Since full solutions are rare, these are often better for learning:
: This is arguably the most comprehensive site, offering detailed, step-by-step solutions for early chapters, including Measure Spaces, Events, and Independence. david williams probability with martingales solutions best
. However, several unofficial, high-quality resources provide solutions to most of the exercises: Recommended Unofficial Solution Guides Ryan McCorvie’s Solutions Since full solutions are rare, these are often
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.” But Williams demands more: “Explain in words why
: Measure Spaces (Ch 1) and Conditional Expectation (Ch 9).
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.
Search for the specific exercise number (e.g., "Williams Probability with Martingales Exercise 13.2").