# American Institute of Mathematical Sciences

October  2013, 18(8): 2083-2100. doi: 10.3934/dcdsb.2013.18.2083

## 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, United Kingdom 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, United Kingdom

Received  April 2012 Revised  February 2013 Published  July 2013

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.
Citation: Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2083-2100. doi: 10.3934/dcdsb.2013.18.2083
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