A Numerical Scheme for Invariant Distributions of Constrained Diffusions

Reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic. Stationary distributions of such models give useful information on the steady state performance of the corresponding stochastic networks and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions. In this work we propose and analyze a Monte-Carlo scheme based on an Euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity. Appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model. Almost sure consistency results are established that in particular show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a.s. to the corresponding integrals with respect to the invariant measure. Proofs rely on constructing suitable Lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of Echeverria's criteria for reflected diffusions. Regularity properties of the underlying Skorohod problems play a key role in the proofs. Rates of convergence for suitable families of test functions are also obtained. A key advantage of Monte-Carlo methods is the ease of implementation, particularly for high dimensional problems. A numerical example of a eight dimensional Skorohod problem is presented to illustrate the applicability of the approach

Some Topics in Large Deviations Theory for Stochastic Dynamical Systems

In this dissertation, we study large deviations problems for stochastic dynamical systems. First, we consider a family of Stochastic Partial Differential Equations (SPDE) driven by a Poisson Random Measure (PRM) that are motivated by problems of chemical/pollutant dispersal. We established a Large Deviation Principle (LDP) for the long time profile of the chemical concentration using techniques based on variational representations for nonnegative functionals of general PRM. Second, we develop a LDP for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. SPDEs driven by PRM have been proposed as models for many different physical systems. The approach taken here, which is based on variational representations, reduces the proof of the LDP to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. Third, we study stochastic systems with two time scales. Such multiscale systems arise in many applications in engineering, operations research and biological and physical sciences. The models considered in this dissertation are usually referred to as systems with "full dependence", which refers to the feature that the coefficients of both the slow and the fast processes depend on both variables. We establish a LDP for such systems with degenerate diffusion coefficients. The last part of this dissertation focuses on numerical schemes for computing invariant measures of reflected diffusions. Reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic. Stationary distributions of such models give useful information on the steady state performance of the corresponding stochastic networks and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions. We propose and analyze a Monte-Carlo scheme based on an Euler type discretization. We prove an almost sure consistency of the appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion to the true diffusion model. Rates of convergence are also obtained for certain class of test functions. Some numerical examples are presented to illustrate the applicability of this approach.Doctor of Philosoph