Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 135, pp. 1-10.
Title: A stochastic control problem
Authors: William Margulies (California State Univ., Long Beach, CA, USA)
Dean Zes (B&Z Engineering Consulting, Long Beach, CA, USA)
Abstract:
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.