Stochastic Processes - Ph.D. course 2018/19

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 Stochastic Processes.
  • Discrete-time Markov chains.
  • Continuous-time Markov chains.
  • Brownian Motion.
    •

    TIME AND LOCATION:

    14 lessons in total

    Every Thursday starting on February 7th and ending on May 23rd

    Day Room Time
    Thursdays 7.3.J08 (Leganés) 10:00-13:00

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganés)

    email: email

    ASSESSMENT CRITERIA:

  • Short partial exam. 30%
  • Short partial exam. 30%
  • Final Project. 40%
    •

    REQUIREMENT:

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

    BASIC BIBLIOGRAPHY:

    1. Bass, R.F. (2011): Stochastic Processes. Cambridge University Press.
    2. Evans, L.C. (2013): An Introduction to Stochastic Differential Equations. American Mathematical Society.
    3. Norris, J.R. (1997): Markov Chains. Cambridge University Press.
    4. Rosenthal, J.S. (2006): A First Look at Rigorous Probability Theory. (2ed) World Scientific Publishing Co.
    5. Ross, S.M. (1996): Stochastic processes. John Wiley & Sons
    6. 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. Durrett, R. (2001): Essentials of stochastic processes. Springer
    3. Durrett, R. (1996): Stochastic calculus: a practical introduction. CRC Press
    4. Feller, W.: An introduction to probability theory and its applications (vol. I & II). John Wiley & Sons
    5. Harrison, J.M.: Brownian motion and stochastic flow systems. R. E. Krieger
    6. Mikosch, T.: Elementary stochastic calculus :with finance in view. World Scientific
    7. Ross, S.M. (2007): Introduction to probability models. Academic Press

    TIME AND LOCATION:

    Day Room Time
    Thursdays 7.3.J08 (Leganés) 10:00-13:00

    MARKS:

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganés)

    email: email

    CHRONOGRAM:

    # Day Content Notes Exercises
    February 2019
    01 07

    Introduction and basic notions. Measurable spaces, σ-algebras, Probability spaces, Semi-algebras, Extension Theorem [R1] (§1, §2). Continuity of probabilities [R1] (§3.3). Random variables [R1] (§3.1). Independence [R1] (§3.2).

    (Discrete time) Stochastic Processes [R1] (§7). Existence of the coin tossing probability space [R1] (§2.6). Elementary existence theorem [R1] (Th. 7.1.1).

    		[N1] (Ex. 1.1.6)
    02 14

    Discrete-time Markov chains. Definitions and basic properties [N1] (§1.1). Examples [N1] 1.1.4. Class structure [N1] (§1.2). Hitting times and absorption probabilities [N1] (§1.3). Examples [N1] 1.3.1.

    03 21

    Discrete-time Markov chains. Hitting times and absorption probabilities [N1] (§1.3, Theorem 1.3.2, 1.3.5). Examples [N1] 1.3.3 (Gamblers' ruin).

    04 28

    Discrete-time Markov chains. Strong Markov property [N1] (§1.4). Recurrence and transience [N1] (§1.5, Lemmas 1.5.1, 1.5.2, Theorem 1.5.3).

    March 2019
    05 07

    Discrete-time Markov chains. Recurrence and transience [N1] (§1.5, Theorems 1.5.4, 1.5.5, 1.5.7). Recurrence and transience of random walks [N1] (§1.6, Theorems 1.6.1, 1.6.2). Invariant distributions [N1] (§1.7, Theorems 1.7.1, 1.7.2, 1.7.5).

    		[N1] (Example 1.7.11)
    06 14

    Discrete-time Markov chains. Invariant distributions [N1] (§1.7, Theorems 1.7.5, 1.7.6, 1.7.7).

    07 21

    Discrete-time Markov chains. Convergence to equilibrium [N1] (§1.8, Theorem 1.8.3, Theorem 1.8.4, 1.8.5).

    		[N1] (Theorem 1.8.4)
    08 28

    Discrete-time Markov chains. Ergodic Theorem [N1] (§1.10) Exercises: solution of First Partial Exam (2018)

    		First Partial Exam (2018)
    April 2019
    09 04 First Partial Exam Exam
    10 11

    Continuous-time Markov chains. Q-matrices and their exponential [N1] (§2.1). Continuous-time random processes [N1] (§2.2). Some properties of the exponential distribution [N1] (§2.3).

    18
    11 25

    Continuous-time Markov chains. Poisson processes [N1] (Theorem 2.4.1, 2.4.3, 2.4.4, 2.4.5, 2.4.6). Jump chain and holding times [N1] (§2.6). Stopping times and strong Markov property [N1] (Lemma 6.5.1 ).

    May 2019
    02
    12 09

    Continuous-time Markov chains. Explosion [N1] (Theorem 2.7.1). Forward and backward equations [N1] (Theorem 2.8.1, 2.8.2, 2.8.3, 2.8.4). Class structure [N1] (§3.2). Hitting times and absorption probabilities [N1] (§3.3). Recurrence and transience [N1] (§3.4). Invariant distributions [N1] (§3.5). Convergence to equilibrium [N1] (§3.6).

    13 16

    Brownian Motion. Introduction and preliminaries [R2] (§8.1) Hitting times [R2] (§8.2) Brownian motion with drift [R2] (§8.4) Backward Diffusion Equations [R2] (§8.5)

    14 23 Second Partial Exam Exam

    BASIC BIBLIOGRAPHY:

    [B1] Bass, R.F. (2011): Stochastic Processes. Cambridge University Press.

    [E1] Evans, L.C. (2013): An Introduction to Stochastic Differential Equations. American Mathematical Society.

    [N1] Norris, J.R. (1997): Markov Chains. Cambridge University Press.

    [R1] Rosenthal, J.S. (2006): A First Look at Rigorous Probability Theory. (2ed) World Scientific Publishing Co.

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

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