This paper concerns the optimal stopping time problem in a finite horizon of a controlled jump diffusion process. We prove that the value function is continuous and is a viscosity solution of the integrodifferential variational inequality arising from the associated dynamic programming. We also establish comparison principles, which yield uniqueness results. Moreover, the viscosity solution approach allows us to extend maximum principles for linear parabolic integrodifferential operators in C 0 ([0; T ] \Theta IR n ) and to obtain C 1;2 ([0; T ) \Theta IR n ) existence result for the associated Cauchy problem in the nondegenerate case. Key words: stochastic control, viscosity solutions, jump-diffusion processes AMS Subject Classifications: 93E20, 49L25, 60J75 1 Introduction In this paper, we investigate the optimal stopping time problem of a controlled jump diffusion process and the associated Bellman variational inequality. Let us briefly recall the stochastic background f...

. We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of H ormander used in the study of hypoelliptic differential operators. 1. Intr...

This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the Fokker-Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This "logit equilibrium" forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.

this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise, and the classical "heat-bath" model of R. Kubo, modeling the motion of a "large" molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde's). We then establish pathwise existence and uniqueness of solutions to these classes of sfde's under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde's is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from 0. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel

Consider a particle moving on the surface of the unit sphere in R 3 and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for various characteristics of the process including expected travel time to a cap, the limiting distribution, the likelihood ratio and some estimates for parameters appearing in the model. KEY WORDS: Drift; great circle path; likelihood ratio; pole-seeking; skew product; spherical Brownian motion; stochastic differential equation; travel time. 1.

Efficient Method of Moments (EMM) is used to estimate and test continuous time diffusion models for stock returns and interest rates. For stock returns, a four-state, two-factor diffusion with one state observed can account for the dynamics of the daily return on the S&P composite index, 1927--1987. This contrasts with results indicating that discrete-time, stochastic volatility models cannot explain these dynamics. For interest rates, a trivariate yield factor model is estimated from weekly, 1962-1995, Treasury rates. The yield factor model is sharply rejected, although extensions permitting convexities in the local variance come closer to fitting the data.

The Linear Quadratic Gaussian Poisson (LQGP) problem de-notes an optimal control problem with linear dynamics and quadratic costs with both Gaussian and Poisson noise distur-bances. The LQGP problem provides a benchmark model with sufficient complexity while permitting formal solutionsfor testing both theoretical and computational methods. The problem is examined and is illustrated with a flexible, multi-stage manufacturing system application.

Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theory of simple jump-diffusions. Next, marked-jump-diffusions are treated using compound Poisson processes to include random marked jump-amplitudes in parallel with the equivalent Poisson random measure formulation. Otherwise, the approach is quite applied, using basic principles with no abstractions beyond Poisson random measure. This treatment is suitable for those in classical applied mathematics, physical sciences, quantitative finance and engineering, but have trouble getting started with the abstract measure-theoretic literature. The approach here builds upon the treatment of continuous functions in the regular calculus and associated ordinary differential equations by adding non-smooth and jump discontinuities to the model. Finally, the stochastic optimal control of marked-jump-diffusions is developed, emphasizing the underlying assumptions. The survey concludes with applications in biology and finance, some of which are canonical, dimension reducible problems and others are genuine nonlinear problems. Key words. Jump-diffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic differential equations, stochastic chain rules, stochastic optimal control AMS subject classifications. 60G20, 93E20, 93E03 1. Introduction. There

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INTRODUCTION When Bellman introduced dynamic programming in his original monograph [8], computers were not as powerful as current personal computers. Hence, his description of the extreme computational demands as the Curse of Dimensionality [9] would not have had the super and massively parallel processors of today in mind. However, massive and super computers can not overcome the Curse of Dimensionality alone, but parallel and vector computation can permit the solution of higher dimension than was previously possible and thus permit more realistic dynamic programming applications. Today such large problems are called Grand and National Challenge problems [45, 46] in high performance computing. Today's availability of high performance vector supercomputers and massively parallel processors have made it possible to compute optimal policies and values of control systems for much larger dimensions than was possible earlier. Advance