Tyrone E. Duncan

On the Calculation of Mutual Information

Abstract : Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannons result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannons result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.

FRACTIONAL BROWNIAN MOTION AND STOCHASTIC EQUATIONS IN HILBERT SPACES

In this paper, stochastic differential equations in a Hilbert space with a standard, cylindrical fractional Brownian motion with the Hurst parameter in the interval (1/2,1) are investigated. Existence and uniqueness of mild solutions, continuity of the sample paths and state space regularity of the solutions, and the existence of limiting measures are verified. The equivalence of the probability laws for the solution evaluated at different times and different initial conditions and the convergence of these probability laws to the limiting probability are verified. These results are applied to specific stochastic parabolic and hyperbolic differential equations. The solution of a specific parabolic equation with the fractional Brownian motion only in the boundary condition is shown to have many results that are analogues of the results for a fractional Brownian motion in the domain.

A Kiefer-Wolfowitz algorithm with randomized differences

A Kiefer-Wolfowitz or simultaneous perturbation algorithm that uses either one-sided or two-sided randomized differences and truncations at randomly varying bounds is given in this paper. At each iteration of the algorithm only two observations are required in contrast to 2l observations, where l is the dimension, in the classical algorithm. The algorithm given is shown to be convergent under only some mild conditions. The rate of convergence and asymptotic normality of the algorithm are also established.

ON THE SOLUTIONS OF A STOCHASTIC CONTROL SYSTEM. II

The control system considered in this paper is modeled by the stochastic differential equation \[dx(t,\omega ) = f(t,x( \cdot ,\omega ),u(t,\omega ))dt + dB(t,\omega ),\] where B is n-dimensional Brownian motion, and the control u is a nonanticipative functional of

Evaluation of likelihood functions

x( \cdot ,\omega )

Adaptive control of continuous-time linear stochastic systems

taking its values in a fixed set U. Under various conditions on f it is shown that for every admissible control a solution is defined whose law is absolutely continuous with respect to the Wiener measure

Adaptive continuous-time linear quadratic Gaussian control

\mu

Adaptive Boundary and Point Control of Linear Stochastic Distributed Parameter Systems

, and the corresponding set of densities on the space C forms a strongly closed, convex subset of

STOCHASTIC INTEGRATION FOR FRACTIONAL BROWNIAN MOTION IN A HILBERT SPACE

L^1 (C,\mu )

Semilinear Stochastic Equations in a Hilbert Space with a Fractional Brownian Motion

. Applications of this result to optimal control and two-person, zero-sum differential games are noted. Finally, an example is given which shows that in the case where only some of the components of x are observed, the set of attainable densities is not weakly closed in