Stochastic Differential Equations

Differential equations are used to describe a wide variety of physical phenomena. However, in many situations, observed phenomena only roughly follow the trajectories of the equations that seem to correspond to them. The possible causes of such behavior can be varied: modeling error, fluctuation of equation parameters over time, presence of observational noise, etc. In these situations, probabilistic approaches naturally find their place, and it can then be interesting to incorporate random terms into differential equations in order to take into account the preceding uncertainties. However, the introduction of these random terms leads to an integration of the equations that does not, in general, correspond to an immediate adaptation of classical differential equation theory.

The first aim of this article is to introduce Itô's calculus, which allows us to tackle stochastic differential equations. We begin with a few reminders and complements of probability theory (section 1). After presenting some important results relating to the Itô calculus, we'll see (section 2) how it can be used to solve stochastic differential equations (SDEs, section 3). As in the case of classical differential equations, it is generally not possible to accurately integrate SDEs. We will therefore present some techniques for obtaining numerical approximations of the trajectories of SDEs (section 4). In many problems, the EDS parameters are not known a priori, and the user needs to estimate them from the parametric model data and the realization of one or more EDS trajectories. A general approach for dealing with this kind of situation will be presented (section 5). We will also see how non-parametric approaches can be used in cases where no a priori parametric model is available to estimate the EDS coefficients. An introduction is also given to Lévy processes, which can be used to extend the study of EDS to the case where the model input has discontinuities (section 6).

Although this article provides only a brief account of the preceding topics, which have been more extensively developed in numerous specialized articles and books, it is nevertheless intended that, through the variety of these topics and the examples proposed, the reader should understand the general philosophy and acquire a practical ability to model and solve various engineering problems that he or she may encounter and for which the use of SDEs may prove useful. The presentation can be supplemented by reading the main references on which it is based, in particular Bernt Øksendal's book Sochastic differential equations. , a...