JournalsrmiVol. 30, No. 2pp. 581–666

Stochastic variational formulas for solutions to linear diffusion equations

  • Joseph G. Conlon

    University of Michigan, Ann Arbor, USA
  • Mohar G. Guha

    University of Michigan, Ann Arbor, USA
Stochastic variational formulas for solutions to linear diffusion equations cover
Download PDF

Abstract

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.

Cite this article

Joseph G. Conlon, Mohar G. Guha, Stochastic variational formulas for solutions to linear diffusion equations. Rev. Mat. Iberoam. 30 (2014), no. 2, pp. 581–666

DOI 10.4171/RMI/794