TY - JOUR

T1 - Constrained stochastic differential games with Markovian switchings and additive structure

T2 - The total expected payoff

AU - Escobedo-Trujillo, Beatris Adriana

AU - López-Barrientos, José Daniel

AU - Garrido, Javier

AU - Colorado-Garrido, Darío

AU - Herrera-Romero, José Vidal

N1 - Publisher Copyright:
© 2023 The Author(s)

PY - 2023/9/1

Y1 - 2023/9/1

N2 - 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.

AB - 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.

KW - Constrained Nash equilibria

KW - Controlled Markov chains

KW - Dynamic programming

KW - Hybrid systems

KW - Total payoff criteria

UR - http://www.scopus.com/inward/record.url?scp=85171690089&partnerID=8YFLogxK

U2 - 10.1016/j.rico.2023.100288

DO - 10.1016/j.rico.2023.100288

M3 - Artículo

AN - SCOPUS:85171690089

SN - 2666-7207

VL - 12

JO - Results in Control and Optimization

JF - Results in Control and Optimization

M1 - 100288

ER -