Martingales: The Next Steps
Discrete-time martingales live under a star of grace. They offer inferences of power, purpose, and surprise, yet they impose little in the way of technical nuisance. In continuous time, martingale theory requires more attention to technical foundations — at a minimum, we need to build up some of the basic machinery of probability spaces and to sharpen our view of conditional expectations. Nevertheless, by focusing on martingales with continuous sample paths, we can keep technicalities to a minimum, and, once we have a stopping time theorem, our continuous-time tools will be on a par with those we used in discrete time. We can then examine three classic martingales of Brownian motion that parallel the classic martingales of simple random walk. These martingales quickly capture the most fundamental information about hitting times and hitting probabilities for Brownian motion.
KeywordsBrownian Motion Conditional Expectation Dominate Convergence Theorem Simple Random Walk Uniform Integrability
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