Exact Finite-Difference Schemes: Exact difference schemes for stochastic differential equations

In this Section we propose a new way to construct explicit numerical methods for stochastic differential equations (SDEs) via the Steklov mean. The foundations for this new family of methods are based on the exact finite difference scheme for the deterministic ver sion of the SDEs. First, we construct a scheme for the scalar case which is named Steklov method and present the principal results about its convergence and stability for multiplicative and additive SDEs. Next, we extend the previous method towards a multidimensional set up with coefficients of the SDEs under locally Lipschitz and monotone growth conditions. This method is constructed on the basis that the drift function can be rewritten in a linearized form, hence its name, the Linear Steklov (LS) method. Also we provide numerical evidence of the accuracy and efficiency of the Steklov family of schemes versus several methods for SDEs.