Stochastic Processes - Ph.D. course 2012

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 14th and ending on April 19th

    Day Room Time
    Tuesdays 11.2.16 (Getafe) 09:30-11:30
    Thursdays 11.2.16 (Getafe) 12:00-14:00

    FINAL EXAM:

    Day Room Time
    April 26, 2012 11.2.17 (Getafe) 10:00-13:00

    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.

    The Final Grade of the course will be equal to 60% of the Continuos Evaluation mark plus 40% of the Final Exam mark.

    REQUIREMENT:

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

    PRACTICAL SESSIONS:

    Three sets of theoretical/practical/simulations exercises 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 11.2.16 (Getafe) 09:30-11:30
    Thursdays 11.2.16 (Getafe) 12:00-14:00

    FINAL EXAM:

    Day Room Time
    April 26, 2012 11.2.17 (Getafe) 10:00-13:00

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganés)

    email: email

    CHRONOGRAM:

    # Day Content Notes Prob. Set Deadline Prob. Sets
    February 2012
    01 Tu - 14

    Introduction and basic notions. [R2] (§1.9)

    The Poisson process.

    		 Notes
    02 Th - 16 The Poisson Process. [R2] (§2.1, §2.2 and §2.3) Notes
    03 Tu - 21 The Poisson Process. [R2] (Example 2.3(B))
    The Renewal Theory. [R2] (§3.1, §3.2 and §3.3)     		Notes #1
    04 Th - 23 Renewal Theory. [R2] (§3.3, Example 3.3(A));
    Wald's Equation. [R2] (§3.3.1);
    The Elementary Renewal Theorem. [R2] (§3.3.4)     		Notes
    05 Tu - 28 The Key Renewal Theorem and Applications. [R2] (§3.4) Notes #1
    March 2012
    06 Th - 01 Alternating Renewal Processes. [R2] (§3.4.1))
    Delayed Renewal Processes. [R2] (§3.5, Theorem 3.5.2)
    07 Tu - 06 Markov Chains. [R2] (§4.1, Examples 4.1(A-B), §4.2, Examples 4.2(A))
    08 Th - 08 Markov Chains. [R2] (§4.3, Example 4.3(A) and §4.4) Exercises
    09 Tu - 13 The Gambler's Ruin Problem. [R2] (§4.4, Example 4.4(A));
    Semi-Markov Processes. [R2] (§4.8);
    The Cramér-Lundberg risk model     		Notes #2
    10 Th - 15 Continuous-Time Markov Chains. [R2] (§5.1 and §5.2)
    Birth-Death processes and the Yule process [R2] (§5.3, Example 5.3(B))
    The Kolmogorov Differential Equations [R2] (§5.4)
    11 Tu - 20 The Kolmogorov Differential Equations [R2] (§5.4, Examples 5.4(A-B))
    Limiting Probabilities [R2] (§5.5, Example 5.5(A))
    The infinitesimal generator.     		Notes #2
    12 Th - 22 Brownian Motion. [R2] (§8.1, §8.2 and §8.3, Example 8.3(A))
    13 Tu - 27 Brownian Motion with drift. [R2] (§8.4, Example 8.4(A) and §8.5)
    		Notes #3
    14 Th - 29 Brownian Motion with drift. [R2] (§8.4, Proposition 8.4.2)
    Stochastic Calculus. [S1] (§6.4, §7.2, §8.0 and §8.4-8.6)     		Notes
    April 2012
    - Tu - 3
    - Th - 5
    15 Tu - 10 Stochastic Calculus. [S1] (§9.0-9.3) Notes
    16 Th - 12 Black-Scholes formula. Notes #3
    - Tu - 17
    17 Th - 19 Optimal dividends' payout for the Brownian motion. Notes

    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.