Since there is no official solution manual released by the author, students often rely on these community-driven or academic sources: University Course Pages: Search for course materials from MIT OpenCourseWare
Williams was not being cruel by omitting answers; he was being respectful. He assumed you were a mathematician, not a tourist. The best "solution" to the book is a well-worn copy, coffee-stained, with your own proofs filling every blank space.
So, go ahead. Find the GitHub repo. Download the PDFs. But then close the laptop. Take out a pen. And prove it yourself. That is the only solution that matters. David Williams Probability With Martingales Solutions
For specific "stuck" points, these platforms offer peer-reviewed proofs and alternative explanations:
To illustrate what a good solution looks like, let’s dissect three classic problem types from Williams. If you are building your own , these are the critical hurdles. Since there is no official solution manual released
Here’s what you should know, and where you can find legitimate help.
I understand you’re looking for solutions to exercises from . This is a classic graduate/probability text, but it’s important to know that official, publicly distributed solution manuals for this book are not legally available. So, go ahead
Williams often leaves parts of proofs as exercises. Completing these "internal" exercises first is essential for following the main text. Use the Doob Decomposition:
A martingale is a sequence of random variables that have the property that the expected value of the next variable in the sequence, given all prior variables, is equal to the current variable. Martingales are used to model a wide range of phenomena, including financial markets, population growth, and random walks.
GitHub Repositories: Several PhD students and researchers have compiled their own handwritten or LaTeX-transcribed solutions into public repositories. These are excellent for checking your logic after you have attempted a proof. Strategies for Solving the Exercises