Stochastic Integrals

The stochastic integrals we will define are of the form ∫ [ 0 , t ] H s d X s https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203739907/d9d2a1de-d0c2-4efc-b75b-ea60041105a0/content/inequ9_1.tif"/> or ∫ 0 t H s d X s https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203739907/d9d2a1de-d0c2-4efc-b75b-ea60041105a0/content/inequ9_2.tif"/> , where both the integrand (H_t ) and the integrator (X_t ) are stochastic processes. In 1944, K. Itô first defined the stochastic integrals of adapted measurable processes w.r.t. a Brownian motion. The key character of this kind of stochastic integrals is that the processes produced by integration are martingales (or, more generally, local martingales). In 1967, H. Kunita and S. Watanabe defined the stochastic integrals of a class of adapted measurable processes w.r.t. general square integrable martingales, and took a crucial step in developing modern theory of stochastic integrals. In 1970, C. Doléans-Dade and P. A. Meyer investigated the stochastic integrals of locally bounded predictable processes w.r.t. local martingales or semimartingales. In 1976, P. A. Meyer discussed the stochastic integrals of optional processes w.r.t. local martingales. In 1979, J. Jacod found the reasonable way to define the stochastic integrals of non-bounded predictable processes w.r.t. semimartingales. In this chapter we introduce the definition and fundamental properties of stochastic integrals (§1–§3). In §4 we present the very useful Lenglart’s inequality, and by means of it study the continuity of stochastic integrals w.r.t. integrand processes. In §5 we deal with the change of variables formula (Itô formula) and Doléans-Dade exponential formula for semimartingales. In §6 we introduce local times of semimartingales and a generalization of Itô formula. In §7 a short discussion on stochastic differential equations, by using Métivier-Pellaumail’s approach, is given.