On the Numerical Stability of Simulation Methods for SDES

Eckhard Platen () and Lei Shi ()

No 234, Research Paper Series from Quantitative Finance Research Centre, University of Technology, Sydney

Abstract: When simulating discrete time approximations of solutions of stochastic differential equations (SDEs), numerical stability is clearly more important than numerical efficiency or some higher order of convergence. Discrete time approximations of solutions of SDEs are widely used in simulations in finance and other areas of application. The stability criterion presented is designed to handle both scenario simulation and Monte Carlo simulation, that is, strong and weak simulation methods. The symmetric predictor-corrector Euler method is shown to have the potential to overcome some of the numerical instabilities that may be experienced when using the explicit Euler method. This is of particular importance in finance, where martingale dynamics arise for solutions of SDEs and diffusion coefficients are often of multiplicative type. Stability regions for a range of schemes are visualized and discussed. For Monte Carlo simulation it turns out that schemes, which have implicitness in both the drift and the diffusion terms, exhibit the largest stability regions. It will be shown that refining the time step size in a Monte Carlo simulation can lead to numerical instabilities.

Keywords: stochastic differential equations; scenario simulation; Monte Carlo simulation; numerical stability; predictor-corrector methods; implicit methods (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-cmp and nep-ore
Date: 2008-10-01
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