P. Sundar

Propagation of chaos and the McKean-Vlasov equation in duals of nuclear spaces

An interacting system ofn stochastic differential equations taking values in the dual of a countable Hilbertian nuclear space is considered. The limit (in probability) of the sequence of empirical measures determined by the above systems asn tends to ∞ is identified with the law of the unique solution of the McKean-Vlasov equation. An application of our result to interacting neurons is briefly discussed. The propagation of chaos result obtained in this paper is shown to contain and improve the well-known finite-dimensional results.

Comparison of solutions of stochastic equations and applications

Pathwise comparison of solutions to a class of stochastic systems of differential equations is proved which extends the existing result of Geiβ and Manthey. When the diffusion coefficients are defferent, the Gal’čhuk-Davis method is extended to establish the comparision results. We illustrate our results with several examples some of which arise in stochastic finance theory

Law of the iterated logarithm for solutions of stochastic differential equations

We prove the law of the iterated logarithm for solutions of Stochastic Differential Equations (SDEs) driven by continuous semiraartingales, under suitable conditions. This extends a result of Kulinich for classical diffusions to solutions of SDEs which are not necessarily Markov

A Technique for Stochastic Control Problems with Unbounded Control Set

We describe a change of time technique for stochastic control problems with unbounded control set. We demonstrate the technique on a class of maximization problems that do not have optimal controls. Given such a problem, we introduce an extended problem which has the same value function as the original problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal controls for the original problem. By this we mean a sequence of controls for which the payoff functions approach the value function.

Stochastic Analysis and Diffusion Processes

1. Introduction to Stochastic Processes 2. Brownian Motion and Wiener Measure 3. Elements of Martingale Theory 4. Analytic Tools for Brownian Motion 5. Stochastic Integration 6. Stochastic Differential Equations 7. The Martingale Problem 8. Probability Theory and Partial Differential Equations 9. Gaussian Solutions 10. Jump Markov Processes 11. Invariant Measures and Ergodicity 12. Large Deviations for Diffusions

Two-dimensional stochastic Navier-Stokes equations with fractional Brownian noise

Abstract. We study the perturbation of the two-dimensional stochastic Navier–Stokes equation by a Hilbert-space-valued fractional Brownian noise. Each Hilbert component is a scalar fractional Brownian noise in time, with a common Hurst parameter H and a specific intensity. Because the noise is additive, simple Wiener-type integrals are sufficient for properly defining the problem. It is resolved by separating it into a deterministic nonlinear PDE, and a linear stochastic PDE. Existence and uniqueness of mild solutions are established under suitable conditions on the noise intensities for all Hurst parameter values. Almost surely, the solutions paths are shown to be quartically integrable in time and space. Whether this integrability extends to the random parameter is an open question. An extension to a multifractal model is given.

The Enskog Process

The existence of a weak solution to a McKean–Vlasov type stochastic differential system corresponding to the Enskog equation of the kinetic theory of gases is established under suitable hypotheses. The distribution of any solution to the system at each fixed time is shown to be unique, when the density exists. The existence of a probability density for the time-marginals of the velocity is verified in the case where the initial condition is Gaussian, and is shown to be the density of an invariant measure.

A note on the solution of a stochastic partial differential equation

Abstract A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.

Two Applications of Reproducing Kernel Hilbert Spaces in Stochastic Analysis

The importance of reproducing kernel Hilbert spaces in the study of Gaussian processes is illustrated in two concrete problems. The first deals with mutual singularity of the law of the solution of a stochastic partial differential equation and the law of the driving process. The second gives a characterization for a Gaussian process to be a semimartingale in terms of reproducing kernel Hilbert spaces. Expansion of filtrations for Gaussian processes is discussed.

Stochastic problems with unbounded control set

Describes a change of time technique for stochastic control problems with unbounded control set. The authors demonstrate the technique on a class of maximization problems that do not have optimal controls. Given such a problem, the authors introduce an extended problem which has the same value function as the original problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal controls for the original problem. By this the authors mean a sequence of controls for which the payoff functions approach the value function.