition. The problems are treated via the backward stochastic differential equations approach, that allows also to solve in mild sense Hamilton Jacobi Bellman equations in Banach spaces. We apply the results to controlled stochastic heat equation, in space dimension 1, with control and noise acting on a subdomain.     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

A Three-Field Finite Element Method for Elliptic Partial Differential Equations Driven by Stochastic Loads

Abstract

This paper is concerned with the application of nonconforming finite element methods to stochastic partial differential equations. We present a mixed formulation of a three-field finite element method applied to an elliptic model problem involving stochastic loads. We then derive the exact form for the expected value and variance of the solution. Additionally, the rate of convergence for the stochastic error is presented. Finally, we demonstrate through numerical experiments that the method is robust and reliable.     

Existence,uniqueness, and numerical approximations for stochastic Burgers equations

Abstract

In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.     

Stochastic Convolution-Type Heat Equations with Nonlinear Drift

Abstract

In this article, we study the solution of a class of stochastic convolution-type heat equations with nonlinear drift. For general initial condition and coefficients, we prove existence and uniqueness by using the characterization theorem and Banach's fixed-point theorem. We also give an implicit solution, which is a well-defined generalized stochastic process in a suitable distribution space. Finally, we investigate the continuous dependence of the solution on the initial data as well as the dependence on the coefficient.     

Optimal Solutions of Fractional Nonlinear Impulsive Neutral Stochastic Functional Integro-Differential Equations

Abstract

In this article, we consider a new class of fractional impulsive neutral stochastic functional integro-differential equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, fractional calculus, analytic α-resolvent operator and suitable fixed point theorems, we prove the existence of mild solutions and optimal mild solutions for these equations. Second, the existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. The results are obtained under weaker conditions in the sense of the fractional power arguments. Finally, an example is given for demonstration.     

The obstacle problem for quasilinear stochastic integral-partial differential equations

ABSTRACT

We prove the existence and uniqueness of solutions to a kind of quasilinear stochastic integral-partial differential equations with obstacles. Our method is based on the probabilistic interpretation of the solutions so that penalization method can be applied to a sequence of backward doubly stochastic differential equations with jumps. Relations between regular potentials and regular measures play an important role.     

A Stochastic Linear Quadratic Optimal Control Problem with Generalized Expectation

Abstract

In this article, we initiate a study on optimal control problem for linear stochastic differential equations with quadratic cost functionals under generalized expectation via backward stochastic differential equations.     

Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

Existence of Random Solutions of a General Class of Stochastic Functional Integral Equations

Abstract

In this paper, we establish the existence of solutions of a more general class of stochastic functional integral equations. The main tools here are the measure of noncompactness and the fixed point theorem of Darbo type. The results of this paper generalize the results of Rao–Tsokos [Rao, A.N.V.; Tsokos, C.P. A class of stochastic functional integral equations. Coll. Math. 1976, 35, 141–146.] and Szynal–Wedrychowicz [Szynal, D.; Wedrychowicz, S. On existence and an asymptotic behaviour of random solutions of a class of stochastic functional integral equations. Coll. Math. 1987, 51, 349–364.].     

The Upper and Lower Solutions Method for Stochastic Inclusions with Discontinuous Multivalued Mappings

Abstract

In this article, we consider a stochastic integral inclusion driven by semimartingale with discontinuous multivalued right hand side. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered sets. The presented studies extend some recent results both for deterministic differential inclusions and stochastic differential equations for increasing operators.     

Small time and semiclassical asymptotics for stochastic heat equations driven by Lévy noise

The paper is devoted to the study of stochastic heat equations driven by Lévy noise. Applying the WKB method, we obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients. The first step in this construction consists in solving the corresponding stochastic Hamilton-Jacobi equations which constitute the "classical part" of the semiclassical approximation. In its turn, the corresponding Hamilton-Jacobi equations can be solved via solutions of the corresponding Hamiltonian systems, which gives rise to the method of stochastic characteristics. The relevant theory of stochastic Hamiltonian systems and stochastic Hamilton-Jacobi equations was developed in our previous papers. Here we put the final rung on the ladder: stochastic Hamiltonian systems, stochastic Hamilton-Jacobi equations, stochastic heat equations.     

New approach to optimal control of stochastic Volterra integral equations

ABSTRACT

We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus.

• We give conditions under which there exist unique solutions of such equations.

• Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus.

• As an application we solve a problem of optimal consumption from a cash flow modelled by an SVIE.

     

Differential equations in spaces of abstract stochastic distributions

We develop the theory of stochastic distributions with values in a separable Hilbert space, and apply this theory to the investigation of abstract stochastic evolution equations with additive noise.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Characterization of Stochastic Viability of Any Nonsmooth Set Involving Its Generalized Contingent Curvature

Abstract

Thanks to the Stroock and Varadhan “Support Theorem” and under convenient regularity assumptions, stochastic viability problems are equivalent to invariance problems for control systems (also called tychastic viability), as it has been singled out by Doss in 1977 for instance. By the way, it is in this framework of invariance under control systems that problems of stochastic viability in mathematical finance are studied. The Invariance Theorem for control systems characterizes invariance through first‐order tangential and/or normal conditions whereas the stochastic invariance theorem characterizes invariance under second‐order tangential conditions. Doss's Theorem states that these first‐order normal conditions are equivalent to second‐order normal conditions that we expect for invariance under stochastic differential equations for smooth subsets. We extend this result to any subset by defining in an adequate way the concept of contingent curvature of a set and contingent epi‐Hessian of a function, related to the contingent curvature of its epigraph. This allows us to go one step further by characterizing functions the epigraphs of which are invariant under systems of stochastic differential equations. We shall show that they are (generalized) solutions to either a system of first‐order Hamilton‐Jacobi equations or to an equivalent system of second‐order Hamilton‐Jacobi equations.     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

Sur l'existence de solutions d'équations différentielles stochastiques progréssives rétrogrades couplées

Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamadème, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amongst others.

Unfortunately, most existence or uniqueness results on solutions of forward–backward stochastic differential equations need regularity assumptions. The coefficients are required to be at least continuous which is somehow too strong in some applications. To the best of our knowledge, our work is the first to prove existence of a solution of a forward–backward stochastic differential equation with discontinuous coefficients and degenerate diffusion coefficient where, moreover, the terminal condition is not necessary bounded.

The aim of this work is to find a solution of a certain class of forward–backward stochastic differential equations on an arbitrary finite time interval. To do so, we assume some appropriate monotonicity condition on the generator and drift coefficients of the equation.

The present paper is motivated by the attempt to remove the classical condition on continuity of coefficients, without any assumption as to the non-degeneracy of the diffusion coefficient in the forward equation.

The main idea behind this work is the approximating lemma for increasing coefficients and the comparison theorem. Our approach is inspired by recent work of Boufoussi and Ouknine, 2003, On a SDE driven by a fractional brownian motion and with monotone drift, Electronic Communications in Probability, 8, 122–134; combined with that of Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569. Pursuing this idea, we adopt a one-dimensional framework for the forward and backward equations and we assume a monotonicity property both for the drift and for the generator coefficient.

At the end of the paper we give some extensions of our result.