Electron. J. Diff. Eqns., Vol. 2004(2004), No. 135, pp. 1-10.

A stochastic control problem

William Margulies, Dean Zes

In this paper, we study a specific stochastic differential equation depending on a parameter and obtain a representation of its probability density function in terms of Jacobi Functions. The equation arose in a control problem with a quadratic performance criteria. The quadratic performance is used to eliminate the control in the standard Hamilton-Jacobi variational technique. The resulting stochastic differential equation has a noise amplitude which complicates the solution. We then solve Kolmogorov's partial differential equation for the probability density function by using Jacobi Functions. A particular value of the parameter makes the solution a Martingale and in this case we prove that the solution goes to zero almost surely as time tends to infinity.

Submitted May 28, 2004. Published November 23, 2004.
Math Subject Classifications: 60H05, 60H07.
Key Words: Stochastic differential equations; control problems; Jacobi functions.

Show me the PDF file (285K), TEX file, and other files for this article.

William Margulies
Department of Mathematics
California State University
Long Beach, CA 90840, USA
email: wmarguls@csulb.edu
Dean Zes
B&Z Engineering Consulting
3134 Kallin Ave, Long Beach, CA 90808, USA
email: deanzes@charter.net

Return to the EJDE web page