Optimal Stochastic Switching and the Dirichlet Problem for the Bellman Equation

Let ${L^i}$ be a sequence of second order elliptic operators in a bounded n-dimensional domain $\Omega$, and let ${f^i}$ be given functions. Consider the problem of finding a solution u to the Bellman equation ${\sup _i}({L^i}u - {f^i}) = 0$ a.e. in $\Omega$, subject to the Dirichlet boundary condition $u = 0$ on $\partial \Omega$. It is proved that, provided the leading coefficients of the ${L^i}$ are constants, there exists a unique solution u of this problem, belonging to ${W^{1,\infty }}(\Omega ) \cap W_{{\text {loc}}}^{2,\infty }(\Omega )$. 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.