These are among the most complete unofficial solution sets available. They cover Chapters 1 through 18, detailing complex proofs regarding uniform integrability, Doob's Decomposition, and the Martingale Convergence Theorem.
Oxford often provides public lecture notes and problem sheet solutions that align directly with Williams’ curriculum. 3. Stack Exchange (Mathematics and Cross Validated) david williams probability with martingales solutions best
Another popular open-source repository focusing on detailed, step-by-step measure theory proofs from Chapters 1 through 5. 2. University Course Websites These are among the most complete unofficial solution
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.) University Course Websites The best solution here is
Many advanced courses use Williams's book as a reference. Searching online will uncover "problem sets" and "solution guides" from universities, which can offer model answers.
For Chinese-speaking readers, there are additional resources to explore. Math StackExchange has some solutions discussed in Chinese contexts. Baidu Baike also provides an overview and discussion of the book in Chinese, which can be helpful for conceptual understanding.