# Stochastic Calculus and Applications (Lent 2020)

Time and Location: M-W-F, 9-10am; lectures in room MR5; example classes see below.

Lectures: Roland Bauerschmidt rb812, Example classes: Daniel Heydecker dh489

This course is an introduction to Itô calculus, in Part III of the Cambridge Tripos.

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## Schedule

• Stochastic calculus for continuous processes. Martingales, local martingales, semi-martingales, quadratic variation and cross-variation, Itô's isometry, definition of the stochastic integral, Kunita-Watanabe theorem, and Itô's formula.
• Applications to Brownian motion and martingales. Lévy characterization of Brownian motion, Dubins-Schwarz theorem, martingale representation, Girsanov theorem, and Dirichlet problems.
• Stochastic differential equations. Strong and weak solutions, notions of existence and uniqueness, Yamada-Watanabe theorem, strong Markov property, and relation to second order partial differential equations.
• Applications and examples.

Notes

## Example Classes

• Example class 1: Mo 10 Feb, 3:30-5:30 MR3 / Tu 11 Feb, 1:30-3:30 MR3
• Example class 2: Mo 24 Feb, 3:30-5:30 MR11 / Tu 25 Feb, 1:30-3:30 MR3
• Example class 3: Mo 9 Mar, 3:00-5:00 MR3 / Tu 10 Mar, 1:30-3:30 MR3
• Example class 4: Mo 20 Apr, 3:30:5:30 online / Tu 21 Apr, 1:30-3:30 online
• Drop-in session 1: Mo 17 Feb, 4:00-5:00
• Drop-in session 2: Mo 2 Mar, 4:00-5:00
• Revision class: Tu 19 May, 2:00-4:00 online

## Approximate list of lectures

1. Introduction. The Wiener Integral.
2. Lebesgue-Stieltjes integration and signed measures.
3. Lebesgue-Stieltjes integration and signed measures.
4. Finite variation processes.
5. Local martingales.
6. L^2 bounded martingales.
9. Kunita-Watanabe inequality, semimartingales, Ito integral for simple processses
10. Ito isometry
11. Consequences of Ito isometry
12. Integration by parts and Ito formula
13. Levy's characterisation of Brownian motion. Dubins-Schwarz theorem.
14. Dubins-Schwarz theorem. Girsanov's theorem.
15. Girsanov's theorem and Cameron-Martin formula.
16. SDEs
17. Strong existence of solutions.
18. Solution map.
19. Examples of SDEs and local solutions.
20. Local solutions and Dirichlet-Poisson problem.
21. Dynkin's formula. Cauchy problem.
22. Markov property.
23. Generator and invariant measures.
24. Convergence to equilibrium.

## References

I will mostly follow the following references:

• J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016.
• D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer.

I will assume the content of these references:

Lecture notes from previous years: