Basic Probability Theory

Long title: Basic Probability Theory: Independent Random Variables and Sample Spaces. Chapters: Elementary Probability - Basic Probability - Canonical Sample Spaces - Working on Probability Spaces - A Solutions to Exercises

Stochastic 2-D Navier-Stokes Equation with Artificial Compressibility

In this paper we study the stochastic Navier-Stokes equation with artificial compressibility. The main results of this work are the existence and uniqueness theorem for strong solutions and the limit to incompressible flow. These results are obtained by utilizing a local monotonicity property of the sum of the Stokes operator and the nonlinearity.Comment: 18 page

On Optimal Ergodic Control of Diffusions with Jumps

Our purpose is to study an optimal ergodic control problem where the state of the system is given by a diffusion process with jumps in the whole space. The corresponding dynamic programming (or Hamilton-Jacobi-Bellman) equation is a quasi-linear integro-differential equation of second order. A key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only locally bounded and Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic HJB equation is established

On Some Reachability Problems for Diffusion Processes

The main purpose of this paper is to discuss the minimization of energy spent in order that a controlled diffusion process reaches a given target, a d-dimensional bounded domain. The exterior Dirichlet problem for the Hamilton-Jacobi-Bellman equation is studied for a class of criteria which includes the case of energy. Extensions to diffusion with jumps, examples and some other reachability problems are considered

Invariant Measure for Diffusions with Jumps

Our purpose is to study an ergodic linear equation associated to diffusion processes with jumps in the whole space. This integro-differential equation plays a fundamental role in ergodic control problems of second order Markov processes. The key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic linear equation are established

Relaxation and Linear Programs on a Hybrid Control Model

Some optimality results for hybrid control problems are presented. The hybrid model under study consists of two subdynamics, one of a standard type governed by an ordinary differential equation, and the other of a special type having a discrete evolution. We focus on the case when the interaction between the subdynamics takes place only when the state of the system reaches a given fixed region of the state space. The controller is able to apply two controls, each applied to one of the two subdynamics, whereas the state follows a composite evolution, of continuous type and discrete type. By the relaxation technique, we prove the existence of a pair of controls that minimizes an incurred (discounted) cost. We conclude the analysis by introducing an auxiliary infinite-dimensional linear program to show the equivalence between the initial control problem and its associated relaxed counterpart

Infinite-Dimensional Hamilton-Jacobi-Bellman Equations in Gauss-Sobolev Spaces

We consider the strong solution of a semi linear HJB equation associated with a stochastic optimal control in a Hilbert space H: By strong solution we mean a solution in a L2(μ,H)-Sobolev space setting. Within this framework, the present problem can be treated in a similar fashion to that of a finite-dimensional case. Of independent interest, a related linear problem with unbounded coefficient is studied and an application to the stochastic control of a reaction-diffusion equation will be given