Stochastic variational formulas for solutions to linear diffusion equations | Zendy

Joseph G. Conlon | Zendy; Mohar Guha | Zendy

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Stochastic variational formulas for solutions to linear diffusion equations

Author(s) -

Joseph G. Conlon,

Mohar Guha

Publication year - 2014

Publication title -

revista matemática iberoamericana

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.569

H-Index - 52

eISSN - 2235-0616

pISSN - 0213-2230

DOI - 10.4171/rmi/794

Subject(s) - mathematics , stochastic control , logarithm , mathematical analysis , limit (mathematics) , diffusion , convergence (economics) , optimal control , mathematical optimization , physics , economics , thermodynamics , economic growth

This paper is concerned with solutions to a one dimensional linear diffusion equation and their relation to some problems in stochastic control theory. A stochastic variational formula is obtained for the logarithm of the solution to the diffusion equation, with terminal data which is the characteristic function of a set. In this case the terminal data for the control problem is singular, and hence standard theory does not apply. The variational formula is used to prove convergence in the zero noise limit of the cost function for the stochastic control problem and its first derivatives, to the corresponding quantities for a classical control problem.

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