Constrained stochastic differential games with Markovian switchings and additive structure: The total expected payoff

The main objective of this work is to give conditions for the existence of Nash equilibria for a nonzero-sum constrained stochastic differential game with additive structure and Markovian switchings. In this type of game, each player is interested only in maximizing their finite-horizon total payoff when an additional cost function of the same type is required to be dominated above by another function (in particular, by a constant). The dynamic system for this game is controlled by two players and evolves according to a Markov-modulated diffusion (also known as switching diffusions or piecewise diffusion or diffusion with Markovian switchings). Given that, each player has to solve an optimization problem with constraints. The existence of a Nash equilibrium is thus proved using the Lagrange multipliers approach combined with standard dynamic programming arguments. The Lagrange approach allows the transformation of a constrained game into an unconstrained game. Therefore, this work gives conditions under which a Nash equilibrium for the unconstrained stochastic differential game is also a Nash equilibrium for the corresponding nonzero-sum constrained stochastic differential game. The theory developed here is illustrated by a pollution accumulation problem with two players. Therein, the evolution is governed by a linear stochastic differential equation with Markovian switching, and the decay pollution rate depends on a Markov chain.