Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations

We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion.     

Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay

In this paper, we prove the existence of mild solutions for a first‐order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay. We consider the cases in which the right hand side is convex or nonconvex valued. The results are obtained by using two different fixed point theorems for multivalued mappings. Copyright © 2015 John Wiley & Sons, Ltd.     

Almost automorphic solutions for stochastic differential equations driven by fractional Brownian motion

This paper concerns a class of stochastic differential equations driven by fractional Brownian motion. The existence and uniqueness of almost automorphic solutions in distribution are established provided the coefficients satisfy some suitable conditions. To illustrate the results obtained in the paper, a stochastic heat equation driven by fractional Brownian motion is considered. 1 1 The abstract section is available on the university repository site at http://math.dlut.edu.cn/info/1019/4511.htm .
     

Uniqueness and explosion time of solutions of stochastic differential equations driven by fractional Brownian motion

In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion time in stochastic differential equations driven by fractional Browmian motion with respect to Hurst parameter more than half with small diffusion.     

Optimal solutions to differential inclusions

We treat a control problem given in terms of a differential inclusion $$\dot x(t) \in E(t,x(t))$$ and develop necessary conditions for a minimum in the problem. These conditions are given in terms of certain normals to arbitrary closed sets, and require no smoothness or convexity in the problem. The results subsume related works that incorporate convexity assumptions.     

Random extremal solutions of differential inclusions

Given a Lipschitz continuous multifunction F on \({\mathbb{R}^{n}}\), we construct a probability measure on the set of all solutions to the Cauchy problem \(\dot{x}\in F(x)\) with x(0) = 0. With probability one, the derivatives of these random solutions take values within the set extF(x) of extreme points for a.e. time t. This provides an alternative approach in the analysis of solutions to differential inclusions with non-convex right hand side.     

Semiconcave solutions of partial differential inclusions

We obtain new results on the propagation of singularities for semiconcave solutions of partial differential inclusions. These results will be used to study the behavior of singularities of the value function for a reflected control problem.     

On nonoscillatory solutions of differential inclusions

This paper introduces a nonoscillatory theory for differential inclusions based on fixed point theory for multivalued maps.

     

Periodic solutions for nonconvex differential inclusions

In this paper we prove the existence of periodic solutions for differential inclusions with nonconvex-valued orientor field. Our proof is based on degree theoretic arguments.

     

Discrete approximations and periodic solutions of differential inclusions

We consider two numerical methods for solving a periodic boundary value problem for a system of differential inclusions, the Galerkin method and the polygon method. To the original problem, we assign a sequence of its discretizations. Conditions under which the existence of solutions of the periodic boundary value problem implies the solvability of its discrete versions are presented. The convergence of the sequence of approximate solutions is analyzed.     

Approximation of slow solutions to differential inclusions

To approach a viable solution of a differential inclusion, i.e., staying at any time in a closed convexK, a sufficient condition is given implying the convergence of an approximation sequence defined from the Euler or Runge-Kutta methods applied to a selection process which corresponds to the slowsolution concept. WhenK is smooth, the convergence condition is satisfied. This proves that the method is implementable on a computer for solving, for instance, differentiable equations with a noncontinuous right-hand side. Since the usual best approximation operator is difficult to implement, we introduce a class of quasi-projectors much more suitable for computation.     

On generalized-periodic solutions of nonautonomous differential inclusions

We introduce the notion of a generalized-periodic solution of a classical nonautonomous differential inclusion with periodic right-hand side. We show that the existence of a bounded solution implies the existence of a generalized-periodic solution, and vice versa.     

On local solutions of degenerate differential inclusions

We study the existence and properties of local solution sets for differential inclusions of the form (Ax)′ ∈ F(t, x), where A is a closed linear surjective operator with nontrivial null space and F is a compact set-valued mapping.     

Periodic solutions of linear impulsive differential inclusions

We establish sufficient conditions for the existence of periodic R-solutions of linear differential inclusions with impulses at fixed times. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1287–1296, September, 2008.