A zero sum differential game with correlated informations on the initial position. A case with a continuum of initial positions

Abstract

<p style='text-indent:20px;'>We study a two player zero sum game where the initial position <inline-formula><tex-math id="M1">\begin{document}$ z_0 $\end{document}</tex-math></inline-formula> is not communicated to any player. The initial position is a function of a couple <inline-formula><tex-math id="M2">\begin{document}$ (x_0,y_0) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ x_0 $\end{document}</tex-math></inline-formula> is communicated to player Ⅰ while <inline-formula><tex-math id="M4">\begin{document}$ y_0 $\end{document}</tex-math></inline-formula> is communicated to player Ⅱ. The couple <inline-formula><tex-math id="M5">\begin{document}$ (x_0,y_0) $\end{document}</tex-math></inline-formula> is chosen according to a probability measure <inline-formula><tex-math id="M6">\begin{document}$ dm(x,y) = h(x,y) d\mu(x) d\nu(y) $\end{document}</tex-math></inline-formula>. We show that the game has a value and, under additional regularity assumptions, that the value is a solution of Hamilton Jacobi Isaacs equation in a dual sense.