A Higher order and stable method for the numerical integration of Random Differential Equations

Abstract

Over the last few years there has been a growing and renovated interest in the numerical study of Random Differential Equations (RDEs). On one hand it is motivated by the fact that RDEs have played an important role in the modeling of physical, biological, neurological and engineering phenomena, and on the other hand motivated by the usefulness of RDEs for the numerical analysis of Ito-stochastic differential equations (SDEs) -via the extant conjugacy property between RDEs and SDEs-, which allows to study stronger pathwise properties of SDEs driven by different kind of noises others than the Brownian. Since in most common cases no explicit solution of the equations is known, the construction of computational methods for the treatment and simulation of RDEs has become an important need. In this direction the Local Linearization (LL) approach is a successful technique that has been applied for defining numerical integrators for RDEs. However, a major drawback of the obtained methods is its relative low order of convergence; in fact it is only twice the order of the moduli of continuity of the driven stochastic process. The present work overcomes this limitation by introducing a new, exponential-based, high order and stable numerical integrator for RDEs. For this, a suitable approximation of the stochastic processes present in the random equation, together with the local linearization technique and an adapted Pad´e method with scaling and squaring strategy are conveniently combined. In this way a higher order of convergence can be achieved (independent of the moduli of continuity of the stochastic processes) while retaining the dynamical and numerical stability properties of the low order LL methods. Results on the convergence and stability of the suggested method and details on its efficient implementation are discussed. The performance of the introduced method is illustrated through computer simulations.