Penalization methods for reflecting stochastic differential equations with jumps

The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J ¹ and so the methods of approximation based on the J ¹-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R⁺,?R ^d ) introduced recently by Jakubowski. The S-topology is weaker than J ¹ but stronger than the Meyer-Zheng topology and shares many useful properties with J ¹.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

A general poisson approximation theorem

We study the rate of convergence in a limit theorem due to Kabanov-Liptser-Shiryayev. We show how the probabilities P(N _t= k) can be computed from the compensator, when it is deterministic.     

A trotter product formula for random matrices

This note contains a generalization of the Trotter product formula to the setting of multiple linear systems of stochastic differential equations. From the result it follows that the solution of a system in Stratonovich form in a Lie algebra lies in the corresponding Lie group     

On the solution of stochastic ordinary differential equations via small delays

dx(t)=g(x{t))dW(t) is proved using an approximating sequence of stochastic delay equationsGeneralizations of the approximation scheme are indicated for the Stratonovich case and when the Brownian motion W is replaced by a continuous semi-martingale.     

On the estimation of nonlinear time series models

In the present paper, a framework for parametric estimation in nonlinear time series is developed. Strong consistency and asymptotic normality of minimum Hellinger distance estimates for a determined class of nonlinear models are investigated. The main Interest for these estimates is motivated by their robustness under perturbations as it has been emphazized in Beran [2]. The first part of the paper is devoted to the study of some probabilistic properties which ensure the existence and the optimal properties of the estimates