The Annals of Applied Probability

Randomized and backward SDE representation for optimal control of non-Markovian SDEs

Marco Fuhrman and Huyên Pham

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We study optimal stochastic control problems for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients and gain functionals are path-dependent, and importantly we do not make any ellipticity assumptions on the SDE. We develop a control randomization approach and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent version of the Hamilton–Jacobi–Bellman equation, and an extension to $G$-expectation.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 2134-2167.

Received: November 2013
Revised: April 2014
First available in Project Euclid: 21 May 2015

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 93E20: Optimal stochastic control

Non-Markovian controlled SDEs randomization of controls dominated measures backward stochastic differential equations


Fuhrman, Marco; Pham, Huyên. Randomized and backward SDE representation for optimal control of non-Markovian SDEs. Ann. Appl. Probab. 25 (2015), no. 4, 2134--2167. doi:10.1214/14-AAP1045.

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