An introduction to numerical methods for stochastic differential equations

Abstract

This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises

“…The initial value and boundary value problems of stochastic partial differential equations (SPDEs) have been studied theoretically in, for example, [5,6,8,10,33]. Various numerical methods and approximation schemes for SDEs have also been developed, analyzed, and tested [1,2,4,7,12,13,14,15,20,25,27,29,28,31,34,35].…”

Discretisation of FBSDEs driven by càdlàg martingales

“…In the following theorem we state the discretisation error of the approximation of (2.4) by (3.1). We refer to Platen [20] for a proof. …”

Polynomial Chaos based on the parallelized ensemble Kalman filter to estimate precipitation states

“…There exist a variety of methods in the literature including the estimation by maximum likelihood ( [35], and [20]); the Monte Carlo Markov Chain techniques (MCMC) ( [26], [34], [42], [43]); and the sequential Monte Carlo algorithms (SMC). Examples of the latter are the Kalman filter, the extended Kalman filter (EKF), the particle filter (PF) and the unscented particle filter (UPF) (see [2], [22], [16], [23], [24], [4], [44], [18], [25], [29], [9], [30], [36], [41], [14], [8], [15], [17], [28], [39], [40], [6], [21], [3] and [38]).…”