An analytical introduction to stochastic differential equations: Part I - the Langevin equation
van Gaans, O. W.
We present an introduction to the theory of stochastic differential equations, motivating and explaining ideas from the point of view of analysis. First the notion of white noise is developed, introducing at the same time probabilistic tools. Then the one dimensional Langevin equation is formulated as a deterministic integral equation with a parameter. Its solution leads to stochastic convolution, which is defined as a Riemann-Stieltjes integral. It is shown that the parameter dependence yields a Gaussian system, of which the means and covariances arde computed. We conclude by introducing briefly the notion of invariant measure and the associated Kolmogorov equations.
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Ph. Clément and O. W. van Gaans, "An analytical introduction to stochastic differential equations: Part I - the Langevin equation", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 32 (2000), pp. 217-269.