Geodesic
A stochastic control problem
Electronic journal of differential equations, Tome 2004 (2004)
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.
Classification : 60H05, 60H07
Keywords: stochastic differential equations, control problems, Jacobi functions 
@article{EJDE_2004__2004__a54,
     author = {Margulies,  William and Zes,  Dean},
     title = {A stochastic control problem},
     journal = {Electronic journal of differential equations},
     year = {2004},
     volume = {2004},
     zbl = {1070.60051},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a54/}
}
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%A Zes,  Dean
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Margulies,  William; Zes,  Dean. A stochastic control problem. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a54/