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Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions


Authors: Emmanuel Gobet and Plamen Turkedjiev
Journal: Math. Comp. 85 (2016), 1359-1391
MSC (2010): Primary 49L20, 62Jxx, 65C30, 93E24
Published electronically: August 6, 2015
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Abstract: We design a numerical scheme for solving the Multi-step Forward Dynamic Programming (MDP) equation arising from the time-discretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical least-squares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and we compute the complexity needed to achieve a given accuracy. Numerical experiments are presented illustrating theoretical convergence estimates.


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Additional Information

Emmanuel Gobet
Affiliation: Centre de Mathématiques Appliquées, Ecole Polytechnique and CNRS, Route de Saclay, F 91128 Palaiseau cedex, France
Email: emmanuel.gobet@polytechnique.edu

Plamen Turkedjiev
Affiliation: Centre de Mathématiques Appliquées, Ecole Polytechnique and CNRS, Route de Saclay, F 91128 Palaiseau cedex, France
Email: turkedjiev@cmap.polytechnique.fr

DOI: https://doi.org/10.1090/mcom/3013
Keywords: Backward stochastic differential equations, dynamic programming equation, empirical regressions, non-asymptotic error estimates
Received by editor(s): June 28, 2013
Received by editor(s) in revised form: July 6, 2013, March 18, 2014, and October 31, 2014
Published electronically: August 6, 2015
Additional Notes: This first author’s research was part of the Chair Financial Risks of the Risk Foundation and of the FiME Laboratory
A significant part of the second author’s research was done while at Humboldt University. This research benefited from the support of the Chair Finance and Sustainable Developement, under the aegis of Louis Bachelier Finance and Sustainable Growth Laboratory, a joint initiative with École Polytechnique.
An earlier unpublished version of this work was circulated under the title “Approximation of discrete BSDE using least-squares regression”, \url{https://hal.archives-ouvertes.fr/hal-00642685v1}. The current version is shorter and includes significantly improved estimates.
Article copyright: © Copyright 2015 American Mathematical Society