The Adjoint Process in Stochastic Optimal Control
ALBERTA UNIV EDMONTON DEPT OF STATISTICS AND APPLIED PROBABILITY
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There have been many proofs of minimum principles in stochastic control. For a small sample see the works of Kushner, Bismut, Haussmann, Davis and Varaiya, and the book by Elliott. In this paper, we consider a diffusion and stochastic open loop controls, that is, controls which are adapted to the filtration of the driving Brownian motion process. For such controls the dynamical equations have strong solutions, and the results on the differentiability of the solution, due originally to Blagovescenskii and Freidlin, can be applied. The work of Kunita and Bismut on stochastic flows enables the variation in the expected cost, due to a perturbation of the optimal control, to be calculated explicitly. The minimum principle follows by differentiating this quantity. If the optimal control is Markov the stochastic integral representation result of another work is applied to give an expression for a quantity associated with the adjoint process. Stochastic calculus is then used to derive the equation satisfied by the adjoint process.
- Statistics and Probability