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Stochastic Differential Equations and Diffusions in a Nutshell

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Inference for Diffusion Processes

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Abstract

Stochastic differential equations (SDEs) are a powerful and natural tool for the modelling of complex systems that change continuously in time. This chapter provides a short introduction to SDEs and their solutions, which under regularity conditions agree with the class of diffusion processes. In particular, it covers the motivation and introduction of stochastic integrals as opposed to the classical Lebesgue-Stieltjes integral, the definition of diffusion processes, key properties and formulas from stochastic calculus, and finally numerical approximation and exact sampling methods. The chapter serves as a basis for the remaining parts of this book and offers a quick access to stochastic calculus.

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Notes

  1. 1.

    Some authors denote by a Wiener process the mathematical description given above while Brownian motion stands for the physical movement of a diffusing particle. In this book, both terms are used interchangeably.

  2. 2.

    This choice of autocorrelation implies a constant nonzero power spectral density of the process, defined as the Fourier transform of its autocorrelation function. That explains the term white noise in analogy to white light, where all visible frequencies occur in equal amounts.

  3. 3.

    Contrarily to common matrix notation, but consistently with the differential equation representation, the scalar Δt k is multiplied with the vector μ from the right—a consetude that will be kept throughout this book.

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Authors and Affiliations

  1. Institute for Bioinformatics and Systems Biology, Helmholtz Zentrum München, Neuherberg, Germany

    Christiane Fuchs

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Fuchs, C. (2013). Stochastic Differential Equations and Diffusions in a Nutshell. In: Inference for Diffusion Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25969-2_3

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