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forward-backward doubly stochastic differential equation

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Differential games of partial information forward-backward doubly SDE and applications ∗
Eddie C.M. Hui, Hua Xiao
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 20 / Issue 1 / January 2014

Published online by Cambridge University Press:

10 October 2013, pp. 78-94

Print publication:

January 2014
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This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. To illustrate some possible applications, an example of linear-quadratic nonzero-sum differential games is worked out. Applying stochastic filtering techniques, we obtain an explicit expression of the equilibrium point.