In the pantheon of probability textbooks, most sit quietly on shelves, offering theorems as tombs and proofs as epitaphs. Then there is David Williams’ Probability with Martingales. It is short, dense, and famously opinionated. To the uninitiated, its exercises look like traps. To the initiated, it is an oracle—but an oracle that demands you learn to listen in a particular way.
This is the story of how one graduate student, call her Elena, learned to find best solutions to Williams’ martingale problems, not by brute force, but by absorbing the book’s hidden philosophy.
Sometimes the best "solution" is a better explanation. If you are stuck, it might be because Williams' definition was too brief.
Take the best solution PDF and add your own marginalia: "Here I forgot to check uniform integrability" or "Alternative: use Jensen for conditional expectation". This transforms someone else’s solution into your understanding.
Not all solution sets are created equal. A quick GitHub search reveals dozens of incomplete, error-ridden, or handwritten PDFs. The best solutions for "Probability with Martingales" share four traits:
David Williams had learned to read the world in probabilities. Growing up in a coastal town where fog rolled thicker than certainty, he found solace in numbers that measured chance—dice, coin flips, and later, conditional expectations that bent the future around present information. By his late twenties he was a young professor with a battered copy of a classic text on his desk and a quiet obsession: martingales. david williams probability with martingales solutions best
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.
Word of his curiosity spread. A student, Mira, arrived one semester having failed an exam but carrying relentless questions. She wanted solutions, not just answers—methods she could reuse. Williams taught her with stories. For optional reading he handed her a slim monograph whose title included “martingales” and “Brownian motion.” He insisted she try to solve problems before looking at solutions, to feel the tug between intuition and rigor.
They began with a puzzle: a gambler’s fortune modeled as a martingale. If the gambler stops when reaching a target or falling to ruin, is the expected fortune at stopping equal to the starting fortune? Williams led Mira through optional stopping—conditions under which the stopping time preserves expectation. They probed counterexamples where stopping could break the equality. Mira wrote her first proof by hand, pausing to imagine each inequality as a physical balance.
Williams favored solutions that told a story. For Doob’s decomposition, he drew two rivers: one steady current (a martingale) and one predictable flow (drift). Together they formed the observed process. In exercises, he asked students to separate these streams. He showed them how every integrable process could be split: the martingale part carrying the “surprises,” the predictable part carrying the “foreseeable.” The classroom filled with diagrams and metaphors—martingales as fair bets, stopping times as referee whistles.
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. The Unlikely Oracle: How David Williams Teaches You
Beyond teaching, Williams wrote solutions—careful, annotated, and practical. He preferred constructions that revealed why a result held, not just that it did. For a tricky problem asking to show that a uniformly integrable martingale converges almost surely and in L1, his solution began with basic lemmas: show convergence in probability using maximal inequalities, then upgrade with uniform integrability to L1. He annotated each step with the intuition: control tail mass, squeeze out oscillation, and lock convergence with integrability.
Mira watched Williams craft these solutions like a composer arranging notes. He introduced optional sampling with precise hypotheses: bounded stopping times or uniformly integrable martingales. He offered counterexamples when hypotheses were weakened—an unbounded fair game where stopping time ruins the expectation. The students learned caution as much as technique.
Outside the classroom, Williams applied martingale methods to problems that once seemed unrelated. In a consulting project with an environmental agency, he modeled pollutant levels as stochastic processes and used stopping rules to design alert thresholds. In probability seminars, his favorite trick was using martingale transforms to bound tail probabilities: turn a process into a supermartingale, apply maximal inequalities, and extract exponential tails. The trick worked like a lens focusing scattered randomness into clear bounds.
One winter, Mira faced her qualifying exam. The final question: Prove that every L2 martingale admits a predictable representation with respect to an orthogonal martingale basis—essentially, decompose increments along uncorrelated directions. She remembered Williams’s voice: “Find the right projection.” Her proof unfolded: project the martingale increments onto the span of basis elements, use orthogonality to get coefficients, and show convergence in L2. Her committee applauded not just the proof but the clarity.
Years later, Williams received a letter from Mira—now a researcher—describing how martingale methods guided her work in randomized algorithms. She credited his solutions for the way they taught her to build arguments: begin with a model, test hypothesis sharpness, craft a stopping time, and use martingale inequalities to get high-probability guarantees. Williams kept that letter pinned above his desk like a theorem with a particularly elegant proof. Annotate Your Copy Take the best solution PDF
His legacy became the solutions themselves: a collection of problem answers that balanced rigor and intuition, each one a map for the next traveler. He emphasized the essential rules: check integrability, verify stopping-time hypotheses, use localization when global bounds fail, and always seek the martingale hidden in a process.
On the last page of his notes, Williams wrote a final challenge: “Find a martingale that tells you more than expectation—one that reveals structure.” He passed that challenge on to a new generation. Students left his course with notebooks full of detailed solutions and a new way of seeing chance: not as chaos, but as a landscape navigable by martingales—fair, precise, and full of hidden paths.
And in that coastal town, where fog still rolled in and out, people began to notice the clarity that mathematics can bring: a method to stop, to check, and to expect rightly. Williams’s solutions had become more than answers; they were a craft, teaching others how to turn problems into proofs and uncertainty into understanding.
Midway through the book, Elena faced a classic:
Simple symmetric random walk, ( T = \minn : X_n = a \text or X_n = -b ). Compute ( \mathbbP(X_T = a) ).
She knew the standard solution: use the martingale ( X_n ) and optional stopping theorem. But Williams’ twist: “Beware — ( T ) is not bounded. Check uniform integrability.” Then, in a footnote, he reminds: “Better: use the bounded martingale ( X_n \wedge T ).”
The best solution here is not the slickest formula, but the one that explicitly verifies the conditions. Williams trains you to treat optional stopping as a precision instrument: check bounded stopping time, or bounded increments + finite expectation, or uniform integrability. Otherwise, you get nonsense (e.g., predicting ( \mathbbE[X_T] = 0 ) when ( T ) is the time to hit ±1 starting from 0 — which is false because ( T=1 ) almost surely? Wait, that’s a trap — actually for symmetric RW starting at 0, ( T ) to hit ±1 has ( \mathbbE[X_T]=0 ) because ( X_T ) is symmetric. Williams loves these subtle checks.)
The “best” solution in his sense is the one that justifies each step with a theorem from earlier in the book, no hand-waving.