From 2018.02.27 to 2018.06.14

Tuesday: 12:55-14:45

Thursday: 16-17:40

Place: Chen Ruiqiu Building, room 223

Office hour: Tuesday from 16:00 to 17:00

Office: Math building, room 2.105

Teaching Assistant: Liming Yin - 殷礼鸣

This lecture is an introduction to the theory of stochastic processes, with a special focus on Martingale Theory. The discrete time framework is first addressed and then the continuous time with the construction of Ito’s integral. One finally address with the help of Ito’s Formula some Stochastic differential equations. Throughout the whole course we illustrate the theory by means of examples within financial mathematics.

Measure and probability theory, topology and analysis.

- Lecture, homework sheet, and exam are in English.
- Every week a lecture note following what is taught in class is provided on the wechat group.
- Every week, a homework sheet has to be solved (by group of 2 or 3 students) and will be graded. The homework sheets are available on this webpage, section Lecture Material. The homework sheet should be handed electronically to the teaching assistant on Friday before the lecture.
- Final exam in June
- Final grade is a pondered average between homework sheet and final exam

- Recall of basics in Probability theory, conditional expectation, independence, uniform integrability. (1 week)
- Martingale Theory (4 weeks)
- Martingale (recall from discrete), Doob’s optional sampling theorem, Doob’s upcrossing’s lemma, doob’s inequality, modification of martingales.
- Stochastic processes, modification, indistiguishability, rcll lcrl, stopping times, hitting times
- Martingale (recall from discrete), Doob’s optional sampling theorem, Doob’s upcrossing’s lemma, doob’s inequality, modification of martingales.
- Browninan Motion, Poisson process, Levy process Process of Bounded Variations, predictable, Doob-Meyer decomposition
- Local Martingales

- Stochastic Integration (4 weeks)
- Lebesgue’s Stieljes integral, Chain-rule Formula
- Quadratic Variations
- Ito-isometrie, Stochastic integral
- Covariations, Watanabe’s inequality
- Semi-martingales, Ito’s Formula
- Martingale representation theorem.
- Girsanov Theorem

- Brownian Motion and Partial Differential Equations. (2 weeks)
- Harmonic functions and Dirichlet problem.
- The heat equation
- Formula of Feynman and Kac

- Stochastic Differential Equations (SDE) (3 weeks)
- Strong solutions, comparison result
- Weak solutions, Martingale problem of Strook and Varadhan
- Linear Equations

- Backward Stochastic Differential Equations (BSDE) (2 weeks)
- Lipschitz case, existence and uniqueness
- Comparison theorem
- Application to finance

The course won’t follow any textbook however you might find some interesting/alternative approaches in

[1] **J.M Steel:** *Stochastic Calculus and financial Applications*, 2nd Ed. Springer Verlag New York (2001) (more easy)

[2] **I. Karatzas S. Schreeve**: *Foundation of Modern Probability* Springer Verlag, New York (1991) (more advanced)

[3] **P. Protter:** *Stochastic Integration and Differential Equations*, 2nd Ed. Springer Verlag Berlin-Heidelberg (2004) (very advanced with discontinuous processes)

[4] **P. Dellacherie, P.A. Meyer:** *Probability and Potentials B*, , 1982 (the most general reference, very hard and build upon Probability and Potentials A)

- Ipython notebooks: