Optimal Stochastic Switching and the Dirichlet Problem for the Bellman Equation

Let Lⁱ be a sequence of second order elliptic operators in a bounded n-dimensional domain Ω, and let fⁱ be given functions. Consider the problem of finding a solution u to the Bellman equation ᵢ(Lⁱu - fⁱ) = 0 a.e. in Ω, subject to the Dirichlet boundary condition $u = 0$ on ∂ Ω. It is proved that, provided the leading coefficients of the Lⁱ are constants, there exists a unique solution u of this problem, belonging to W1,∞(Ω ) ∩ W_loc2,∞(Ω ). The solution is obtained as a limit of solutions of certain weakly coupled systems of nonlinear elliptic equations; each component of the vector solution converges to u. Although the proof is entirely analytic, it is partially motivated by models of stochastic control. We solve also certain systems of variational inequalities corresponding to switching with cost.