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The field of stochastic stability is surveyed, with emphasis on the invariance theorems and their potential application to systems with randomly varying coefficients. Some of the basic ideas are reviewed, which underlie the stochastic Liapunov function approach to stochastic stability. The invariance theorems are discussed in detail.

A problem is considered which arises in the numerical analysis of the degenerate partial differential equations of stochastic control theory. The relationship between diffusions and finite state Markov chains is considered. The role of probability theory in numerical analysis is illustrated.

Accelerated procedures for Markov chain model optimal control problems solution with computation time advantage over usual dynamic programming

Convergence of identification scheme for scalar- input-scalar-output linear systems with random inputs, applying invariant set theorem for random systems

Partial differential equations finite difference approximation, applying convergence theorems with probabilistic method

Nonlinear filtering for linear parabolic distributed parameter systems with white noise, considering stochastic boundary value problem

Necessary conditions for discrete parameter stochastic optimization problems

Linear and dynamic programming and Markov chain optimal control

Riccati-like linear functional differential equation with quadratic cost, analyzing feedback control solution existence and uniqueness

Local stochastic stability property used to prove stability and existence of diffusions with rapidly growing drift terms

Accelerated procedures for solution of discrete Markov control problems

Abstract variational theory application to continuous parameter stochastic optimization problems to derive maximum principles in linear programming

Probability limit theorems applied to convergence of finite difference approximations of partial differential equations

Randomly sampled linear systems stability with linear or nonlinear feedback loops, using stochastic Liapunov function method

Existence theorem for Markov chain finite state stochastic games applied to saddle point and optimal strategy or epsilon-optimal strategy pairs

Linear functional differential equations control with quadratic cost

Scheme for choosing approximating system of difference equations for degenerate linear and nonlinear elliptic boundary value problems

Optimal control of system governed by linear parabolic equation with white noise inputs, using mathematical model to generate distributed system analog

Optimal control problems for Markov chains solved by iterative method, using nonlinear finite difference equations to approximate degenerate elliptic functions

Stability of stochastic processes defined by difference differential equations, noting properties of solution, weak infinitesimal operator, etc

Probability limit theorems and convergence of finite difference approximations of partial differential equations

Iterative method for optimal control problems for Markov chains, discussing degenerate nonlinear elliptic equations solution in optimal stochastic control theory

Filtering for linear distributed parameter systems

Identification scheme for scalar input and scalar output linear systems for wide class of random inputs

Stability of randomly sampled linear systems studied by Liapunov function method

Numerical solution for variety of nonlinear degenerate elliptic boundary value problems, and convergence of associated iterative procedures

Optimal cost discounted stochastic control for diffusion processes

Approximations to optimal nonlinear filters for problems involving signal and noise processes

Exact dynamical equation derived for conditional density mode representing stochastic process

Stability of processes defined by stochastic difference and differential equations

Existence theorems and computational procedures for finite state stochastic games