# Part III - Stochastic Calculus and Applications

## Lectured by R. Bauerschmidt, Lent 2018

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 The Lebesgue–Stieltjes integral
- 2 Semi-martingales
- 2.1 Finite variation processes
- 2.2 Local martingale
- 2.3 Square integrable martingales
- 2.4 Quadratic variation
- 2.5 Covariation
- 2.6 Semi-martingale
- 3 The stochastic integral
- 3.1 Simple processes
- 3.2 Itô isometry
- 3.3 Extension to local martingales
- 3.4 Extension to semi-martingales
- 3.5 Itô formula
- 3.6 The Levy characterization
- 3.7 Girsanov's theorem
- 4 Stochastic differential equations