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1. Stochastic Calculus

Welcome to the world of randomness in motion! This section explores the mathematical framework for modeling continuous-time random processes — essential for everything from derivatives pricing to optimal control theory and reinforcement learning.

Think of this as your reference guide where we break down the heavy machinery of stochastic calculus. We'll dive into Brownian motion, Itô calculus, martingales, and the powerful theorems that let us work with uncertainty in continuous time. Instead of just memorizing formulas, I'll try to show where they come from for having a better assessment in some models.

I'm covering the essential toolkit: stochastic integrals, Itô's lemma, change of measure techniques, and the mathematical foundations that make modern quantitative finance possible. Rigorous when needed, intuitive when helpful.

1.1 Topics Covered

  • Itô's Isometry — The fundamental property relating stochastic integrals to their variance and the bridge between Itô calculus and L² spaces

1.2 References

Below a non-exhaustive list of books I've found really good to read for learning about the topic. I'll add others soon.

Resource Description Personal Rating
1 Accessible introduction to stochastic processes with clear examples ⭐⭐⭐⭐⭐
2 Comprehensive introduction to stochastic differential equations with applications ⭐⭐⭐⭐
3 Standard reference for continuous-time financial mathematics ⭐⭐⭐⭐
4 Rigorous treatment of Brownian motion and stochastic calculus theory ⭐⭐⭐⭐
5 Advanced reference on martingales and Brownian motion ⭐⭐⭐⭐

  1. Gregory F. Lawler. Introduction to Stochastic Processes. Chapman and Hall/CRC, 2nd edition, 2006. doi:10.1201/9781315273600. ↩

  2. Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications. Springer, 6th edition, 2013. URL: https://link.springer.com/book/10.1007/978-3-642-14394-6. ↩

  3. Steven E. Shreve. Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004. URL: https://link.springer.com/book/10.1007/978-1-4757-4296-1. ↩

  4. Ioannis Karatzas and Steven E. Shreve. Brownian Motion and Stochastic Calculus. Springer, 2nd edition, 1991. ↩

  5. Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, 3rd edition, 1999. URL: https://link.springer.com/book/10.1007/978-3-662-06400-9. ↩