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A stochastic differential game for the inhomogeneous \(\infty \)-Laplace equation. (English) Zbl 1192.91025

The authors study a zero-sum stochastic differential game with terminal payoff \(h\in C(\overline G,\mathbb{R}\setminus\{0\})\) and running payoff \(q\in C(\partial G,\mathbb{R})\), where \(G\subset\mathbb{R}^m\) is a bounded \(C^2\) domain. The game is played until the state process exits the domain. The main result establishes a characterization of the game value as the unique viscosity solution \(u\) of the equation \(-2\Delta_\infty u= h\) in \(G\) with boundary data \(q\).

MSC:

91A15 Stochastic games, stochastic differential games
91A23 Differential games (aspects of game theory)
35J70 Degenerate elliptic equations
49L20 Dynamic programming in optimal control and differential games
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References:

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