Optimal stopping for measure-valued piecewise deterministic Markov processes

This paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.

Looking forward approach with random horizon in cooperative differential games

In the paper authors present a new approach to determination and computation of a solution for differential games with prescribed duration in the case when players lack certain information about the dynamical system and payoff function on the whole time interval on which the game is played. At each time instant players receive information about dynamical system and payoff functions, however the duration of the period of this information is unknown and can be represented as a random variable with known parameters. At certain periods of time the information is updated. A novel solution is described as a combination of imputation sets in the truncated subgames that are analyzed using Looking Forward Approach with random horizon. A resource extraction game serves as an illustration in order to compare a cooperative trajectory, imputations, and imputation distribution procedure in the game with Looking Forward Approach and in the original game with prescribed duration. Looking Forward Approach is used for constructing game theoretical models and defining solutions for conflict-controlled processes where information about the process updates dynamically.

Two-player zero-sum stochastic differential games with random horizon

We consider a two-player zero-sum stochastic differential game with a random planning horizon and diffusive state variable dynamics. The random planning horizon is a function of a non-negative continuous random variable, which is assumed to be independent of the Brownian motion driving the state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial differential equation of Hamilton–Jacobi–Bellman–Isaacs type.