Articlegreen2014

Two Person Zero-Sum Game in Weak Formulation and Path Dependent Bellman--Isaacs Equation

107 citations0/0 files storedSIAM Journal on Control and Optimization

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Game Theory and Applications

Abstract

In this paper we study a two person zero sum stochastic differential game in weak formulation. Unlike the standard literature, which uses strategy type controls, the weak formulation allows us to consider the game with control against control. We shall prove the existence of game value under natural conditions. Another main feature of the paper is that we allow for non-Markovian structure, and thus the game value is a random process. We characterize the value process as the unique viscosity solution of the corresponding path dependent Bellman--Isaacs equation, a notion recently introduced by Ekren et al. [Ann. Probab., 42 (2014), pp. 204--236] and Ekren, Touzi, and Zhang [Stochastic Process., to appear; preprint, arXiv:1210.0006v2; preprint, arXiv:1210.0007v2].

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References

2019Nonlinear Expectations and Stochastic Calculus under Uncertainty 2011Wellposedness of second order backward SDEs 2013Functional Itô calculus and stochastic integral representation of martingales 2006Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs 1995Zero-sum stochastic differential games and backward equations 2012Contract Theory in Continuous-Time Models 2008Stochastic Differential Games and Viscosity Solutions of Hamilton–Jacobi–Bellman–Isaacs Equations 2003BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations

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