The local linearization scheme for nonlinear diffusion models with discontinuous coefficients
Consider the stochastic differential equation in \(\mathbb R^d\) \[ dX(t)=b(X(t)) dt+\sigma(X(t)) dB(t), \] where \(B\) is a standard Brownian motion. The author studies a numerical scheme called local linearisation to approximate the solution, which is different from the usual Itô-Taylor numerical schemes. On a time interval \([kh,(k+1)h]\), the diffusion coefficient is assumed to be constant, and the drift is linearised as follows: From Itô formula, \(db(X(t))=J(X(t)) d(X(t))+M(X(t)) dt\), for some \(J\) and \(M\), one can approximate \[ b(X(t))\approx J(X(kh))X(t)+M(X(kh))t+b(X(kh))-J(X(kh))X(kh)-M(X(kh))kh , \] thus obtaining a linear s.d.e. that can be explicitly solved in each interval. The paper gives two conditions for the weak convergence of the approximated diffusion to the solution of the original equation. The coefficients need not be continuous, though it must be assumed that a solution exists and is unique. Essentially, the drift can be piecewise twice-differentiable, and the diffusion coefficient can be only piecewise continuous. Nothing is said about the order of convergence, but some experiments are shown where this scheme behaves better than the Euler scheme. The experiments are performed on continuous-time threshold autoregressive models.
continuous-time threshold autoregressive model
scientific article; zbMATH DE number 1395128