Stochastic Processes - Ph.D. course 2017/18

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.
  • Discrete-time Martingales.
  • Brownian Motion.
  • Continuous-time Martingales.
  • Stochastic integrals.
    •

    TIME AND LOCATION:

    16 lessons in total

    Every Thursday starting on February 1st and ending on May 24th

    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 2018
    01
    01 08

    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).

    02 15

    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.

    		[N1] (Ex. 1.1.6)
    03 22

    Discrete-time Markov chains. Examples [N1] 1.3.3 (Gamblers' ruin), 1.3.4 (Birth-and-death chain). Strong Markov property [N1] (§1.4).

    March 2018
    04 1

    Discrete-time Markov chains. Recurrence and transience [N1] (§1.5). Recurrence and transience of random walks [N1] (§1.6).

    05 8

    Discrete-time Markov chains. Invariant distributions [N1] (§1.7).

    		[N1] (Example 1.7.11)
    06 15

    Discrete-time Markov chains. Convergence to equilibrium [N1] (Theorem 1.8.3, Theorem 1.8.4, 1.8.5). Ergodic Theorem [N1] (§1.10)

    		[N1] (Theorem 1.8.4)
    07 22 First Partial Exam Exam
    29
    April 2018
    5
    08 12

    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).

    09 19

    Continuous-time Markov chains. Poisson processes [N1] (Theorem 2.4.1, 2.4.3, 2.4.4, 2.4.5, 2.4.6).

    10 26

    Continuous-time Markov chains. Jump chain and holding times [N1] (§2.6). Stopping times and strong Markov property [N1] (Lemma 6.5.1, 6.5.2, 6.5.3, Theorem 6.5.4).

    May 2018
    11 3

    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).

    12 4

    Continuous-time Markov chains. Invariant distributions [N1] (§3.5). Convergence to equilibrium [N1] (§3.6). Ergodicity [N1] (§3.8).

    13 10

    Brownian Motion. Definition and basic properties [B1] (§2.1. Proposition 2.2, 2.3, Theorem 2.4) Minimal augmented filtration [B1] (§3.1. only mentioned) Stopping times [B1] (§3.3. Proposition 3.8) Markov properties [B1] (§4.1. Theorem 4.1, 4.2, Corollary 4.3) Backward and Forward Diffusion Equations [R2] (§8.5)

    14 11

    Discrete-time Martingales. Definitions [N1] (§4.1). The optional stopping theorem [N1] (Theorem 4.1.1)

    Continuous-time Martingales. Doob’s inequalities [B1] (§3.2) The optional stopping theorem [B1] (§3.4) Some applications of martingales [B1] (§3.6)

    15 17

    Stochastic integrals. Stochastic integrals [B1] (§10.1) Extensions [B1] (§10.2) Itô’s formula [B1] (§11)

    		Notes Notes
    16 24 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.