Multivalued Stochastic Differential Equations Research Articles

Multivalued Impulsive SDEs Driven by G-Brownian Noise: Periodic Averaging Result

This paper aims to study two approximation theorems in view of the periodic averaging results for non-Lipschitz multivalued stochastic differential equations with impulses and G-Brownian motion (MISDEGs). By adopting G-Itô’s formula and non-Lipschitz condition, the solutions to the simplified MSDEGs without impulses may replace those of the initial MISDEGs in view of approximation in L 2 -sense and capacity. Finally, we bring a couple of two examples to enhance our theoretical results.

Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations

In this paper we derive error estimates of the backward Euler–Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler–Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525–589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.

Large deviations for invariant measures of multivalued stochastic differential equations

In this paper, the problem of the large deviations for the invariant measures of the multivalued stochastic differential equations is considered. Under the assumptions of diffusion coefficient being non-Lipschitz and elliptic, we establish the large deviation principle for the invariant measures of the solutions to the multivalued stochastic differential equations. The proof is based on the work of large deviations and invariant measures for the solutions to the multivalued stochastic differential equations.

Lp-Variational solutions of multivalued backward stochastic differential equations

We prove the existence and uniqueness of the Lp-variational solution, with p > 1, of the following multivalued backward stochastic differential equation with p-integrable data: {−dYt + ∂yΨ(t,Yt)dQt∋H(t,Yt,Zt)dQt−ZtdBt,0≤t<τ, Yτ = η, where τ is a stopping time, Q is a progressively measurable increasing continuous stochastic process and ∂yΨ is the subdifferential of the convex lower semicontinuous function y↦Ψ(t, y). In the framework of [14] (the case p ≥ 2), the strong solution found it there is the unique variational solution, via the uniqueness property proved in the present article.

Multi-valued backward stochastic differential equations with regime switching

The paper considers a class of multi-valued backward stochastic differential equations with subdifferential of a lower semi-continuous convex function with regime switching, whose generator is a continuous-time Markov chain with a finite state space. Firstly, we get the existence and uniqueness of the solution by the penalization method. Secondly, we prove that the solution of the original system is weakly convergent. Finally, we give an application to the homogenization of a class of multi-valued PDEs with Markov chain.

Moderate deviation principle for multivalued stochastic differential equations

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

Optimal Mild Solutions for a Class of Nonlocal Multi-Valued Stochastic Delay Differential Equations

In this paper, we introduce a new class of impulsive multi-valued stochastic delay differential equations with nonlocal conditions in separable Hilbert spaces. Using stochastic analysis, analytic semigroup, fractional powers of closed operators and suitable fixed point theorems, we prove the existence of optimal mild solutions for these systems in the $$\alpha $$ -norm. An example is provided to illustrate the applicability of our results.

On Uniform Large Deviations Principle for Multi-valued SDEs via the Viscosity Solution Approach

This paper deals with the uniform large deviations for multivalued stochastic differential equations (MSDEs for short) by applying a stability result of the viscosity solutions of second order Hamilton-Jacobi-Belleman equations with multivalued operators. Moreover, the large deviation principle is uniform in time and in starting point.

On approximations of the Euler-Peano scheme for multivalued stochastic differential equations

ABSTRACTIt is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. Krylov's conditions put forward in On Kolmogorov's equations for finite-dimensional diffusions, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math., 1715, Springer, Berlin, 1999, pp. 1–63). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.

Stochastic averaging principles for multi-valued stochastic differential equations driven by poisson point Processes

ABSTRACTThe purpose of this article is to investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes. The solutions to MSDEs driven by Poisson point processes can be approximated by solutions to averaged MSDEs in the sense of both convergence in mean square and convergence in probability. Finally, an example is presented to illustrate the averaging principle.

Strong convergence rate for multivalued stochastic differential equations via stochastic theta method

In this paper we study the stochastic theta method for multivalued stochastic differential equations driven by standard Brownian motions and obtain the strong convergence rate of this numerical scheme.

L^{p}(pgeq2)-strong convergence in averaging principle for multivalued stochastic differential equation with non-Lipschitz coefficients

We investigate the averaging principle for multivalued stochastic differential equations (MSDEs) driven by a random process under non-Lipschitz conditions. We consider the convergence of solutions in L^{p}~(pgeq2) and in probability between the MSDEs and the corresponding averaged MSDEs.

Multivalued stochastic delay differential equations and related stochastic control problems

We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type):Specify that in this case the coefficients at time t depends also on previous values of X(t) through Y (t) and Z(t). Also X is constrained with the help of a bounded variation feedback law K to stay in the convex set .Afterwards we consider optimal control problems where the state X is a solution of a controlled delay stochastic system as above. We establish the dynamic programming principle for the value function and finally we prove that the value function is a viscosity solution for a suitable Hamilton-Jacobi-Bellman type equation.

Multivalued monotone stochastic differential equations with jumps

We study multivalued stochastic differential equations (MSDEs) with maximal monotone operators driven by semimartingales with jumps. We discuss in detail some methods of approximation of solutions of MSDEs based on discretization of processes and Yosida approximation of the monotone operator. We also study the general problem of stability of solutions of MSDEs with respect to the convergence of driving semimartingales.

Exponential ergodicity for non-Lipschitz multivalued stochastic differential equations with Lévy jumps

We study the exponential ergodicity of diffusions generated by a multivalued stochastic differential equation with Lévy jumps when the coefficients are non-Lipschitz continuous by proving that the transition semigroup is strongly Feller and irreducible, and that it admits a unique invariant measure. This is obtained through an [Formula: see text]-convergence result, Girsanov’s theorem, coupling method combined and a stopping argument.

English

We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded.

Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems

ABSTRACTIn this paper, we prove the existence and uniqueness of a solution for a class of multi-valued stochastic differential equations driven by G-Brownian motion (MSDEG) by means of the Yosida approximation method. Moreover, we set up an optimality principle of stochastic control problem and prove the value function of the control problem is the unique viscosity solution of a class of nonlinear partial differential variational inequalities.

Convergence of invariant measures for multivalued stochastic differential equations

This article is concerned with the weak convergence of invariant measures associated with multivalued stochastic differential equations in the finite dimensional space.

An Averaging Principle for Multivalued Stochastic Differential Equations

The averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated. An averaged MSDEs for the original MSDEs is proposed, and their solutions are quantitatively compared. Under suitable assumptions, it is shown that the solution of the MSDEs converges to that of the original MSDEs in the sense of mean square and also in probability. Two examples are presented to illustrate the averaging principle.

Well-posedness of Stratonovich multi-valued SDEs driven by semimartingales

In this paper we consider Stratonovich type multi-valued stochastic differential equations (MSDEs) driven by general semimartingales. Based on an existence and uniqueness result for MSDEs with respect to continuous semimartingales, we apply the random time change and approximation technique to prove existence and uniqueness of solutions to Stratonovich type multi-valued SDEs driven by general semimartingales with summable jumps.