Numerical analysis of stochastic differential equations without tears (Q2784993)

This article presents a compendium of definitions and results in the area of numerics for stochastic ordinary differential equations (SODEs). The list of references alone consists of 570 entries (somewhat irritating is that papers by the same author appearing in the same year do not get the usual a's and b's to distinguish between them). After the setting of the scene for SODEs and an introduction to Itô-Taylor expansions, a toolbox of numerical methods for approximating solutions of SODEs is provided. The remainder of the article can be roughly divided into three parts. The first part is concerned with, what the author calls ``the main principles of numerics'', i.e. definitions of consistency, convergence etc. are given and several convergence results are stated. The second part deals with qualitative properties of the solutions of the SODEs and the solutions of various numerical methods. Theorems concerning, for example, stability or stationarity are given. The last part treats implementation issues, after presenting some interesting test cases. These issues range from pseudo-random number generation, via adaptive step-size strategies to variance reduction. Several comments concerning future developments conclude the article.NEWLINENEWLINEFor the entire collection see [Zbl 0979.00017].