Continuous Runge-Kutta Methods for Stratonovich Stochastic Differential Equations

Stochastic differential equations (SDEs) are applied in many disciplines like physics, biology or mathematical finance in order to describe dynamical systems disturbed by random effects. Approximation methods for the strong as well as for the weak time discrete approximation have been proposed in recent years (see, e.g., [BBOO, DR06, KP99, KMS97, Mil95, MT04, New91, Roe04, Roe06b, TVA02] and the literature therein) converging with some given order at the discretization points. However, there is still a lack of higher order continuous time approximation methods guaranteeing uniform orders of convergence not only at the discretization points but also at any arbitrary time point within the approximation interval. Classical time discrete methods are inefficient in this case where the number of output points has to be very large because this forces the step size to be very small. Therefore, we develop a continuous extension of the class of stochastic Runge-Kutta (SRK) methods introduced in [Roe06c] for the weak approximation which provides continuous time approximations of the solution of Stratonovich SDE systems with uniform order two in the weak sense. Such methods are also called dense output formulas [HNW93]. The main advantage of the presented continuous extension of the SRK methods is their negligible additional computational complexity compared to the time discrete SRK methods. Especially, we are interested in continuous sample trajectories of the applied SRK method. For example, an SRK method with continuous sample trajectories allows the use of an individual discretization for each sample trajectory which needs not necessarily to contain some common discretization points for all trajectories in order to be able to calculate the expectation at these common time points. Further, in future research SRK methods with continuous sample trajectories may be applied for the numerical treatment of stochastic delay differential equations like in the deterministic setting where continuous Runge-Kutta methods are already successfully applied.