Survey and Introduction to Modern Martingale Theory (Stopping Times, Semi-Martingales and Stochastic Integration)
Technical rept. Feb 1985-May 1986
ARMY BALLISTIC RESEARCH LAB ABERDEEN PROVING GROUND MD
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This document is primarily about the work of the Strasbourg group led by Meyer and Delacherie and the development of the modern theory or semi-martingales and stochastic integration. These developments have occurred over the last two decades and extend the theories of Doob and Ito. Unlike the sources, the first chapter introduces the subject in terms of a stochastic calculus for discrete parameter processes. In particular, discrete parameters point processes are defined and the discrete form of stochastic calculus is applied to them to obtain the nonlinear filtering formulas. There are two reasons for including the material in Chapter 1 First, discrete parameter point processes provide a useful tool for modeling a wide variety or dynamical systems second, this material provides an elementary introduction to the remaining chapters. Chapter 2 introduces the notion of a filtered probability space stopping times relative to a filtration, stochastic intervals and defines previsible, optional and progressive stochastic processes. Stochastic point processes are considered in Chapter 3 along with Lebesgue-Stieltjes stochastic integrals, the simplest of the stochastic integrals. Chapter 4 contains one of the most basic concepts of modern martingale theory the dual previsible projection and local martingales are defined in Chapter 5. The construction of the stochastic integral of bounded previsible processes relative to semi-martingales is given in the final Chapter along with some applications to Brownian motion.
- Statistics and Probability