Elżbieta Z. Ferenstein

Randomized stopping games and Markov market games

We study nonzero-sum stopping games with randomized stopping strategies. The existence of Nash equilibrium and ɛ-equilibrium strategies are discussed under various assumptions on players random payoffs and utility functions dependent on the observed discrete time Markov process. Then we will present a model of a market game in which randomized stopping times are involved. The model is a mixture of a stochastic game and stopping game.

On Randomized Stopping Games

The paper is concerned with two-person nonzero-sum stopping games in which pairs of randomized stopping times are game strategies. For a general form of reward functions, existence of Nash equilibrium strategies is proved under some restrictions for three types of games: quasi-finite-horizon, random-horizon and infinite-horizon games.

Trajectory Controllability of Semilinear Systems with Delay in Control and State

In this paper we consider the finite-dimensional dynamical control system described by scalar semilinear ordinary differential state equation with variable delay. The semilinear state equation contains both pure linear part and nonlinear perturbation. We extend the concept of controllability on trajectory controllability for systems with point delay in control and in nonlinear term. Moreover, we present remarks and comments on the relationships between different concepts of controllability. Finally we propose the possible extensions.

Optimal Stopping of a Risk Process with Disruption and Interest Rates

It is a standard approach in classical risk theory to assume a claim process which does not change throughout the whole observation period. Most commonly, encountered models deal with compound Poisson processes. It would be beneficial to investigate more general classes of claim processes with arbitrary distributions of random variables governing inter-occurrence times between losses and loss severities. Further generalization of such framework would be a model allowing for disruptions i.e. changes of such distributions according to some unobservable random variables, representing fluctuating environmental conditions. The question of providing the company with tools allowing for detection of such change and maximizing the returns leads to an optimal stopping problem which we solve explicitly to some extent. Moreover, we provide references to previously examined models as well as numerical examples emphasizing the efficiency of the suggested method.

On the Reducibility of the Discrete Linear Time-Varying Systems

For the discrete linear time-varying systems we present basic facts and definitions concerning the Lyapunov transformation, kinematic similarity and reducibility in the context of stability and Lyapunov exponents theory. Moreover, the paper contains the original result giving the necessary and sufficient conditions for the reducibility of a system to system with identity matrix.

ON OPTIMAL STOPPING OF RISK PROCESSES WITH REGIME SWITCHING

Abstract In the paper we solve a problem of optimal stopping of a risk process in two alternative settings. We assume that the main characteristics of the risk process change according to unobservable random variable. In the first model we assume that the post-disorder distributions are not known a’priori and are randomly chosen from a finite set of admissible distributions. The second model concentrates on a situation when more than one disorder is possible. For both models optimal stopping rules with respect to given utility function are constructed using dynamic programming methodology.

Nash Equilibrium in a Game Version of the Elfving Problem.

Multi-person stopping games with a finite and infinite horizon, players’ priorities, and observed rewards at jump times of a Poisson process are considered. The existence of a Nash equilibrium is proved and its explicit form is obtained for special classes of reward sequences. Our game is a generalization of the Elfving stopping time problem to the case of many players and modification of multi-person stopping games with priorities.