Un survol de la theorie de l'integrale stochastique

AbstractThe objective of this paper is to present the principal results of a large part of stochastic calculus in a manner that should be comprehensible to readers having only the general notions of stochastic processes. Not all the theorems are proved in detail, but all the fundamental theorems are explained with clarity and precision, and with special attention to the motivations behind them.Given two real valued stochastic processes X and Y, the basic problem is to give a meaning to Z = ∫ Y dX in such a way that the integral sign is not misused. If X is a process whose paths are of bounded variation, then Z should coincide with the ordinary Lebesgue-Stieltjes integral taken path by path. If Y is a left continuous step function, then Z should coincide with the obvious choice: if Y is constant on ]t, u], then Zu-Zt is that constant times Xu-Xt. And finally, the Lebesgue dominated convergence theorem should hold: if the processes Yn converge to Y and all the Yn are dominated by a process Y' for which ∝ Y' dX is well defined, then Zn = ∫ Yn dX should converge to Z = ∫ Y dX in some sense.Starting with these requirements, it is shown that, if ∫ Y dX is defined for all predictable Y, then X must be a semimartingale. Conversely, the integral is well defined for all predictable Y and all semimartingales X.With the integrals defined, a number of their important properties are discussed. In particular, the integral Z is a semimartingale, and a change of variable formula (Ito's formula) holds for ⨍(Z). Finally, stochastic integral equations are introduced, and a general theorem is given on the existence and uniqueness of solutions.A bibliography with commentaries supplements the text for the benefit of those who would like to go deeper into the subject