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.
By the end of the book, Elena had a method, distilled from Williams’ marginal notes and problem design: david williams probability with martingales solutions best
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.
It is a common oversight that Williams provides solutions or strong hints to a number of exercises directly in the text. While not formally published, a typeset PDF often
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:
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. How to find it legally : Check with
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.
When looking for solutions, your best strategy is to look for course materials from universities that use this text.