Divergence of the multilevel Monte Carlo method
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Abstract
One of the most promising methods for simulating moments or other expectations of functionals of the solution process of a stochastic differential equation is the multilevel Monte Carlo Euler method in [Giles, M. B., Multilevel Monte Carlo Path Simulation, Operations Research 56 (2008), no. 3, 606-617]. This method has been shown to converge if the coefficients of the SDE are globally Lipschitz continuous. In this article we establish that the multilevel Monte Carlo Euler method does, in general, not converge if the coefficients of the SDE are not globally Lipschitz continuous. More precisely, the main result of this article proves divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. This divergence result reveals that the multilevel Monte Carlo Euler method often produces completely wrong values in the case of SDEs with superlinearly growing coefficients. Additionally, this divergence of the new multilevel Monte Carlo Euler method is in fundamental contrast to the convergence of the classical Monte Carlo Euler method for such SDEs. Nonetheless, the multilevel Monte Carlo method combined with a strongly convergent numerical approximation method does converge. In particular, we show that the multilevel Monte Carlo Euler method combined with a tamed Euler method converges for such SDEs.Keywords
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