An analytical introduction to stochastic differential equations. I: The Langevin equation (Q2764722)

The authors focus on an example with a simple structure, the Langevin equation, to present an introduction to the theory of stochastic differential equations, developing the ideas from the viewpoint of analysis. The program is to begin with the deterministic equation NEWLINE\[NEWLINEdX(t)/dt = c(\theta -X(t)),\tag{1}NEWLINE\]NEWLINE discretise it (with implicit Euler), add ``noise'' in the sense of a random walk, develop the notion of convergence in distribution and arrive at NEWLINE\[NEWLINEX(t) = x + c\int_0^t (\theta -X(t)) ds + \sigma W(t).\tag{2}NEWLINE\]NEWLINE On the way they provide the necessary background from probability theory, they introduce Stieltjes integrals, Gaussian systems, Brownian motion, the central limit theorem, Volterra integral equations and stochastic convolution. Finally they consider stochastic initial conditions for (2), invariant measures and Kolmogorov equations. The article is very readable and presents the material in an interesting and clear fashion.