Deterministic quadrature formulas for SDEs based on simplified weak Itô-Taylor steps
A deterministic algorithm for the weak approximation of the solution of a stochastic differential equation (SDE) is constructed. For the time-stepping, a simplified weak order \(\gamma\) Ito-Taylor method is used. After every time step, the diameter and cardinality of the support of the discrete measure are reduced to avoid an explosion of the computational cost. Under suitable smoothness assumptions on the coefficients of the SDE, the resulting algorithm yields the optimal rate of convergence of the error in terms of complexity, i.e., \(r/d\), where \(d\) is the dimensionality of the problem and the payoff function has bounded derivatives up to order \(r\). Results of numerical simulations in one space dimension are given. In higher space dimensions, however, the support cardinality reduction step becomes computationally expensive. Hence it is not clear whether the algorithm as written in the paper can be practically employed for \(d\geq 2\).
scientific article; zbMATH DE number 6695908
scientific article