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

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Diffusion Approximations for Multiscale Stochastic Networks in Heavy Traffic

Applications arising from computer, telecommunications, and manufacturing systems lead to many challenging problems in the simulation, stability, control, and design of stochastic models of networks. The networks are usually too complex to be analyzed directly and thus one seeks suitable approximate models. One class of such approximations are diffusion models that can be rigorously justified when networks are operating in heavy traffic, i.e., when the network capacity is roughly balanced with network load. We study stochastic networks with time varying arrival and service rates and routing structure. Time variations are governed, in addition to the state of the system, by two independent finite state Markov processes X and Y. Transition times of X are significantly smaller than the typical interarrival and processing times whereas the reverse is true for the Markov process Y. We first establish a diffusion approximation for such multiscale queueing networks in heavy traffic. The result shows that, under appropriate heavy traffic conditions, properly normalized queue length processes converge weakly to a Markov modulated reflected diffusion process. More precisely, the limit process is a reflected diffusion with drift and diffusion coefficients that are functions of the state process, the invariant distribution of X and a finite state Markov process which is independent of the driving Brownian motion. We then study the stability properties of such Markov modulated reflected diffusion processes and establish positive recurrence and geometric ergodicity properties under suitable stability conditions. As consequences, we obtain results on the moment generating function of the invariant probability measure, uniform in time moment estimates and functional central limit results for such processes. We also study relationship between invariant measures of the Markov modulated constrained diffusion processes and that of the underlying queueing network. It is shown that, under suitable heavy traffic and stability conditions, the invariant probability measure of the queueing process converges to that of the corresponding Markov modulated reflected diffusion. The last part of this dissertation focuses on ergodic control problems for discrete time controlled Markov chains with a locally compact state space and a compact action space under suitable stability, irreducibility and Feller continuity conditions. We introduce a flexible family of controls, called action time sharing (ATS) policies, associated with a given continuous stationary Markov control. It is shown that the long term average cost for such a control policy, for a broad range of one stage cost functions, is the same as that for the associated stationary Markov policy. Through examples we illustrate the use of such ATS policies for parameter estimation and adaptive control problems