Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs

Mireille Bossy (1) , Héctor Olivero Quinteros (2)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 DIM - Departamento de Ingeniería Matemática [Santiago]

In this paper we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE X_t = x_0 + \int_{0}^t{b(X_s)ds}+\int_{0}^t sigma |X_s|^\alpha dW_s, x_0>0, sigma>0, alpha in[1/2,1). Assuming b(0)/\sigma^2 big enough, and b smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downside, our hypothesis on the critical parameter value b(0)/\sigma^2 is more restrictive than others available in the literature. Some numerical experiments and comparison with various other schemes complement our theoretical analysis that also applies for the simple projected Milstein scheme with same convergence rate.

Mots clés: CEV models; CIR model; Milstein scheme; multilevel Monte Carlo; stochastic differential equations; strong error analysis; non-Lipschitz diffusion coefficient

Domaines: Analyse numérique [math.NA]

Mireille Bossy, Héctor Olivero Quinteros. Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs. Bernoulli, 2018, 24 (3), pp.1995-2042. ⟨10.3150/16-BEJ918⟩. ⟨hal-01185353⟩