Nash equilibrium payoffs for stochastic differential games with reflection

Abstract

In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.

Nash Equilibrium Payoffs for Stochastic Differential Games with two Reflecting Barriers

“…Recently, Buckdahn et al [3] studied Nash equilibrium payoffs for stochastic differential games with linear cost functionals. Lin [11], [12] generalizes the earlier result in [3]. In Lin [11], [12], 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.…”

“…Lin [11], [12] generalizes the earlier result in [3]. In Lin [11], [12], 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. Moreover, the cost functionals are defined with the help of BSDEs, and, thus, they are nonlinear.…”

“…By means of the arguments [5] and [12], we can obtain the following theorem. We omit the proof here.…”

“…in Dynkin games and mixed zero-sum games. We shall study Nash equilibrium payoffs for nonzero-sum stochastic different 356 Q. LIN games, which are different from the earlier ones [3], [11], and [12]. We shall also study Nash equilibrium payoffs for nonzero-sum stochastic differential games with more general running cost functionals, which are defined with the help of RBSDEs with two reflecting barriers.…”

“…In comparison with [11] and [12], this paper has the following improvements and advantages. First, the cost functionals of both players are defined by BSDEs without reflecting barriers as in [11] and BSDEs with one reflecting barriers as in [12]. In this paper, the cost functionals of both players are defined by BSDEs with two reflecting barriers.…”