David Williams' Probability with Martingales is a celebrated textbook in measure-theoretic probability, renowned for its lively, witty style and focus on discrete-time martingales. However, the book itself does not include an official solutions manual

Review the "A" Exercises first: These are the foundations. If you can't solve these without help, you likely need to re-read the preceding chapter.

Finding reliable solutions for David Williams ' Probability with Martingales can be challenging because there is no official solutions manual. Instead, students rely on high-quality unofficial guides from the community. 🏆 Top Recommended Solution Guides

2. The Second Lesson: Optional Stopping is a Scalpel, Not a Hammer

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) ).

Advanced Tools: Uniform Integrability (Ch 13) and Central Limit Theorem (Ch 18).

By the martingale property, we have $\mathbbE[X_n+1 | \mathcalF_n] = X_n$. Taking expectations, we get:

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.

7. Final Pro Tip

If you want the absolute best single resource:
Download the GitHub repository by “probability-martingales” (search that exact phrase). It contains:

Before diving into the best solution resources, it is important to understand why this specific book remains a staple in graduate-level mathematics: