Lectures

Lectures

• LECTURE 0. A preliminary meeting before the start of the class. We discussed students background and history of Mathematical Finance. I recalled some basic facts about probability theory. See e.g. Ширяев or your favorite textbook on Probability Theory.

• LECTURE 1. Stochastic Processes: general definitions (adapted processes and controls, progressive measurability). The standard hypotheses for filtrations. Continuous and cadlag versions with examples. Kolmogorov extension and continuity theorems. Brownian Motion: informal and rigorous derivations. Beyond informal discussions and examples, all the material is found in Baudoin 1.1, 1.2, 1.3, 1.4, 1.5, 2.1. The Skorohod space of cadlag functions can be found in Billingsley 3.12.

• LECTURE 2. Continuity theorem and unicity of versions. Brownian Motions and its characterizations. Martingales and main examples. Baudoin 1.5, 1.7. Protter I.1-I.2-I.3.

• LECTURE 3. Stochastic integration with respect to semimartingales I.  Protter II.1-II.4.

• LECTURE 4. Stochastic integration with respect to semimartingales II.  Protter II.1-II.4.

• LECTURE 5. General integration of caglad process w.r.t. semimartingales. Stability of semimartingales. Quadratic variation. Ito formula. Protter II.4-II.7.

• LECTURE 6. Financial markets, portfolios and arbitrage. Oksendal 12.1 (more precisely, generalization to semimartingales).

• LECTURE 7. No arbitrage principle. General Girsanov Theorem. Oksendal 12.2, for the Girsanov theorem in our framework, see this post.

• LECTURE 8. Exercises only.

• LECTURE 9. Pricing contingent contracts with fixed maturity time.

• LECTURE 10. Pricing in a Markovian Market and discrete time. Oksendal 12, Shiriayev VI.1

• LECTURE 11. Review, and shortcominsg of the models.

• LECTURE 12. Examples of pricing for markets with jumps. Lamberton 7.1-7.2-7.3-7.4.

• LECTURE 13. Optimal control theory in a Markov environment. Fleming-Soner IV.1-IV.2-IV.3

• LECTURE 14. Viscosity solutions. Fleming-Soner IV.1-IV.2-IV.3 and this note. There were technical issues during the lecture, you can get a recording here, and here.

• LECTURE 15. Viscosity solutions. Recording here, and here.

• LECTURE 16. Optimal control, mean-field theories. Carmona-Delarue.

• LECTURE 17. Mean-Field Games. Carmona-Delarue.

ReFERENCES

1. Akyıldırım,  Soner - A brief history of mathematics in finance, Borsa _Istanbul Review 14 (2014) 57e63.

2. Baudoin F., Diffusion processes and stochastic calculus. European Mathematical Society, 2014.

3. Bäuerle N., Rieder U., Markov decision processes with applications to finance. Springer Science & Business Media, 2011.

4. Billingsley P., Convergence of probability measures. John Wiley & Sons, 2013.

5. Brigo D., Mercurio  F., Interest Rate Models - Theory and Practice. Springer, 2006.

6. Fleming W.H., Soner H.M., Controlled Markov processes and viscosity solutions. Springer Science & Business Media, 2006.

7. Oksendal B., Stochastic differential equations: an introduction with applications. Springer Science & Business Media, 2013.

8. Protter P., Stochastic integration and differential equations. Springer, 2005.

9. Shiryaev A.N., Essentials of stochastic finance: facts, models, theory. World scientific, 1999.

10. Carmona R., Delarue F., Probabilistic Theory of Mean Field Games with Applications, Springer, 2018.

11. Ширяев А.Н., Вероятность-1/2. МЦНМО, 2007.