Place: Chen Rui Qiu Building

: Course name: Stochastic Processes Credit hours: 48/3 Semester: Spring Category: Master Degree Department: Mathematics Course Nature/Object: Foundation of Stochastic Processes/Graduate students or advanced undergraduates. Prerequisite: Measure, Integration and Probability Theory

- I. Karatzas S. Schreeve:
*Foundation of Modern Probability*, Springer Verlag, New York (1991) - J.M Steel:
*Stochastic Calculus and financial Applications*, 2nd Ed. Springer Verlag New York (2001) - P. Protter:
*Stochastic Integration and Differential Equations*, 2nd Ed. Springer Verlag Berlin-Heidelberg (2004)

- Recap of Probability and Measure Theory and Conditional expectation. (2 weeks)
- Measure Theory, Integration
- Lp Spaces, Inequalities
- Radon Nykodym derivative measure change
- Conditional Expectation
- Independence

- Discrete time martingales, stopping times, Doob’s theorems, convergence of martingales. (2 weeks)
- Stochastic processes, filtration, stopping times
- Martingale
- Doob’s optional sampling theorem
- Doob’s upcrossing’s lemma, convergence of martingales and applications (Borel Cantelli martingale version)
- Doob’s maximal inequalities, $L^p$ convergence and applications (weak law of large numbers).
- Fubini-Tonelli, Uniform integrability
- $L^1$ martingale convergence

- Markov Processes, Markov Chains, Brownian Motion (2 weeks)
- Definition of Markov Processes, Kolmogorov Extension Theorem
- Kolmogorov extension theorem, Construction of Markov Probability measures
- Markov property, Chapman-Kolmogorov Theorem
- Recurrence/transience and applications
- Stationarity, invariance, irreducibility
- Construction of the Brownian motion (Kolmogorov-Centov theorem) and first properties.

- Continuous time Processes, Martingales, Lebesgues-Stieljes, Path regularity (2 Weeks)
- Stochastic processes, modification, indistinguishability, rcll lcrl
- stopping times, hitting times, stopped processes
- Lebesgue Stieljes Integration
- Martingale convergence theorem, path regularity
- Process of Bounded Variations, predictable, Doob-Meyer decomposition
- Poisson process, Brownian motion

- Stochastic integral (2 weeks)
- Quadratic Variations
- Ito-isometrie, Stochastic integral
- Covariations, Watanabe’s inequality
- Semi-martingales
- Ito’s Formula and applications
- Ito-Meyer formula and local times

- Stochastic Exponential and Girsanov Transformation (2 weeks)
- Exponential of a local martingale
- Uniform integrability, Kasamaki theorem
- Girsanov Theorem
- Levy characterization of BM

- Stochastic Differential Equations (SDE) (2 weeks)
- Strong solutions, comparison result
- Comparison theorem, stability
- Weak solutions, Martingale problem of Strook and Varadhan (bonus)

- Markov processes continuous time, Feynman-Kac and PDE (1-2 weeks)
- Markov processes in continuous time (Brownian setting)
- Blumenthal 0-1 law and the continuity of the BM filtration
- Feynman Kac formula
- Application to PDE

- Martingale representation theorem and BSDEs (1-2 weeks) (Bonus)
- Martingale representation theorem
- Linear BSDEs
- Lipschitz case, existence and uniqueness
- Comparison theorem
- Application to finance