First order strong approximations of scalar SDEs defined in a domain

Abstract

We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general result can be applied to many SDEs we encounter in mathematical finance and bio-mathematics. We will demonstrate flexibility of our approach by analysing classical examples of SDEs with sublinear coefficients (CIR, CEV models and Wright-Fisher diffusion) and also with superlinear coefficients (3/2-volatility, Ait-Sahalia model). Our goal is to justify an efficient Multi-Level Monte Carlo (MLMC) method for a rich family of SDEs, which relies on good strong convergence properties.L p -error criterion. This strategy was also suggested by Alfonsi in [1]. Here, we extend that work in several ways:• Considering the maximum error in the discretization points, we prove that the drift-implicit Euler-Maruyama scheme for the CIR process strongly converges with rate one under slightly more restrictive conditions on the parameters of the process than in [6].• We provide a general framework for the strong order one convergence of the BEM scheme for SDEs with constant diffusion and one-sided Lipschitz drift coefficients.• Using this framework we present a detailed convergence analysis for several SDEs with sub-and super-linear coefficients.• We also show that BEM for the transformed SDE is closely related to a drift-implicit Milstein scheme for the original SDE, which has been introduced in [13]. In the case of the CIR process we provide a sharp error analysis for this scheme.

Simple Generation of Gamma, Gamma–Gamma, and K Distributions With Exponential Autocorrelation Function

“…However, the theoretical (and practical) limit for the positive solution of (7) is given by α > 0.5 and does not depend on the θ and τ c values [26].…”

“…(17)]. The stable solution of such a process is currently the subject of significant interest [26]- [28].…”

“…Following [26], we used the implicit Milstein scheme that preserves the SDE solution domain, while converging to the corresponding numerical solution. The implicit Milstein scheme may be described by [26,Eq. (40)]…”

Multilevel Dimension Reduction Monte-Carlo Simulation for High-dimensional Stochastic Models in Finance

“…Similar observations are made in [6] in the context of Asian options. Now recall the coefficients F {νt} , G {νt} , and H {νt} of the payoff P , which are given in (9). To compute these coefficients, we need to evaluate the following stochastic integrals:…”

“…For the simulation of ν(t), we use a drift-implicit Milstein scheme that preserves the positivity of the original CIR model (1d), and has a good strong convergence property, recently established in [9]. More specifically, given a timestep size h = T /N, the Milstein discretization of (1d) iŝ…”