This page is an index into the various stochastic calculus posts on the blog.
I decided to use this blog to post some notes on stochastic calculus, which I started writing some years ago while learning the subject myself. The aim was to introduce the theory of stochastic integration in as direct and natural way as possible, without losing any of the mathematical rigour. The required background for properly understanding these notes is measure theoretic probability theory. These notes are currently in progress, and are being updated regularly.
In addition to the notes listed above, I am also starting to post examples demonstrating the various results and techniques of stochastic calculus, together with counterexamples to show how they can fail if the necessary conditions are not met. In stochastic process theory, in particular, there are often measurability or integrability conditions required which, if they are not met, can cause the expected results to fail in quite subtle ways. The aim is to build up a collection of examples showing what can go wrong, and to help understand the limits of the standard theory.
- Stochastic Calculus Examples and Counterexamples
- Failure of Pathwise Integration for FV Processes
- Failure of the Martingale Property
- The Optimality of Doob’s Maximal Inequality
- Martingales with Non-Integrable Maximum
- Failure of the Martingale Property For Stochastic Integration
- A Martingale Which Moves Along a Deterministic Path
- Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample
Posts on stochastic calculus which do not fit into the categories above are listed here.