On the speed of convergence of Picard iterations of backward stochastic differential equations

Martin Hutzenthaler; Thomas Kruse; Tuan Anh Nguyen

doi:10.3934/puqr.2022009

Article Contents

2022, Volume 7, Issue 2: 133-150. Doi: 10.3934/puqr.2022009

This issue Previous Article Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise Next Article

On the speed of convergence of Picard iterations of backward stochastic differential equations

1.
Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, Essen, Nordrhein-Westfalen 45127, Germany
2.
Institute of Mathematics, University of Giessen, Arndtstraße 2, Giessen 35392, Germany

thomas.kruse@math.uni-giessen.de
thomas.kruse@math.uni-giessen.de

Received: July 06, 2021

Accepted: June 22, 2022

Early access: June 2022

Published: June 2022

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Grant No. HU1889/7-1).

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Abstract

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is z-independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.

Keywords:

Backward stochastic differential equation,
Picard iteration,
A priori estimate,
Semilinear parabolic partial differential equation.

Mathematics Subject Classification: 65C99, 60H99, 60G99.

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