The numerical analysis of stochastic differential equations (SDEs) differs significantly from that of ordinary differential equations. This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations.

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"The authors draw upon their own research and experiences in obviously many disciplines... considerable time has obviously been spent writing this in the simplest language possible." --ZAMP

The aim of this book is to provide an accessible introduction to stochastic differ­ ential equations and their applications together with a systematic presentation of methods available for their numerical solution. During the past decade there has been an accelerating interest in the de­ velopment of numerical methods for stochastic differential equations (SDEs). This activity has been as strong in the engineering and physical sciences as it has in mathematics, resulting inevitably in some duplication of effort due to an unfamiliarity with the developments in other disciplines. Much of the reported work has been motivated by the need to solve particular types of problems, for which, even more so than in the deterministic context, specific methods are required. The treatment has often been heuristic and ad hoc in character. Nevertheless, there are underlying principles present in many of the papers, an understanding of which will enable one to develop or apply appropriate numerical schemes for particular problems or classes of problems.

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The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations due to peculiarities of stochastic calculus. This book provides an introduction to stochastic calculus and stochastic differential equations, in both theory and applications, emphasising the numerical methods needed to solve such equations. It assumes of the reader an undergraduate background in mathematical methods typical of engineers and physicists, though many chapters begin with a descriptive summary. The book is also accessible to others who only require numerical recipes. The stochastic Taylor expansion provides the basis for the discrete time numerical methods for differential equations. The book presents many new results on high-order methods for strong sample path approximations and for weak functional approximations, including implicit, predictor-corrector, extra-polation and variance-reduction methods. Besides serving as a basic text on such methods, the book offers the reader ready access to a large number of potential research problems in a field that is just beginning to expand rapidly and is widely applicable. To help the reader to develop an intuitive understanding of the underlying mathematics and hand-on numerical skills, exercises and over 100 PC-Exercises are included.

1 Probability and Statistics 2 Probability Theory and Stochastic Processes 3 Ito Stochastic Calculus 4 Stochastic Differential Equations 5 Stochastic Taylor Expansions 6 Modelling with Stochastic Differential Equations 7 Applications of Stochastic Differential Equations 8 Time Discrete Approximation of Deterministic Differential Equations 9 Introduction to Stochastic Time Discrete Approximation 10 Strong Taylor Approximations 11 Explicit Strong Approximations 12 Implicit Strong Approximations 13 Selected Applications of Strong Approximations 14 Weak Taylor Approximations 15 Explicit and Implicit Weak Approximations 16 Variance Reduction Methods 17 Selected Applications of Weak Approximations