# American Institute of Mathematical Sciences

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A sparse Markov chain approximation of LQ-type stochastic control problems
September  2016, 6(3): 391-406. doi: 10.3934/mcrf.2016008

## On the convergence of the Sakawa-Shindo algorithm in stochastic control

 1 INRIA-Saclay and Centre de Mathématiques Appliquées, Ecole Polytechnique and Laboratoire de Finance des Marchés d'Énergie, 91128 Palaiseau, France 2 CIFASIS - Centro Internacional Franco Argentino, de Ciencias de la Información y de Sistemas, CONICET - UNR - AMU, S2000EZP Rosario, Argentina 3 Institut de recherche XLIM-DMI, UMR-CNRS 7252, Faculté des sciences et techniques, Université de Limoges, 87060 Limoges, France

Received  May 2015 Revised  August 2015 Published  August 2016

We analyze an algorithm for solving stochastic control problems, based on Pontryagin's maximum principle, due to Sakawa and Shindo in the deterministic case and extended to the stochastic setting by Mazliak. We assume that either the volatility is an affine function of the state, or the dynamics are linear. We obtain a monotone decrease of the cost functions as well as, in the convex case, the fact that the sequence of controls is minimizing, and converges to an optimal solution if it is bounded. In a specific case we interpret the algorithm as the gradient plus projection method and obtain a linear convergence rate to the solution.
Citation: J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
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