A zero-sum stochastic differential game problem on infinite horizon with continuous and impulse controls is studied. We obtain the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities. We also obtain a verification theorem which provides an optimal strategy of the game.

A zero-sum stochastic differential game problem on infinite horizon with continuous and impulse controls is studied. We obtain the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities. We also obtain a verification theorem which provides an optimal strategy of the game.

In some preceding works we consider a class $\mathcal{O}\mathcal{P}$ of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line $l=l_{\gamma (\cdot )}$, regarded as determined by its (variable) curvature function $\gamma (\cdot )$ of domain $\left[s_0,s_1\right]$. Assume that the problem $\tilde{\mathcal{P}}\in \mathcal{O}\mathcal{P}$ is regular but has an impulsive monotone character in the sense that near each of some points ${\delta }_1$ to $\delta {}_{\nu }\gamma (\cdot )$ is monotone and $\left|\gamma {}^{\prime }\right(\cdot \left)\right|$ is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify $\tilde{\mathcal{P}}$ into a structurally...