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

TL;DR

This paper proves that Picard iterations for backward stochastic differential equations converge at a rate at least square-root factorially fast, revealing a phase transition in convergence speed depending on the nonlinearity.

Contribution

It establishes a new lower bound on the convergence speed of Picard iterations and identifies conditions for even faster factorial convergence.

Findings

Convergence is at least square-root factorially fast for general nonlinearities.

No higher convergence speed than square-root factorially fast is possible in general.

Factorial convergence occurs when the nonlinearity is z-independent.

Abstract

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearity 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.