On a class of stochastic differential equations arising from the stochastic approximation theory

The paper dealt with generalized stochastic approximation procedures of Robbins-Monro type. We consider these procedures as strong solutions of some stochastic differential equations with respect to semimartingales and investigate their almost sure convergence and mean square convergence     

Concentration inequalities for stochastic differential equations of pure non-Poissonian jumps

We provide concentration inequalities for solutions to stochastic differential equations of pure not-necessarily Poissonian jumps. Our proofs are based on transportation cost inequalities for square integrable functionals of point processes with stochastic intensity and elements of stochastic calculus with respect to semi-martingales. We apply the general results to solutions of stochastic differential equations driven by renewal and non-linear Hawkes point processes.     

On-line solution of a stochastic pursuit-evasion game

In this paper, the concepts of prior-commitment and delayed-commitment strategies for zero-sum, linear-quadratic differential games with noise-corrupted measurements are applied to a detailed example of a pursuit-evasion game. Quasilinearization is used to solve the nonlinear two-point boundary-value problem of the prior-commitment game. A closed form solution is obtained for the delayed-commitment strategies. Comparison of the payoff functionals confirms the relationships discussed in Ref. 1.     

On the existence of the stabilizing solution of generalized Riccati equations arising in zero-sum stochastic difference games: the time-varying case

In this paper, a large class of time-varying Riccati equations arising in stochastic dynamic games is considered. The problem of the existence and uniqueness of some globally defined solution, namely the bounded and stabilizing solution, is investigated. As an application of the obtained existence results, we address in a second step the problem of infinite-horizon zero-sum two players linear quadratic (LQ) dynamic game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such an optimal control problem, a crucial role is played by the unique bounded and stabilizing solution of the considered class of generalized Riccati equations.     

Fejér processes in theory and practice: Recent results

In this paper we briefly survey the recent results of the theory of Fejér mappings and processes as applied to solving various mathematical problems, including structured systems of linear and convex inequalities, operator equations, as well as problems of linear and quadratic programming which are not necessarily solvable (improper ones).     

Two unsolved problems in the stability theory of stochastic differential equations with delay

Two unsolved problems of the stability theory for stochastic differential equations with delay are offered for consideration.     

Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion

In this work, we establish a new concept of pseudo almost periodic processes in p-th mean sense using the measure theory. We use the μ-ergodic process to define the spaces of μ-pseudo almost periodic process in the p-th mean sense. We establish many interesting results on the functional space of such processes like completeness and composition theorems. The main objective of this paper is to use those results and some stochastic analysis approaches to study the existence, the uniqueness and the global attractiveness for a μ-pseudo almost periodic mild solution to a class of abstract stochastic evolution equations driven by fractional Brownian motion. We provide an example to illustrate our results.