Ebook: Stochastic integrals

This text introduces the basic concepts of the theory of stochastic integrals for semimartingales. Having introduced martingales and local martingales, the stochastic integral is defined for locally uniform limits of elementary proceeses. This corresponds to the Riemann integral in one-dimensional analysis, and it suffices for the study of stochastic differential equations and diffusions, including the Feynman-Kac formula and the Stroock-Varadhan martingale problem approach. Predictability is introduced mainly as a tool for the structure theory of semimartingales which culminates in the Dellacherie-Bichteler characterization theorem. Besides these abstract parts, the material of the text is designed for a one-semester course.