David Williams Probability With Martingales Solutions Best May 2026

First, let's appreciate the beast. Williams writes with a witty, almost conversational style—rare for rigorous probability. But don't let the charm fool you. The exercises are deliberately sparse in hinting and heavy in synthesis.

Unlike modern textbooks that separate "warm-up" from "challenge" problems, Williams’ exercises are integrated into the narrative. A typical exercise might ask you to prove a lemma that he will use two pages later. If you skip it, you lose the thread.

The core difficulties include:

Without high-quality solutions, a student can spend a week stuck on a single problem, mistaking a typo in their reasoning for a lack of ability.

If you are studying advanced probability theory, there is one name that inevitably invokes a mix of reverence and terror: David Williams.

His book, Probability with Martingales, is considered a masterpiece of mathematical literature. It is concise, rigorous, and beautifully written. However, it is also notorious for its "terse" style. Williams often leaves significant gaps for the reader to fill, and the exercises can be brutally challenging. david williams probability with martingales solutions best

If you found yourself searching for "David Williams Probability with Martingales solutions best," you are likely stuck on a problem, frustrated by a lack of hints, or simply trying to ensure your understanding is on the right track.

Because official solution manuals for this text are scarce or non-existent, students often feel stranded. In this post, we break down the best strategies and resources to find solutions and master this essential text.

Probability with Martingales is a standard text for graduate-level courses in Stochastic Analysis at top universities (Cambridge, Oxford, MIT, etc.).

Elena’s first encounter was Exercise 4.3 (paraphrased):
Let ( X_n ) be a symmetric random walk. Show that ( X_n^3 - 3nX_n ) is a martingale.

Her instinct was to expand and condition blindly. She wrote pages of algebra, got lost, and peeked at the back—where Williams often writes not a full solution, but a mocking or encouraging remark. For this exercise? “Use the ‘increment trick’ and the fact that ( X_n^2 - n ) is a martingale.” First, let's appreciate the beast

She realized: Williams doesn’t give solutions. He gives hints that teach you a method. The method here: express a candidate martingale ( M_n = f(X_n) - A_n ) where ( A_n ) is compensator. For a random walk with variance 1 per step, ( \mathbbE[X_n+1^3 \mid \mathcalFn] = X_n^3 + 3X_n ). So to cancel the drift, subtract ( 3nX_n ). The best solution is the one that generalizes: find ( A_n ) such that ( \mathbbE[Mn+1 \mid \mathcalF_n] = M_n ). That is the martingale problem in embryo.

Key takeaway from Williams: A martingale is a fair game relative to the past. To construct one, compute the conditional expectation of the next step and remove the predictable part. That is the Doob decomposition in disguise.

If you are looking for the "best" source for solutions:

Remember: The value of Probability with Martingales lies in the struggle with the measure-theoretic rigor. A solution manual is a crutch; community discussion is a classroom.

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) ). Without high-quality solutions, a student can spend 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.

If the best solution uses a lemma (e.g., the "Scheffé’s lemma" for $L^1$ convergence), and you don't recognize it, stop and go back to Williams or another reference (e.g., Durrett). The goal is to fill gaps, not to memorize.

While not formally published, a typeset PDF often attributed to various authors (most coherently D. R. Wood) circulates in academic circles. It covers roughly 80% of the exercises in Chapters 4–14. Its quality is high because it:

How to find it legally: Check with your university library’s digital repository or ask a course instructor. Some professors keep a copy for teaching assistants.