Viscosity solutions associated with switching game for piecewise-deterministic processes

We consider the existence and uniqueness of viscosity solutions to a system of integro-differential equations with bilateral implicit obstacles. The system is the dynamic programming equation associated with a switching game for two players, in which the underlying state processes are piecewise-deterministic processes. We prove the probabilistic representation of the viscosity solution as the saddle point of the game for the Rn case

[1]  Erhan Çinlar,et al.  A stochastic integral in storage theory , 1971 .

[2]  J. Michael Harrison,et al.  The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule , 1976, Math. Oper. Res..

[3]  Avner Friedman,et al.  Optimal stochastic switching and the Dirichlet problem for the Bellman equation , 1979 .

[4]  On first order singular bellman equation , 1981 .

[5]  Walter A. Rosenkrantz,et al.  Some martingales associated with queueing and storage processes , 1981 .

[6]  Zvi Rosberg,et al.  Optimal control of service in tandem queues , 1982 .

[7]  On the control of jump processes , 1982 .

[8]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[9]  N. Yamada,et al.  A system of elliptic variational inequalities associated with a stochastic switching game , 1983 .

[10]  S. Lenhart,et al.  A System of Nonlinear Partial Differential Equations Arising in the Optimal Control of Stochastic Systems with Switching Costs , 1983 .

[11]  A. Hordijk,et al.  Average optimal policies in Markov decision drift processes with applications to a queueing and a replacement model , 1983, Advances in Applied Probability.

[12]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[13]  S. Lenhart,et al.  Integro-differential equations associated with optimal stopping time of a piecewise-deterministic process , 1985 .

[14]  D. Verms.  Optimal control of piecewise deterministic markov process , 1985 .

[15]  Mark H. A. Davis.  Control of piecewise-deterministic processes via discrete-time dynamic programming , 1986 .

[16]  U. S. Gugerui.  Optimal stopping of a piecewise-deterministic markov process , 1986 .

[17]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[18]  H. Ishii.  Perron’s method for Hamilton-Jacobi equations , 1987 .

[19]  M. Dempster,et al.  Optimal capacity expansion under uncertainty , 1987, Advances in Applied Probability.

[20]  Suzanne Lenhart,et al.  Switching control of piecewise-deterministic processes , 1988 .

[21]  Suzanne Lenhart,et al.  Viscosity solutions associated with impulse control problems for piecewise-deterministic processes. , 1989 .

[22]  A remark on a system of inequalities with bilateral obstacles , 1989 .

[23]  Juan Ye.  Optimal control of piecewise deterministic Markov processes. , 1990 .

[24]  S. Lenhart,et al.  Viscosity Solutions for Weakly Coupled Systems of Hamilton-Jacobi Equations , 1991 .