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Computation 2018, 6(3), 41; https://doi.org/10.3390/computation6030041

Singularly Perturbed Forward-Backward Stochastic Differential Equations: Application to the Optimal Control of Bilinear Systems

1
Laboratory of Statistics and Random Modeling, University of Abou Bekr Belkaid, Tlemcen 13000, Algeria
2
Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046 Cottbus, Germany
*
Author to whom correspondence should be addressed.
Received: 12 February 2018 / Revised: 22 June 2018 / Accepted: 27 June 2018 / Published: 28 June 2018
(This article belongs to the Special Issue Computation in Molecular Modeling)
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Abstract

We study linear-quadratic stochastic optimal control problems with bilinear state dependence where the underlying stochastic differential equation (SDE) has multiscale features. We show that, in the same way in which the underlying dynamics can be well approximated by a reduced-order dynamics in the scale separation limit (using classical homogenization results), the associated optimal expected cost converges to an effective optimal cost in the scale separation limit. This entails that we can approximate the stochastic optimal control for the whole system by a reduced-order stochastic optimal control, which is easier to compute because of the lower dimensionality of the problem. The approach uses an equivalent formulation of the Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs (FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares Monte Carlo algorithm and show its applicability by a suitable numerical example. View Full-Text
Keywords: linear quadratic stochastic control; bilinear systems; slow-fast dynamics; model reduction; forward-backward stochastic differential equations; least squares Monte Carlo linear quadratic stochastic control; bilinear systems; slow-fast dynamics; model reduction; forward-backward stochastic differential equations; least squares Monte Carlo
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Kebiri, O.; Neureither, L.; Hartmann, C. Singularly Perturbed Forward-Backward Stochastic Differential Equations: Application to the Optimal Control of Bilinear Systems. Computation 2018, 6, 41.

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