Ergodicity of robust switching control and nonlinear system of quasi variational inequalities

Erhan Bayraktar (1) , Andrea Cosso (2) , Huyên Pham (3, 4)

1 Department of Mathematics, University of Michigan
2 POLIMI - Politecnico di Milano [Milan]
3 CREST - Centre de Recherche en Économie et Statistique
4 LPSM - Laboratoire des Protéines et Systèmes Membranaires

We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [3]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.

Mots clés: stochastic games; ergodic problem; randomization; system of quasi variational inequalities; Optimal switching

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Erhan Bayraktar, Andrea Cosso, Huyên Pham. Ergodicity of robust switching control and nonlinear system of quasi variational inequalities . SIAM Journal on Control and Optimization, 2017, 55 (3), pp.1915-1953. ⟨10.1137/15M1023014⟩. ⟨hal-01104773v2⟩