Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients (Q1805773)

Numerical solution to the stochastic differential equation of the form \[ dX(t)=b(X(t))dt+\sigma (X(t)) dB(t),\tag{1} \] is studied where \(B\) denotes an \(r\)-dimensional Brownian motion and the coefficients \(b:\mathbb R^n\to \mathbb R^n\) and \(\sigma : \mathbb R^n \to \mathbb R^n \otimes \mathbb R^r\) may be discontinuous. Under suitable regularity conditions on the set of discontinuities and assuming uniqueness of a solution to the equation (1) the weak convergence \(X_n \to X\) is shown, where \(X_n\) solve the equation discretized by the Euler scheme. Convergence of the moments is proved as well. The results are applied to the threshold autoregressive moving average models.