Multilevel Monte Carlo method with applications to stochastic partial differential equations

Abstract

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h ⁻²) if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.

Keywords:

• multilevel Monte Carlo
• stochastic partial differential equations
• stochastic finite element methods
• stochastic parabolic equation
• stochastic hyperbolic equation
• multilevel approximations

2010 AMS Subject Classifications:

• 41A25
• 60G60
• 60H15
• 65C05
• 65C30

Acknowledgements

This research was supported in part by the European Research Council under grant ERC AdG 247277. The authors thank the anonymous referee for very helpful comments.