Brownian games : uniqueness and regularity issues

This paper extends the analysis of the dual Brownian game [Gamma]*(x,T) initiated in [2]. The existence of a value [psi]*(x,T) for [Gamma](x,T) as well as the existence of optimal strategies was proved there. In this paper we will prove successively that player 2’s optimal strategy is unique, that it depends continuously (even in an Holderian way) on x, and that, under a strict ellipticity condition, the mapping [psi]*(•, T ) is C2, [alpha] for a strictly positive [alpha]. Brownian games were essentially introduced to prove the existence of a solution to a non linear elliptic PDE problem. The regularity of [psi]*proved here joint to the results proved in [2] indicates that [psi]* is the solution to this PDE problem.