Tail estimates for positive solutions of stochastic heat equation

In this paper, we study the solution of a class of stochastic heat equations of convolution type. We give an explicit solution X _t using two basic tools: the characterization theorem for generalized functions and the convolution calculus. For positive initial condition f and coefficients processes Vt, Mt, we prove that the corresponding solution X _t admits an integral representation by a certain measure. Finally, we compute the tail estimate for the obtained solution and its expectation.     

Study of Dependence for Some Stochastic Processes

Abstract

This article is concerned with studying the following problem: Consider a multivariate stochastic process whose law is characterized in terms of some infinitesimal characteristics, such as the infinitesimal generator in case of finite Markov chains. Under what conditions imposed on these infinitesimal characteristics of this multivariate process, the univariate components of the process agree in law with given univariate stochastic processes. Thus, in a sense, we study a stochastic processe' counterpart of the stochastic dependence problem, which in case of real valued random variables is solved in terms of Sklar's theorem.     

Reflected BSDE's with discontinuous barrier and application

We deal with reflected backward stochastic differential equations with right continuous and left limited barrier. We show the existence and uniqueness of the solution and we give a comparison theorem. As an application, we study the link between such an equations with stochastic mixed control problems.     

On a McKean‐Vlasov Stochastic Integro‐differential Evolution Equation of Sobolev‐Type

Abstract

We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.     

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.     

One-Dimensional Nonlinear Stochastic Wave Equation and Triviality Effect

Abstract

The limiting behavior of solutions to stochastic wave equations with singularities represented by stochastic terms is considered. In cases when the initial data are certain functionals of the smoothed white noise process, it is proved that the triviality effect appears. At the end of the paper, a concrete application of the smoothed positive noise is given.     

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.     

Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

Abstract

A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order.     

Existence and uniqueness of solutions to stochastic heat equations with Markovian switching and Feller property

Abstract

In this paper, we focus on two-component Markov processes which consist of continuous dynamics and discrete events. Using the classical fixed point theorem for contractions to investigate the existence and uniqueness of solutions of stochastic heat equations with Markovian switching, then developing the corresponding Feller property of the solution.     

On Generalized Regular Stochastic Differential Delay Systems with Time Invariant Coefficients

Abstract

In this article, we consider the generalized linear regular stochastic differential delay system with constant coefficients and two simultaneous external differentiable and non differentiable perturbations. These kinds of systems are inherent in many application fields; among them we mention fluid dynamics, the modeling of multi body mechanisms, finance and the problem of protein folding. Using the regular Matrix Pencil theory, we decompose it into two subsystems, whose solutions are obtained as generalized processes (in the sense of distributions). Moreover, the form of the initial function is given, so the corresponding initial value problem is uniquely solvable. Finally, two illustrative applications are presented using white noise and fractional white noise, respectively.     

The density evolution of the killed McKean–Vlasov process

ABSTRACT

The density evolution of McKean–Vlasov stochastic differential equations in the presence of an absorbing boundary is analysed where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the solution of an integro-differential Fokker–Planck equation with Cauchy–Dirichlet data. This problem arises naturally within mean field game theory.     

The Mean-Variance Hedging of a Defaultable Option with Partial Information

Abstract

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q ⁰, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.     

Integral representation of generalized grey Brownian motion

ABSTRACT

In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein–Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case.     

Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces

The stochastic integrals of M- type 2 Banach valued random functions w.r.t. compensated Poisson random measures introduced in (Rüdiger, B., 2004, In: Stoch. Stoch. Rep., 76, 213–242.) are discussed for general random functions. These are used to solve stochastic integral equations driven by non Gaussian Lévy noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local Lipshitz conditions for the drift and noise coefficients on M-type 2 as well as general separable Banach spaces. The continuous dependence of the solution on the initial data as well as on the drift and noise coefficients are shown. The Markov properties for the solutions are analyzed.     

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.     

Existence of Solution of Nonlinear Neutral Stochastic Differential Inclusions with Infinite Delay

Abstract

This article is concerned with the existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay in a Hilbert Space. Sufficient conditions for the existence are obtained by using a fixed point theorem for condensing maps due to Martelli.     

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.     

Optimal investment strategy for asset-liability management under the Heston model

ABSTRACT

This paper focuses on an asset-liability management problem for an investor who can invest in a risk-free asset and a risky asset whose price process is governed by the Heston model. The objective of the investor is to find an optimal investment strategy to maximize the expected exponential utility of the surplus process. By using the stochastic control method and variable change techniques, we obtain a closed-form solution of the corresponding Hamilton–Jacobi–Bellman equation. We also develop a verification theorem without the usual Lipschitz assumptions which can ensure that this closed-form solution is indeed the value function and then derive the optimal investment strategy explicitly. Finally, we provide numerical examples to show how the main parameters of the model affect the optimal investment strategy.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II

This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181–204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an ω-wise stochastic viscosity solution, which will lead to the uniqueness results. As the byproducts we extend the measurable section theorem of Dellacherie and Meyer (1978), and a fundamental lemma of Crandall et al. (1992)     

Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Abstract

In the paper we compute the explicit form of the fractional chaos decomposition of the solution of a fractional stochastic bilinear equation with the drift in the fractional chaos of order one and initial condition in a finite fractional chaos. The large deviations principle is also obtained for the one-dimensional distributions of the solution of the equation perturbed by a small noise.