Impulsive control of symmetric Markov processes and quasi-variational inequalities

Abstract

By introducing the notion of impulsive control of a diffusion process A. Bensoussan—J.L. Lions ([!]) showed that if the solutoin of a quasi-variational inequality has sufficient regularity (twice differentiability and continuity), it turns out to be a pay-off function. Furthermore they constructed the optimal strategy out of the solution. But the regularity problems remained open. On the other hand M. Robin ([7]) has set up an impulsive control problem of a general Markov process with a Feller transition semi-group and has constructed the optimal strategy out of the pay-off function which was characterized however in terms of the semi-group rather than the generator of the basic Markov process. As for the characterization by means of the quasi-variational inequality the regularity of the solution was still assumed in order to identify the solution with the pay-off function like that of Bensoussan-Li ons. Regularity problems of elliptic or parabolic quasi-variational inequalities have been studied by L.A. Cafarelli—A. Friedman and others (cf. [2], [5]) under the condition that the diffusion and drift coefficients have sufficient regularity. Cafarelli-Friedmans' work, combined with Robin's, establishes completely the relationship between impulsive control problems and quasi-variational inequalities with respect to nice diffusion processes. Our objective is to extend this relationship to general symmetric Markov processes associated with regular Dirichlet spaces. We prove that the pay-off function is characterized by the weak (maximum) solution of the quasi-variational inequality defined on the Dirichlet space (Theorem 2 in §2). Since we assume only that the Dirichlet space is regular, Theorem 2 establishes the relationship for a wide class of processes. It applies as well to symmetric diffusion process with measurable coefficients and symmetric Markov processes with non local generators (cf. [4]). Our approach is more potential theoretic than others and accordingly the regularity questions can be dispensed with. Indeed we use the potential theory of Dirichlet spaces and Markov processes developed in [4]. The same method has been used in [6] to establish the relationship between variational inequalities