A stochastic control problem

• Main Authors: William Margulies, Dean Zes
• Format: Article
• Language: English
• Published: Texas State University 2004-11-01
• Series: Electronic Journal of Differential Equations
• Subjects: Stochastic differential equations; control problems; Jacobi functions.
• Online Access: http://ejde.math.txstate.edu/Volumes/2004/135/abstr.html

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.