Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the local times of the unknown process

In this paper, we consider both, the strong and weak convergence of the Euler–Maruyama approximation for one-dimensional stochastic differential equations involving the local times of the unknown process. We use a transformation in order to remove the local time Lta${L_{t}^{a}}$ from the stochastic differential equations of typeXt=X0+∫0tφ⁢(Xs)⁢𝑑Bs+∫ℝν⁢(d⁢a)⁢Lta.$X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})\,dB_{s}+\int_{\mathbb{R}}\nu(da)L_{t}^{% a}.$Here B is a one-dimensional Brownian motion, φ:ℝ→ℝ${\varphi:\mathbb{R}\rightarrow\mathbb{R}}$ is a bounded measurable function, and ν is a bounded measure on ℝ${\mathbb{R}}$. We provide the approximation of Euler–Maruyama for the stochastic differential equations without local time. After that, we conclude the approximation of Euler–Maruyama Xtn${X_{t}^{n}}$ of the above mentioned equation, and we provide the rate of strong convergence Error=𝔼⁢|XT-XTn|${\operatorname{Error}=\mathbb{E}\lvert X_{T}-X_{T}^{n}\rvert}$, and the rate of weak convergence Error=𝔼⁢|G⁢(XT)-G⁢(XTn)|${\operatorname{Error}=\mathbb{E}\lvert G(X_{T})-G(X_{T}^{n})\rvert}$, for any function G:ℝ→ℝ${G:\mathbb{R}\rightarrow\mathbb{R}}$ of bounded variation.