Semi-discrete approximations for stochastic differential equations and applications

In this paper, we propose a new point of view in numerical approximation of stochastic differential equations. By using Ito-Taylor expansions, we expand only a part of the stochastic differential equation. Thus, in each step, we have again a stochastic differential equation which we solve explicitly or by using another method or a finer mesh. We call our approach as a semi-discrete approximation. We give two applications of this approach. Using the semi-discrete approach, we can produce numerical schemes which preserves monotonicity so in our first application, we prove that the semi-discrete Euler scheme converge in the mean square sense even when the drift coefficient is only continuous, using monotonicity arguments. In our second application, we study the square root process which appears in financial mathematics. We observe that a semi-discrete scheme behaves well producing non-negative values.