Convergence, non-negativity and stability of a new Milstein scheme with applications to finance

Desmond J. Higham; Xuerong Mao; Lukasz Szpruch

doi:10.3934/dcdsb.2013.18.2083

Article Contents

2013, Volume 18, Issue 8: 2083-2100. Doi: 10.3934/dcdsb.2013.18.2083

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Convergence, non-negativity and stability of a new Milstein scheme with applications to finance

1.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH
2.
Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland
3.
School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3JZ

Received: March 31, 2012

Revised: January 31, 2013

Published: September 2013

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Abstract

We propose and analyse a new Milstein type scheme for simulating stochastic differential equations (SDEs) with highly nonlinear coefficients. Our work is motivated by the need to justify multi-level Monte Carlo simulations for mean-reverting financial models with polynomial growth in the diffusion term. We introduce a double implicit Milstein scheme and show that it possesses desirable properties. It converges strongly and preserves non-negativity for a rich family of financial models and can reproduce linear and nonlinear stability behaviour of the underlying SDE without severe restriction on the time step. Although the scheme is implicit, we point out examples of financial models where an explicit formula for the solution to the scheme can be found.

Keywords:

Milstein scheme,
implicit schemes stochastic differential equation,
stability,
strong convergence,
non-negativity.

Mathematics Subject Classification: Primary: 60H10, 60H35; Secondary: 65J15.

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