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On the speed of convergence of Picard iterations of backward stochastic differential equations
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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 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.
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Cited by 1 Pith paper
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Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem
For a simple HJB control example, MLP approximations have L2 error growing like d^{n/2}/(c^n sqrt(n!)) - 1, so no polynomial-in-dimension error bound can hold uniformly as the number of levels grows.
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