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If L is the (possibly degenerate) differential generator of a diffusion process whose measures converge to a unique invariant measure μ, then formally the value γ in F V(x)+k(x)−γ=0 is ∝ k(x) μ(dx). A finite difference approximation is used to solve the differential equation. The coefficients in the finite difference equation are one-step transition probabilities for some Markov chain whose (suitable) continuous time interpolations converge weakly to the diffusion. Under reasonable conditions, the invariant measure of the sequence of chains converges weakly to the weak sense density of μ. as the finite difference intervals go to zero. The approximating measure can be taken to be the invariant measure (or Cesaro sum of the n-step transition probabilities for the chain), of the chain, suitably weighted.

In the paper, the authors prove several convergence results for a numerical method in optimal stochastic control, based on finite difference approximations to the non-linear Bellman partial differential equation for the optimal cost. As a very useful by-product to the main development, the authors obtain new existence results for optimal controls, and interesting results on the approximation of a controlled diffusion by a controlled Markov chain. The methods and results are new, and depend on results concerning weak convergence of probability measures. The methods seem to be quite powerful, and have applications to many other problems in approximation and control. (Author)

The paper applies some results on weak convergence of probability measures to the problem of convergence of finite difference approximations to a broad class of degenerate elliptic and parabolic partial differential equations. The equations are of type which arise in stochastic control theory and, generally, have only weak solutions. Interest in the problem stems from interest in stochastic control theory, and in numerical methods for the solution of stochastic control problems. The results are of interest in stochastic control theory and in numerical analysis. Some conditions are stated, the problem is discussed along with its probabilistic interpretation. (Author)