The Order 1.5 Approximation for Solutions of Jump-Diffusion Equations

Abstract

Using in Itô equation, only information from increments of a driving Wiener process make it possible to make such Runge-Kutta approximations in which the error in ℒ2-norm is of an order not greater than 1 in general case. This convergence order can be greater only when functional coefficients from the equation satisfy a certain additional condition. However, it is obvious that in general case information only from increments of the driving Wiener process does not allow to compute trajectories of the solution at discretization points, even though the analytical formula of this solution is known. In this article we show that if values of increments of the driving Wiener process are sufficient to calculate values of a solution trajectory at discretization points, then the condition mentioned above, which allows to make an order approximation, also is fulfilled. Moreover, for a certain class of scalar jump-diffusion equations a numerical method is described which converges with uniform mean square error of order .