Text on Probability Theory applied to Actuarial Science

Question

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life contingencies exam in April.

For my final semester as an undergraduate, I am doing an independent study to learn about measure-theoretic probability theory in the context of actuarial science. I am especially interested in learning the theory behind Ito calculus and the proof of the Black-Scholes equation and formula, rather than just doing routine calculations using these formulas (like in exam MFE).

The current plan is to start off with A Probability Path and Adventures in Stochastic Proccesses both by Resnick, but none of these cover Ito calculus. They touch on Brownian motion and martingales for a little bit, but not very much of it.

I have taken two semesters of real analysis (we covered everything up to complete metric spaces and integration and differentiation in $\mathbb{R}^n$) and will be taking my second semester in abstract algebra (vector spaces, group actions, Sylow $p$-groups, and some other stuff I don't know about). Is there a text that we can use during this independent study that would be accessible to me that pertains to Ito calculus and its applications to finance (and/or actuarial science)?

Edit: Two texts that I have found in my research are Brzezniak and Zastawniak and Øksendal. Does anyone have a particular preference of one of these over the other? Are there any other texts you would recommend?

Answer 1

The books you found are very good.

A classical is Steven Shreve's masterful two-volume text, Stochastic Calculus for Finance, which introduces students to stochastic calculus as a tool for financial derivative pricing

• Stochastic Calculus for Finance I - The Binomial Asset Pricing Model
• Stochastic Calculus for Finance II - Continuous-Time Models Shreve's much longer second volume, Continuous-Time Models, is a self-contained introduction to stochastic calculus and its applications to financial modeling. In my view, there is no better introductory treatment of the topic.

The celebrated books of L.C.G. Rogers, D. Williams, Diffusions, Markov processes, and martingales (my favourites)

• Diffusions, Markov Processes, and Martingales - Volume 1. Foundations
• Diffusions, Markov Processes and Martingales - Volume 2. Itô Calculus