Stochastic Processes


Place: Chen Rui Qiu Building 312

Course Objective


: 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


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