Levy Processes

Textbook: A. Kyprianou, Introductory Lectures on Fluctuations of Levy Processes with Applications, Springer 2006.

The first 31 pages are available from

http://www.springer.com/math/probability/book/978-3-540-31342-7?detailsPage=samplePages

  • 09.02.10 material covered: sections 1.1, 1.2.1, 1.2.2

    Homework (to be returned 16.02.10, each question = 1 point):

    1. Show directly from the definition that a compound Poisson process is a Levy process.

    2. Prove the last displayed formula on p. 7.

    3. Work out Exercise 1.1.

    4. Work out Exercise 1.2.

  • 16.02.10 material covered: Sections 1.2.4, 1.2.5. General features of Levy measures, forms of the Levy-Khintchine representation in special cases, definition of subordinators and the `Laplace form' of the Levy-Khintchine representation.

    Homework (to be returned 23.02.10):

    1. (1 point) Work out (carefully!) exercise 1.5

    2. (1 point) Derive the characteristic (and/or Laplace) exponent in Exercise 1.7

    3. (2 points) Show that the uniform distribution on $[0,1]$ is not infinitely divisible.

  • 23.02.10 material covered: Poisson approximation and the CLT reviewed. Infinite divisible distributions as limits in the `triangular scheme'. Stable distributions: definition, classification and features. Stable subordinators. Section 1.2.6.

    Homework (to be returned 9.03.10):

    1. (1 point) Let X have stable distribution. Determine when (a) the expected value of X is finite, (b) the variance of X is finite.

    2. (1 point) Decide if the function cos \theta is the characteristic function of some infinitely divisible distribution.

    3. (1 point) Assume that X_1,X_2,\dots are iid random variables with finite mean and variance. Suppose further that X has the same distribution as X_1, and also that X has the same distribution as (X_1+\dots+X_n)/(\sqrt{n}) for every n. Show that the distribution of X is standard normal.

  • 9.03.10. Sections 2.1 and 2.2. Poisson random measures (PRM) in general spaces, behavior of PRM's under maps, and marked Poisson processes.

    1. (1 point) Let N be a PRM with mean measure \mu. Suppose every point of N is marked 1 with probability p, and marked 0 with probability 1-p, independently for all points. Let N_1, N_0 be the corresponding integer-valued random measures. Show that N_1, N_0 are independent PRM's, and determine their mean measures.

    2. (1 point) Let U_1,U_2,\dots be independent uniform [0,1] random variables. Define N(A) to be the number of integers j such that U_1 \dots U_j \in A (product of j uniform variables is in A), where A is an arbitrary Borel subset of the positive halfline (0 , \infty). Show that N is a PRM and find the mean measure of N. Determine the distribution of N(A) for A=[0.25 , 0,75].

    3. (1 point) Do exercise 2.2 using the multinomial property of the Poisson random measures. Note that the last displayed formula on p 37 is a special case of the property.

  • 16.03.10. pages 41-49

    1. (1 point) Generalise Theorem 2.7 (iii) to arbitrary powers $X^k, k=1,2,\ldots$

    2. (1 point) Generalise the result of Exercise 2.6 to arbitrary functions $f$ in place of $x^n$

    3. (1 point) Prove Lemma 2.12.

  • 23.03.10. pages 50-56, 67-69.

    1. (1 point) Characterise all Levy processes representable as difference of two independent subordinators.

    2. (1.5 point) Let {\cal F}_t for t>0 be the natural filtration of a Levy process. Show that the intersection of all these sigma-algebras over t>0 is a trivial sigma-algebra.

  • 30.03.10 page 70, Thm 3.3, pp. 75-80

    (2 points) Consider the Esscher-transformed distribution of a Levy process. Along the lines of the proof on page 79, derive the joint characteristic function for the increments over three or more disjoint intervals.

  • 06.04.10. pages 80-82, 111-113.
  • 4.05.10. Section 5.5

    1. (1 point) Exercise 5.3

    2. (1 point) Exercise 5.8

  • 11.05.10 Sections 3.2, 6.1.

    (1 point) Exercise 6.1

  • 18.05.10 pp 147-155