Stochastic Processes - Ph.D. course 2010

OBJECTIVES:

The aim is to provide theoretical knowledges of Stochastic Processes together with the know-how for modeling and solving actual problems by stochastic techniques.

PROGRAMME:

  • Introduction and basic notions.
  • Discrete-time and Continuous-time Markov Chains.
  • Renewal Theory.
  • Brownian Motion and Introduction to Stochastic Calculus.
    •

    TIME AND LOCATION:

    17 lessons in total

    Every Tuesday and Thursday starting on February 23th and ending on April 15th

    No class on March 30th and April 1st

    Day Room Time
    Tuesdays 4.0.D01 (Leganés) 09:30-11:30
    Thursdays 4.1.E06 (Leganés) 12:15-14:15

    FINAL EXAM:

    Day Room Time
    May 3rd, 2010 1.0.C02 (Leganés) 09:30-12:30

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganés)

    email: email

    ASSESSMENT CRITERIA:

    Continuous evaluation by mean of 3 homeworks (theoretical and/or applied) and one final exam to be done at the end of the course.

    REQUIREMENT:

    Background in Mathematics, Probability and Statistics at Science/Technical graduate level.

    PRACTICAL SESSIONS:

    Three sets of theorical/practical/simulations exercices will be proposed.

    BASIC BIBLIOGRAPHY:

    1. Durrett, R. (2001): Essentials of stochastic processes. Springer
    2. Durrett, R. (1996): Stochastic calculus : a practical introduction. CRC Press
    3. Ross, S.M. (2007): Introduction to probability models. Academic Press
    4. Ross, S.M. (1996): Stochastic processes. John Wiley & Sons
    5. Steele, J.M. (2000): Stochastic Calculus and Financial Applications. Springer.

    ADDITIONAL BIBLIOGRAPHY:

    1. Brémaud, P.: Markov chains: gibbs fields, Monte Carlo simulation and queues. Springer-Verlag
    2. Feller, W.: An introduction to probability theory and its applications (vol. I & II). John Wiley & Sons
    3. Harrison, J.M.: Brownian motion and stochastic flow systems. R. E. Krieger
    4. Mikosch, T.: Elementary stochastic calculus :with finance in view. World Scientific

    TIME AND LOCATION:

    Day Room Time
    Tuesdays 4.0.D01 (Leganés) 09:30-11:30
    Thursdays 4.1.E06 (Leganés) 12:15-14:15

    FINAL EXAM:

    Day Room Time
    May 3rd, 2010 1.0.C02 (Leganés) 09:30-12:30

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganés)

    email: email

    CHRONOGRAM:

    # Day Content Notes Prob. Set Deadline Prob. Sets
    February 2010
    01 Tu - 23 Introduction and basic notions. [R2] (§1.1, §1.2, §1.3 and §1.4)
    Probability spaces, Random Variables, Expactations.
    02 Th - 25 Introduction and basic notions. [R2] (§1.5)
    Random vectors, Independence, Conditional Expectation. One-Step analysis.
    March 2010
    03 Tu - 02 Introduction and basic notions. [R2] (§1.8, §1.9) Stochastic Processes.
    Queueing systems, Kendall's notation. Cramér-Lundberg risk model.
    The Poisson process. [R2] (§2.1 and §2.2)    		 #1
    04 Th - 04 The Poisson process. [R2] (§2.3)
    Renewal Theory. [R2] (§3.1; §3.2; §3.3.0, §3.3.1)    		 Comments #1
    05 Tu - 09 Renewal Theory. [R2] (§3.3.2; §3.4.0, §3.4.1; §3.5) Comments #3
    06 Th - 11 Renewal Theory. [R2] (§3.5)
    HMC. Homogeneous Markov Chains. [R2] (§4.1; §4.2)    		 Comments #4
    07 Tu - 16 HMC. [R2] (§4.3) #5
    08 Th - 18 HMC. [R2] (§4.3; §4.4)
    Semi-Markov Processes. [R2] (§4.8)    		 #6 #1,2,3
    09 Tu - 23 Continuous-Time Homogeneous Markov Chain.
    [R2] (§5.1; §5.2; §5.3; §5.4; §5.5)    		 #7
    10 Th - 25 HMC. [R2] (§5.8) Cramér-Lundberg risk model. Ruin Probability.
    M/G/1 queue. The stationary waiting time distribution.     		Notes #8
    - Tu - 30
    April 2010
    - Th - 01
    11 Tu - 06 Brownian Motion. [R2] (§8.1; §8.2; §8.3) #9
    12 Th - 08 Brownian Motion. [R2] (§8.4; §8.5;)
    Stochastic Integral. Introduction and definition.    		 Notes #4,5,6,7
    13 Tu - 13 Stochastic Integral. Itô's formula.
    Stochastic Differential Equations. Some examples.    		 Notes #10
    14 Th - 15 Black-Scholes formula. Notes
    - Tu - 20 #8,9,10

    BASIC BIBLIOGRAPHY:

    [D1] Durrett, R. (2001): Essentials of stochastic processes. Springer

    [D2] Durrett, R. (1996): Stochastic calculus : a practical introduction. CRC Press

    [R1] Ross, S.M. (2007): Introduction to probability models. Academic Press

    [R2] Ross, S.M. (1996): Stochastic processes. John Wiley & Sons

    [S1] Steele, J.M. (2000): Stochastic Calculus and Financial Applications. Springer.