I have decided to use my blog to post some notes that I initially made on stochastic calculus when learning the subject myself. I wrote these after reading through some books which took an unnecessarily long and difficult route to get to the interesting stuff which I was interested in. Complicated and rather obscure subjects such as optional and predictable projection and a lot of theory of continuous-time martingales were dealt with at length before getting round to the general theory of stochastic integration. Consequently, I decided to go through the theory myself in a more direct way, while still working out rigorous proofs of all the more useful theorems which I was interested in learning. The result was three small notepads containing the following.
- Basic definitions regarding continuous-time filtrations, adapted processes, predictable processes, stopping times, martingales, etc.
- Some useful elementary results such as the debut theorem for right continuous processes and the existence of cadlag versions of martingales
- Definition of stochastic integration and elementary properties.
- Definition and elementary properties of quadratic variations.
- Ito’s formula, including the generalized Ito formula for non-continuous processes.
- Stochastic integration with respect to martingales.
- The Doob-Meyer decomposition.
- Quasimartingale decompositions.
- Decompositions of semimartingales.
- Decompositions and integration with respect to special semimartingales.
This covers a lot of the general underlying theory required. Of course, being able to apply this to practical applications requires further knowledge of stuff like stochastic differential equations. Time permitting, I’ll start to post these notes here.As I have learned much more since originally making these notes, I will attempt to simplify or improve on the originals where possible.
The prerequisite knowledge required to properly understand these notes is measure theoretic probability theory (e.g., properties of the Lebesgue integral such as dominated convergence, Fubini’s theorem, spaces, convergence in probability, etc.).