Stochastic Processes - Ph.D. course 2016/17

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:

    14 lessons in total

    Every Tuesday and Thursday starting on February 7th and ending on March 23rd

    Day Room Time
    Tuesdays 7.3.J08 (Leganés) 09:30-11:30
    Thursdays 7.3.J08 (Leganés) 09:30-11:30

    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.

    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. 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
    Tuesdays 7.3.J08 (Leganés) 09:30-11:30
    Thursdays 7.3.J08 (Leganés) 09:30-11:30

    MARKS:

    LECTURER:

    Bernardo D'Auria

    Room: 7.3.J29 (Juan Benet, Leganés)

    email: email

    CHRONOGRAM:

    # Day Content Notes Exercises Deadline Prob. Sets
    February 2017
    01 Tu - 7

    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).

    		Notes
    02 Th - 9

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

    		Notes
    03 Tu - 14

    Discrete-time Markov chains. Hitting times and absorption probabilities [N1] (§1.3). Strong Markov property [N1] (§1.4). Recurrence and transience [N1] (§1.5). Recurrence and transience of random walks[N1] (§1.6).

    		Notes [N1] 1.3.3, 1.4.1, 1.6.2
    04 Th - 16

    Discrete-time Markov chains. Invariant distributions [N1] (§1.7). Convergence to equilibrium [N1] (Theorem 1.8.3).

    		Notes [N1] 1.6.1, 1.7.1, 1.7.2, 1.7.4
    05 Tu - 21

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

    		[N1] (Theorem 1.8.4)
    [N1] (Example 1.7.11) by Abel
    06 Th - 23

    Continuous-time Markov chains. Q-matrices and their exponentials [N1] (§2.1). Continuoues-time random processes [N1] (§2.2). Some properties of the exponential distribution [N1] (§2.3). Poisson processes [N1] (Theorem 2.4.1, 2.4.3)

    07 Tu - 28

    Continuous-time Markov chains. Poisson processes [N1] (Theorem 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, 6.5.2).

    March 2017
    08 Th - 02

    Continuous-time Markov chains. Stopping times and strong Markov property [N1] (Lemma 6.5.3, Theorem 6.5.4). Explosion [N1] (§2.7). Forward and backward equations [N1] (Theorem 2.8.1, 2.8.2).

    		#1
    09 Tu - 07

    Continuous-time Markov chains. Forward and backward equations [N1] (Theorem 2.8.2, 2.8.3, 2.8.4, 2.8.6). Class structure [N1] (§3.2). Hitting times and absorption probabilities [N1] (§3.3). Recurrence and transience [N1] (§3.4). Invariant distributions [N1] (§3.5). Convegence to equilibrium [N1] (§3.6). Ergodicity [N1] (§3.8).

    10 Th - 09

    Discrete-time Martingales. [N1] (§4.1). Potential theory. [N1] (Example 4.2.1, 4.2.2. Theorem 4.2.3, 4.2.4.).

    		#1
    11 Tu - 14 Brownian Motion. Definition and basic properties [B1] (§2.1. Proposition 2.2, 2.3, Theorem 2.4) Markov properties [B1] (§4.1. Theorem 4.1, 4.2, Corollary 4.3, Proposition 4.5) Construction [E1] (Lemma 3.3, 3.4, Theorem 3.1) Notes
    Th - 16
    12 Tu - 21 Discrete-time Martingales. Basic inequalities [B1] (Theorem A.5, A.6, A.21, A.32, A.33) Martingale convergence theorem [B1] (Theorem A.34, A.35, A.36) Continuous-time Martingales. Minimal augmented filtration [B1] (§3.1) Doob’s inequalities [B1] (§3.2) Stopping times [B1] (§3.3) The optional stopping theorem [B1] (§3.4) Some applications of martingales [B1] (§3.6)
    13 Th - 23 Discrete-time Martingales. Uniform integrability [B1] (Lemma A.15, Proposition A.16, A.17, Theorem A.19) Martingale convergence theorem [B1] (Theorem A.37) Continuous-time semi-Martingales. Total and Quadratic variations [B1] (§9.1) Square integrable martingales [B1] (§9.2) Quadratic variation [B1] (§9.3) The Doob–Meyer decomposition [B1] (§9.4. Theorem 9.12) #2
    14 Tu - 28 Stochastic integrals. Stochastic integrals [B1] (§10.1) Extensions [B1] (§10.2) Itô’s formula [B1] (§11)
    April 2017
    Th - 06 #2
    Wd - 19 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.