Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Abstract

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.     

Stochastic reaction diffusion equations on an infinite interval with reflection

We consider a time evolution of random fields with non-negative values on the real line. Such evolution is described by an infinite dimensional stochastic differential equation of Skorokhod's type, which is a stochastic partial differential equation (SPDE) of parabolic type with reflection. We shall show the existence of the solution, and its uniqueness when the diffusion coefficient is constant.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

Stationary Conjugation of Flows for Parabolic SPDEs with Multiplicative Noise and Some Applications

Abstract

The purpose of this paper is to transform a nonlinear stochastic partial differential equation of parabolic type with multiplicative noise into a random partial differential equation by using a bijective random process. A stationary conjugation is constructed, which is of interest for asymptotic problems. The conjugation is used here to prove the existence of the stochastic flow, the perfect cocycle property and the existence of the random attractor, all nontrivial properties in the case of multiplicative noise.     

A novel approach for stochastic solutions of wick-type stochastic time-fractional Benjamin–Bona–Mahony equation for modeling long surface gravity waves of small amplitude

Objectives: In the paper, two new reliable analytical methods have been devised for getting new exact analytical solutions of wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Moreover, the Hermite transform and inverse Hermite transform have been utilized for converting fractional stochastic differential equation to deterministic fractional partial differential equation and vice versa respectively. Here for reducing fractional partial differential equations (FPDE) to the ordinary differential equation (ODE), fractional complex transform has been utilized.

Methods: The authors have used a newly proposed method and Kudryshov method for getting the solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation.

Results: By using two reliable methods, here, the authors find the new exact solutions for the governing equations.

Conclusion: Two new approaches to find solutions of the aforementioned equation have been established. Also, the new exact solutions have been obtained for stochastic differential equation by using two methods.     

Estimation for Translation of a Process Driven by Fractional Brownian Motion

Abstract

We investigate the general problem of estimating the translation of a stochastic process governed by a stochastic differential equation driven by a fractional Brownian motion. The special case of the Ornstein-Uhlenbeck process is discussed in particular.     

An Optimal Control Problem Associated with SDEs Driven by Lévy-Type Processes

Abstract

In this article, we consider an optimal control problem associated with jump type stochastic differential equations driven by Lévy-type processes. The problem arises from portfolio optimization for the pair of the wealth process and the cumulative consumption process in (incomplete) financial market models. We establish the existence and the uniqueness of (constrained) viscosity solutions to the associated the integro-differential Hamilton–Jacobi–Bellman equation.     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

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.     

A Note on the Solution of a Stochastic Partial Differential Equation

Abstract

A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.     

Computational scheme for solving nonlinear fractional stochastic differential equations with delay

Abstract

This paper studies the numerical solution of fractional stochastic delay differential equations driven by Brownian motion. The proposed algorithm is based on linear B-spline interpolation. The convergence and the numerical performance of the method are analyzed. The technique is adopted for determining the statistical indicators of stochastic responses of fractional Langevin and Mackey-Glass models with stochastic excitations.     

Elements of stochastic analysis for the case of Grassmann variables. III. Correlation functions

The present paper continues earlier studies [1,2], in which analogs were proposed in the case of Grassmann variables for concepts such as classical stochastic analysis, stochastic integrals, random processes, and stochastic partial differential equations and their solutions. This was done for the special case when the classical objects are functionals of a so-called smoothed Wiener process onR ₊×R ^v . In the present paper, the correlation functions of the solution of a stochastic partial differential equation are studied together with some applications.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 323–335, December, 1993.     

Necessary conditions for optimality in relaxed stochastic control problems

In this paper, we are concerned with optimal control problems where the system is driven by a stochastic differential equation of the Ito type. We study the relaxed model for which an optimal solution exists. This is an extension of the initial control problem, where admissible controls are measure valued processes. Using Ekeland's variational principle and some stability properties of the corresponding state equation and adjoint processes, we establish necessary conditions for optimality satisfied by an optimal relaxed control. This is the first version of the stochastic maximum principle that covers relaxed controls.     

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.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

On a general result for backward stochastic differential equations

The existence of the solution of a general infinite dimensional backward stochastic differential equation is discussed. In our setting, we generalize many works concerning the existence problem (by a new approach).     

The Bismut-Elworthy formula for backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces

We consider a forward-backward system of stochastic evolution equations in a Hilbert space. Under nondegeneracy assumptions on the diffusion coefficient (that may be nonconstant) we prove an analogue of the well-known Bismut-Elworthy formula. Next, we consider a nonlinear version of the Kolmogorov equation, i.e. a deterministic quasilinear equation associated to the system according to Pardoux, E and Peng, S. (1992). "Backward stochastic differential equations and quasilinear parabolic partial differential equations". In: Rozowskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and Their Applications , Lecture Notes in Control Inf. Sci., Vol. 176, pp. 200-217. Springer: Berlin. The Bismut-Elworthy formula is applied to prove smoothing effect, i.e. to prove existence and uniqueness of a solution which is differentiable with respect to the space variable, even if the initial datum and (some) coefficients of the equation are not. The results are then applied to the Hamilton-Jacobi-Bellman equation of stochastic optimal control. This way we are able to characterize optimal controls by feedback laws for a class of infinite-dimensional control systems, including in particular the stochastic heat equation with state-dependent diffusion coefficient.