Stochastic Processes

---

When/Where

Place: Chen Rui Qiu Building

Course Objective

Information

: 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

Textbooks/References

• 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)

Course Description

1. 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
2. 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
3. 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.
4. 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
5. 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
6. Stochastic Exponential and Girsanov Transformation (2 weeks)
   • Exponential of a local martingale
   • Uniform integrability, Kasamaki theorem
   • Girsanov Theorem
   • Levy characterization of BM
7. Stochastic Differential Equations (SDE) (2 weeks)
   • Strong solutions, comparison result
   • Comparison theorem, stability
   • Weak solutions, Martingale problem of Strook and Varadhan (bonus)
8. 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
9. Martingale representation theorem and BSDEs (1-2 weeks) (Bonus)
   • Martingale representation theorem
   • Linear BSDEs
   • Lipschitz case, existence and uniqueness
   • Comparison theorem
   • Application to finance