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

Chloé Jimenez

doi:10.3934/jdg.2021009

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2021, Volume 8, Issue 3: 233-266. Doi: 10.3934/jdg.2021009

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

Chloé Jimenez

Univ Brest, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS-UMR 6205, 6, avenue Victor Le Gorgeu, CS 93837, 29238 Brest cedex 3, France

Received: June 2020

Early access: March 2021

Published: July 2021

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Abstract

We study a two player zero sum game where the initial position $ z_0 $ is not communicated to any player. The initial position is a function of a couple $ (x_0,y_0) $ where $ x_0 $ is communicated to player Ⅰ while $ y_0 $ is communicated to player Ⅱ. The couple $ (x_0,y_0) $ is chosen according to a probability measure $ dm(x,y) = h(x,y) d\mu(x) d\nu(y) $. 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.

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Mathematics Subject Classification: 49N70, 49L25, 91A23, 49Q20.

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