On weak approximations of (a, b)-invariant diffusions

We consider scalar stochastic differential equations of the formdXt=μ(Xt)dt+σ(Xt)dBt,X0=x0,where B is a standard Brownian motion. Suppose that the coefficients are such that the solution X possesses the (a, b)-invariance property for some interval (a,b)⊂R:Xt∈(a,b) for all t≥0 if X0=x0∈(a,b). The aim of this paper is constructing weak approximations of X that preserve the above property. The main idea is splitting the equation into two equations (deterministic and stochastic parts) dX˜t=μ(X˜t)dt and X¯t=σ(X¯t)dBt. If the exact solution of one of these equations is known, we use it as the initial condition for the approximate integration of the second one. Though the idea of splitting is not new and is rather widely used for ‘domain-invariant’ strong approximations, it seems to be not yet well developed for weak approximations.