Convergence, Non-negativity and Stability of a New Milstein Scheme with Applications to Finance

Details Convergence, Non-negativity and Stability of a New Milstein Scheme with Applications to Finance Convergence, Non-negativity and Stability of a New Milstein Scheme with Applications to Finance

2012-04-07

arxiv.org/abs/1204.1647v1

Mathematics

Numerical Analysis

Scientific paper

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.

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