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.