[Right from the preface of the book, Prof. David Williams emphasizes that intuition is more important than rigour. The definition of probability in terms of long term frequency is fatally flawed and hence the author makes it very clear in the preface that “probability works only if we do not define probability in the way we talk about probability in the real world”. Meaning colloquial references to probability gives rise to shaky foundations.

Solving an SDE analytically can be done only in few instances(toy SDEs). For the majority of the cases, one solves it numerically. Having said that, this book can be read by anyone who is interested in understanding SDEs better. Simulation is a great way to understand many aspects of Stochastic processes. For example, you can read through Girsanov theorem for change of measure, but by visualizing it through a few sample paths, you have a deeper understanding .

Lloyd N. Trefethen

This book gives a non-rigorous treatment to Brownian motion and its applications to finance. Let me summarize a few points from various chapters. Chapter 1 : Brownian motion This chapter starts off by specifying Brownian motion by the properties of its increments such as independence, first and second moments, transition density etc. A discrete approximation of BM is shown via a binomial tree. Covariance of BM process is derived. A way to manufacture correlated BM is shown.