A new weak approximation scheme of stochastic differential equations and the Runge–Kutta method

The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, anODE-valued randomvariablewhose average approximates the given stochastic differential equation is constructed by using the notion of free Lie algebra. It is proved that the classical Runge–Kutta method for ODEs is directly applicable to the drawn ODE from the random variable. In a numerical experiment, this is applied to the problem of pricing Asian options under the Heston stochastic volatility model. Compared with some other methods, this algorithm gives significantly faster calculation times.