A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals

Abstract

In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of a doubly controlled backward stochastic differential equation. Our results extend former ones by Buckdahn, Cardaliaguet and Rainer [3] and are based on a backward stochastic differential equation approach.

Nash equilibrium payoffs for stochastic differential games with reflection

“…Recently, Lin [9,10] studied Nash equilibrium payoffs for stochastic differential games whose cost functionals are defined by doubly controlled BSDEs. Lin [9,10] generalizes the earlier result by Buckdahn, Cardaliaguet and Rainer [2]. In [9,10], the admissible control processes can depend on events occurring before the beginning of the stochastic differential game, thus, the cost functionals are not necessarily deterministic.…”

Bang–bang-type Nash equilibrium point for Markovian nonzero-sum stochastic differential game

“…One method is related to partial differential equation (PDE) theory. Some of the results show that the payoff function of the game is the unique viscosity solution of a related Hamilton-Jacobi-Bellman-Isaacs equation, e.g., [26]. Other works make use of the Sobolev theory of PDEs (see [4,5,16,27], etc.)…”

Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients

“…In these works, the objectives are various and so are the approaches, usually based on partial differential equations (PDEs) ( [11,24]) or backward SDEs ( [16,17,18,23,22]). On the other hand, it should be pointed out that the frameworks in those papers are not the same.…”

“…Some of them consider strategies as control actions for the players (e.g. [4], [23], [28]) while others deal with the control against control setting (e.g. [13], [16,17,18]).…”