The Existence of Value in Stochastic Differential Games

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Abstract
Using the techniques of Davis and Varaiya [3], [4] a two-person zero sum differential game is considered, whose dynamics are interpreted using the Girsanov measure transformation method. If the Isaacs condition holds it is shown that the upper and lower values of the game are equal and there is a saddle point in feedback strategies. The central point of the mathematics is that analogues of the time derivative and gradient of the upper value function are constructed using martingale methods; because the Hamiltonian satisfies a saddle condition at each point these then also give the lower value.