Strong Predictor–corrector Euler Methods for Stochastic Differential Equations

Abstract

Abstract. This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor-corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the proposed symmetric predictor-corrector Euler methods.

On the numerical stability of simulation methods for SDEs under multiplicative noise in finance

“…This will provide us for p close to 0 with the widest stability region. This region will turn out to be close to the one that refers to the type of asymptotic stability discussed in Bruti-Liberati and Platen (2008). The following analysis provides some reasonable guidelines for the choice of a particular scheme and time step size.…”

“…Typical martingale SDEs have no drift and their diffusion coefficients are often level dependent in an approximately multiplicative manner. For the study of the error propagation of corresponding numerical schemes, test SDEs with multiplicative noise have been suggested in real and complex valued form, see Saito and Mitsui (1993a, b), Platen (1994, 1996), Saito and Mitsui (1996), Higham (2000) and Bruti-Liberati and Platen (2008).…”

“…The discovery of balanced implicit methods in Milstein et al (1998) and Alcock and Burrage (2006) helped to improve the numerical stability in such a situation. Predictor-corrector methods proposed in Platen (1995) and Bruti-Liberati and Platen (2008) can also improve the numerical stability. There has not been a simple methodology that allows one to visualize and understand the limitations of given schemes for simulation problems in finance, in particular with a focus on multiplicative noise.…”

Quasi-exact approximation of hidden Markov chain filters

“…In particular, implicit or predictor-corrector methods are used to control the propagation of errors. We refer for this approach to papers by [1], [2], [7], [8], [11], [12], [15], [16], [20], [21] and [22]. There are various numerical schemes that perform well on some SDEs for certain parameter ranges and sufficiently small step sizes.…”

Exact scenario simulation for selected multi-dimensional stochastic processes

“…In particular, implicit or predictor-corrector methods are used to control the propagation of errors. We refer here to papers by [1], [9], [22], [23], [24], [30], [31], [36], [38], [40], [42] and [44]. The issue of numerical stability can be circumvented when it is possible to simulate exact solutions.…”