David Williams Probability With Martingales Solutions | Best
Williams loves problems where the solution hinges on choosing $T = \minS_n$ or similar. The best solutions explain why that stopping time works, not just that it does. They also check integrability conditions for optional stopping.
By the end of the book, Elena had a method, distilled from Williams’ marginal notes and problem design:
Chapter 8: Martingale convergence. Exercise 8.7:
Let ( M_n ) be a nonnegative martingale. Show that ( M_\infty = \lim M_n ) exists a.s. and ( \mathbbE[M_\infty] \le \mathbbE[M_0] ). Give an example where inequality is strict.
Standard answer: Doob’s forward convergence theorem (upcrossings). But Williams demands more: “Explain in words why ( \mathbbE[M_\infty] < \mathbbE[M_0] ) means ‘mass escaping to infinity’ — e.g., the martingale that is 1 initially, then with probability 1/2 doubles, with probability 1/2 goes to 0, and so on — the ‘Pólya’s urn’ type? No, that’s bounded. Better: ( M_n = 2^n \cdot 1_[0,2^-n] ) on [0,1] with uniform distribution? That’s not a martingale in the usual filtration.”
Actually, Williams’ own famous example: ( M_n = \prod_i=1^n (1 + X_i) ) where ( X_i ) are independent with mean 0 but ( \mathbbE[X_i^2] ) small? No — that explodes. The clean one: ( M_n = ) number of female births in branching process? Not quite.
But the best solution here is not the example — it’s the insight: strict inequality means some probability mass is lost in the limit because ( M_n ) is not uniformly integrable. Williams wants you to feel the difference between a.s. convergence and ( L^1 ) convergence. david williams probability with martingales solutions best
Based on extensive student feedback from mathematics forums (MathStackExchange, Reddit’s r/math, and The GradCafe), here is the current ranking of solution sources:
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.
Here is the paradox: having the best solutions can ruin your learning if used carelessly. To avoid that:
Elena eventually became a researcher. Years later, she recalled Williams’ own words from the preface:
“I have tried to show that martingales are not just a subject, but a way of thinking.”
The “best” solution in Probability with Martingales is not the shortest, nor the one with the cleverest trick. It is the one that reveals the structure: Williams loves problems where the solution hinges on
In that sense, David Williams’ book doesn’t give you answers. It gives you a pair of glasses through which random processes reveal their fair-game essence. And once you see that, every problem’s solution becomes a small act of discovery — not a computation, but a proof that the world, properly conditioned, plays fair.
Finding complete official solutions for David Williams' Probability with Martingales
is rare, as the textbook is designed for students and emphasizes that exercises "play a vital role". However, several high-quality community resources and student-led solution sets are widely recognized as the "best" alternatives for self-study. Amazon.com Top Solution Resources dbFin Exercise Solutions
: This is one of the most structured resources, providing organized links to answers for early chapters (Chapter 0 through Chapter 4). Visit dbFin - Williams Solutions for these categorized notes. Ryan McCorvie’s Solutions
: This resource covers more advanced chapters, including detailed breakdowns for Chapter 12 In that sense, David Williams’ book doesn’t give
problems (e.g., Branching processes and Kronecker’s Lemma). Access them at martingale.ai Probability99 (WordPress)
: A community blog that features long-form discussions and solutions for tricky sections like Exercises G Chapter 10 (Optimal Stopping). Check out the Williams Exercises Discussion for intuitive explanations. Stack Exchange (Mathematics)
: For specific problems (e.g., Exercise 4.1 or 9.2), Math Stack Exchange contains detailed community-vetted proofs and clarifies the "hints" provided in the textbook. Search for
"Williams Probability with Martingales" on MathStackExchange Mathematics Stack Exchange Content Navigational Guide
The book is famous for its lively, selective style rather than being encyclopedic. If you are self-studying, keep these points in mind: Google Books Williams 'Probability with martingales' E9.2
This book (often called "PWM") is a classic but famously terse. The exercises are non-trivial, and official solutions do not exist. The "best" solutions, therefore, are those that are rigorous, well-explained, and community-vetted.
Probability with Martingales is a standard text for graduate-level courses in Stochastic Analysis at top universities (Cambridge, Oxford, MIT, etc.).