One year the department organized a reading seminar on Brownian motion and stochastic integration. Williams chose problems that tested limits: martingales in continuous time, quadratic variation, and the Itô isometry. He demonstrated a technique he loved—localization—by telling a fable about explorers who map a continent piecemeal, using compact maps to piece together the whole. Students learned to replace global assumptions with local boundedness, then stitch results together. When students encountered a stubborn integral, Williams nudged them toward stopping sequences and dominated convergence, turning an analytic wall into stepping stones.
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Uniform integrability is the magic ingredient that turns modern probability into a precise science. It dictates exactly when convergence in probability implies convergence in mean.
Williams does not write standard, algorithmic textbook problems. His exercises require a blend of rigorous analysis and creative problem-solving. One year the department organized a reading seminar
He first met martingales on a rain-slick afternoon in the university library. A graduate student left an open notebook on a table; inside were crisp proofs and diagrams under the heading “Stopping Times.” Williams sat down and traced the argument: a fair game whose expected value, given the present, stayed the same. The simple definition hid power. Martingales were threads that wove past and future into a single fabric, and Williams wanted to pull that fabric apart.
theorem to prove uniqueness of measures. Mastering this technique early will unlock half the proofs in Part A. Students learned to replace global assumptions with local
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The hints provided in the appendix are often cryptic, sometimes consisting of a single word or a suggestive drawing.