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Nash equilibrium payoffs for nonzero-sum stochastic differential games. (English) Zbl 1101.91010
From the text: Existence and characterization of Nash equilibrium payoffs are proved for stochastic nonzero-sum differential games.
The dynamic \(X^{t,x,u,v}\) of the controlled system is the solution of the following stochastic differential equation: \[ dX_s=f(s,X_s,u_s,v_s)\,ds+\sigma(s,X_s,u_s,v_s)\,dB_s \text{ for } s\in [t,T], \text{ and } X_s=x \text{ for } s\leq t \] in which the functions \(f\) and \(\sigma\) are bounded and Lipschitz in \(x\) uniformly in \((t,u,v)\). Here \(u:=(u_s)_{s\in [t,T]}\) and \(v:=(v_s)_{s\in [t,T]}\) are the control actions of the two agents \(c_1\) and \(c_2\). The interventions of the players generate payoffs, \(J_1(t,x,u,v)=E[g_1(X^{t,x,u,v}_T)]\) for \(c_1\) and \(J_2(t,x,u,v)=E[g_2(X^{t,x,u,v}_T)]\) for \(c_2\); the functions \(g_1\) and \(g_2\) are taken to be bounded and Lipschitz. A Nash equilibrium payoff for this game is a pair \((e_1,e_2)\) which can be \(\epsilon\)-nearly reached by \((u^\epsilon, v^\epsilon)\), an admissible strategy for \(c_1\) and \(c_2\). In addition if one of the players deviates from his strategy (\(u^\epsilon\) or \(v^\epsilon\)) then he is penalized.

91A23 Differential games (aspects of game theory)
49N10 Linear-quadratic optimal control problems
91A15 Stochastic games, stochastic differential games
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