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

2012/04/07 by Desmond J. Higham, Xuerong Mao, Higham, Desmond J. +3 · 3 citations
Economics, Econometrics and Finance · #Credit Risk and Financial Regulations #FOS: Mathematics #Financial Risk and Volatility Modeling #Monetary Policy and Economic Impact #Numerical Analysis (math.NA) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1204.1647

openalex publication_date 2012/04/07 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

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

Cited by

• Adaptive Multilevel Splitting: First Application to Rare-Event Derivative Pricing
• First order strong approximations of scalar SDEs with values in a domain
• Lattice Random Walk Discretisations of Stochastic Differential Equations

Related

• Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients
• On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients
• An extended Milstein scheme for effective weak approximation of diffusions
• Positivity-preserving truncated Euler and Milstein methods for financial SDEs with super-linear coefficients
• Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method