Abstract
The Euler scheme is one of the standard schemes to obtain numerical approximations of solutions of stochastic differential equations (SDEs). Its convergence properties are well known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior, which results in SDE models with discontinuous drift coefficient. In this work, we analyze the long time properties of the Euler scheme applied to SDEs with a piecewise constant drift and a constant diffusion coefficient and carry out intensive numerical tests for its convergence properties. We emphasize numerical convergence rates and analyze how they depend on the properties of the drift coefficient and the initial value. We also give theoretical interpretations of some of the arising phenomena. For application purposes, we study a rank-based stock market model describing the evolution of the capital distribution within the market and provide theoretical as well as numerical results on the long time ranking behavior.
Highlights
In recent years, many applications related to stochastic differential equations (SDEs) with discontinuous drift coefficient have emerged
We focus on numerical approximations of SDEs in the presence of a piecewise constant drift and a constant diffusion coefficient
We focus on the strong convergence rate of the Euler scheme, which, for a general SDE
Summary
Many applications related to stochastic differential equations (SDEs) with discontinuous drift coefficient have emerged. For the numerical analysis of SDEs with discontinuous drift and/or diffusion coefficient, the situation is more involved In this manuscript, we focus on the strong convergence rate of the Euler scheme, which, for a general SDE dXt = f (Xt) dt + g(Xt) dWt, t ∈ [0, T], X0 = ξ , where f and g are such that a unique strong solution exists, is given by xekx+p1E = xekxp E + f xekxp E + g xekxp E (W(k+1) – Wk ), k = 0, . Note that in the standard setting of an SDE with additive noise, where the drift coefficient is sufficiently smooth, the Euler scheme has an exact strong convergence order of 1, see, e.g., [6] and [19, p. In the remainder of this chapter, we present and discuss some key results of the simulation studies
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
