Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

Abstract

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewiseconstant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton-Jacobi-Bellman equation in our non-Markovian framework.

Systems of Fully Nonlinear Parabolic Obstacle Problems with Neumann Boundary Conditions

“…Instead much focus has been directed to unbounded domains (essentially ≡ R n ) and often linear PDEs. In this setting the literature is, on the contrary, rather rich, see, e.g., El-Asri-Hamadene [11], Djehiche-Hamadene-Popier [8], Hu-Tang [14], Biswas-Jakobsen-Karlsen [4], Hamadene-Morlais [13], Lundström-Nyström-Olofsson [19,20], Lundström-Olofsson-Önskog [21], Fuhrman-Morlais [12], Reisinger-Zhang [24] and the many references listed therein. Extensions to more general operators have been studied in Klimsiak [18].…”

Systems of fully nonlinear parabolic obstacle problems with Neumann boundary conditions

“…Instead much focus has been directed to unbounded domains (essentially Ω ≡ R n ) and often linear PDEs. In this setting the literature is, on the contrary, rather rich, see, e.g., El-Asri-Hamadene [8], Djehiche-Hamadene-Popier [11], Hu-Tang [13], Biswas-Jakobsen-Karlsen [4], Hamadene-Morlais [12], Lundström-Nyström-Olofsson [18,19], Lundström-Olofsson-Önskog [20], Fuhrman-Morlais [9], Reisinger-Zhang [23] and the many references listed therein. Extensions to more general operators have been studied recently in Klimsiak [17].…”