Differential games of inf-sup type and Isaacs equations

Motivated by the work of Fleming [5], we shall provide the general framework to associate inf-sup type values with the Isaacs equations. We shall show that upper and lower bounds for the generators of inf-sup type are upper and lower Hamiltonian in differential games respectively. In particular, lower (resp.upper) bound corresponds to progressive (resp. strictly progressive) strategy. Under Dynamic Programming Principle and identification of the generator, we can prove that the inf-sup type game is characterized as the unique viscosity solution of the Isaacs equation. We also discuss about the Isaacs equation with Hamiltonian of convex combination between lower and upper Hamiltonians.