Algorithms for Linear Stochastic Delay Differential Equations

Abstract

AbstractModels consisting of linear, N-dimensional stochastic delay differential equations present a particular set of challenges for numerical simulation. While the user often seeks the probability density function of the solution, currently available methods rely on Monte Carlo sampling to generate sample paths, from which a density function must be estimated statistically. In the present work, we derive a new algorithm to compute the density function of the solution with no sampling. By discretizing the stochastic equation in time, we bypass closure issues that prevent a Fokker–Planck approach. We carry out a number of numerical tests to compare the algorithm against Monte Carlo methods, to judge its behavior as the time step decreases, and to check its capabilities of handling fully vectorial systems, both with and without time delays. The results indicate that the algorithm is a fast, accurate alternative to existing methods.KeywordsProbability Density FunctionSample PathVector CaseMarkov Chain MethodMonte Carlo VarianceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.