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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.

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