ABSTRACT
K. Itô invented his famous stochastic calculus on Brownian motion in 40's. In the same period, J.L. Doob developed a martingale theory and related stochastic processes to an increasing family of a-algebras (Tt) of events, where Tt expresses the information avilable until time t. From 60's to 70's the "Strasbourg school", headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic caluculs on semimartingales. It turned out soon that semimartingales constitute the largest class of right-continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes, but also has broad applications to many branches of mathematics (e.g. partial differential equations, differential geometry, stochastic control), physics, engineering, mathematical finance and all other domains in which random dynamic structures are involved.
