Stochastic Processes - Ph.D. course 2009

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:

    10 lessons in total

    Every Tuesday starting on February 24th and ending on May 12ve

    No class on March 24th and April 7th

    Room: 7.1.J07 (Juan Benet, Leganes) Time: 16:00-19:00

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganes)

    email: email

    ASSESSMENT CRITERIA:

    Continuous evaluation by mean of 3 homeworks (theoretical and/or applied) and one final project to be done during 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.: Essentials of stochastic processes. Springer
    2. Durrett, R.: Stochastic calculus : a practical introduction. CRC Press
    3. Ross, S.M.: Introduction to probability models. Academic Press
    4. Ross, S.M.: Stochastic processes. John Wiley & Sons

    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:

    Room: 7.1.J07 (Juan Benet, Leganes) Time: 16:00-19:00

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganes)

    email: email

    # Day Content Prob. Set Deadline Prob. Sets *
    February 2009
    01 24 Introduction and basic notions. [R2] (§1.4, §1.5 and §1.9) #1
    02 The Poisson process. [R2] (§2.1 and §2.2)
    March 2009
    03 03 The Poisson process. [R2] (§2.2 and §2.3) #2
    04 Homogeneous Markov Chains. [R2] (§4.1 and §4.2)
    05 10 Homogeneous Markov Chains. [R2] (§4.2) and [D1] (§1.3) #3
    06 Simple Random Walk. [R2] (Examples 4.2(A) and 4.1(D))
    07 17 HMC. [D1] (§1.3)
    Stopping times, Strong Markov Property, Total variation.    		 #4
    08 HMC. [R2] (§4.3) and [D1] (§1.4 and §1.8)
    Convergence Theorem and stationary distribution.
    - 24 #1,2,3
    09 31 Martingales. [D1] (§2.2) #5
    10 Optional Stopping Theorem with applications.
    [D1] (§2.3 and §2.4)
    April 2009
    - 07
    11 14 Martingales. Convergence Theorem [R2] (§6.4) #6
    12 Renewal Theory.
    [R2] (§3.1, §3.2 and §3.3) and [D1] (§5.1 and §5.2)
    13 21 Semi-Markov Processes. [R2] (§4.8)
    Continuous-time Markov Chains [R2] (§5.1, §5.2 and §5.3)    		 #7 #4,5,6
    14 Continuous-time Markov Chains [R2] (§5.4 and §5.5)
    Brownian Motion and Brownian Bridge [R2] (§8.1)
    15 28 Variations on Brownian Motion [R2] (§8.3 and §8.4)
    Hitting times, Maximum value and Arc Sine Law [R2] (§8.2)    		 #8
    16 Definition of the Stochastic Integral. Itô's formula. [D2]
    May 2009
    - 05
    17 12 Regulated Brownian Motion [H1] (§1.9 and §5.6)
    Change of Drift as Change of Measure [H1] (§1.7 and §1.8)    		 #7,8
    18 Optimal dividens: Analysis with Brownian Motion [GS]
    20 Final Project - Presentations
    * For each Problem Set it is required the solution of at least two problems among the proposed ones.

    BASIC BIBLIOGRAPHY:

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

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

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

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

    ADDITIONAL REFERENCES:

    [GS] Gerber, H.U. and Shiu, S.W. (2004). Optimal dividens: Analysis with Brownian Motion
    North American Actuarial Journal, vol 8(1)

    [H1] Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. Robert E. Krieger Publishing Company

    REPORT SUBMISSION:

    May 15th, 2009

    PRESENTATIONS:

    May 20th, 2009

    Room: 7.3.J08 (Juan Benet, Leganes) Time: 10:30-15:30

    GROUPS:

    # Names Reference To be graded by Group #
    01 Huong Nguyen Thu
    Manh Tran Sy    		 Baccelli and Makowski (1989) 03
    02 Leonardo Martin Berbotto
    Alberto Martin Utrera    		 Goodman (1953) 04
    03 Raluca Ioana Gui
    Nicola Mingotti    		 Feller (1951) 05
    04 Joanna Virginia Rodriguez Cesar Black and Scholes (1973) 01
    05 Javier Arriero Pais
    Cristina Garcia de la Fuente    		 Black and Scholes (1973) 02

    REFERENCE LIST:

      Finance

    1. E. Barndorff-Nielsen and N. Shephard (2003).
      Integrated OU Processes and Non-Gaussian OU-based Stochastic Volatility Models.
      Scandinavian Journal of Statistics, 30(2), pp. 227-295.
    2. F. Black and M. Scholes (1973).
      The Pricing of Options and Corporate Liabilities.
      The Journal of Political Economy, 81(3), pp. 637-654.
    3. R. C. Merton (1973).
      Theory of Rational Option Pricing.
      The Bell Journal of Economics and Management Science, 4(1), pp. 141-183.
    4. H. L. Steven (1993).
      A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.
      Review of Financial Studies, 6(2), pp. 327-343.

      5. Genetics

    5. W. Feller (1951).
      Diffusion Processes in Genetics.
      Proc. Second Berkeley Symp. on Math. Statist. and Prob., Univ. of Calif. Press, pp. 227-246.
    6. L. A. Goodman (1953).
      Population Growth of the Sexes.
      Biometrics, 9(2), pp. 212-225.

      7. Risk Theory

    7. B. Sundt and J. L. Teugels (1995).
      Ruin estimates under interest forces.
      Insurance: Mathematics and Economics, 16(1), pp. 7-22.
    8. H. Yuanjiang, L. Xucheng and J. Zhang (2003).
      Some Results of Ruin Probability for the Classical Risk Process.
      Journal of Applied Mathematics and Decision Sciences, 7(3), pp. 133-146.

      9. Telecommunications and queues

    9. F. Baccelli and A. M. Makowski (1989).
      Dynamic, Transient and Stationary Behavior of the M/GI/1 Queue Via Martingales.
      Ann. Probab., 17(4), pp. 1691-1699.
    10. B. D'Auria (2007).
      Stochastic decomposition of the M/G/∞ queue in a random environment.
      Oper. Res. Lett., 35, pp. 805-812.
    11. M. S. Taqqu, W. Willinger and R. Sherman (1997).
      Proof of a Fundamental Result in Self-Similar Traffic Modeling.
      Computer Communication Review, 27, pp. 5-23.
    12. J. M. Harrison and M. I. Taksar (1983).
      Instantaneous Control of Brownian Motion.
      Mathematics of Operations Research, 8(3), pp. 439-453.
    13. L. M. Wein (1990).
      Optimal Control of a Two-Station Brownian Network.
      Mathematics of Operations Research, 15(2), pp. 215-242.

    MASTER EN ECONOMÍA DE LA EMPRESA Y MÉTODOS CUANTITATIVOS

    Name PS1 PS2 PS3 PS4 PS5 PS6 PS7 PS8
    Leonardo Martin Berbotto C A C A A A A B
    Raluca Ioana Gui B A B A A A B B
    Alberto Martin Utrera C C B A A A A B
    Huong Nguyen Thu D B E B B B B A
    Sy Manh Tran E B E B A A B A

    MASTER EN INGENIERÍA MATEMÁTICA

    Name PS1 PS2 PS3 PS4 PS5 PS6 PS7 PS8
    Javier Arriero Pais B A C A B E B C
    Cristina Garcia de la Fuente A A C C A C C D
    Nicola Mingotti C C* C*
    Joanna Virginia Rodriguez Cesar C C C B A B C A
    Omar Ruben Romero Lugo D C D