The Euler scheme with irregular coefficients
The author considers a \(d\)-dimensional system of stochastic differential equations, driven by Brownian motion \(B\) in \(R^r\), \[ X_t = X_0 + \int_0^t b(s,X_s) ds + \int_0^t \sigma(s,X_s) dB_s,\tag{1} \] where \(X_0\) is an \(R^d\)-valued random variable, independent of \(B\). For the coefficient functions he assumes necessary and sufficient conditions such that equation (1) has a unique weak solution, based on a result by \textit{H. J. Engelbert} and \textit{W. Schmidt} [in: Stochastic differential systems. Lect. Notes Control Inf. Sci. 69, 143-155 (1985; Zbl 0583.60052)]. He then considers the continuous Euler scheme \[ X_t^n = X_{\tau_k^n}^n + b(\tau_k^n,X_{\tau_k^n}^n)(t-\tau_k^n) + \sigma(\tau_k^n,X_{\tau_k^n}^n)(B_t - B_{\tau_k^n}),\tag{2} \] for \({\tau_k^n} \leq t \tau_{k+1}^n, k=0,1,\ldots,n\), where \(0=\tau_0^n \leq \tau_1^n \leq \ldots \leq \tau_n^n = T\) is a sequence of random partitions of \([0,T]\). The aim of the paper is to show weak convergence of the solution of the continuous Euler scheme (2) to the unique weak solution of (1) under only mild smoothness conditions on the coefficient functions \(b\) and \(\sigma\). Under additional conditions the author proves results concerning the rate of convergence in the one-dimensional case.
scientific article; zbMATH DE number 1906082
scientific article