Stochastic Differential Equations

Math 6397, section 30599 - Fall 2009

• Classes: 2:30-4:00 pm TuTh, 345 PGH
• Instructor: Andrew Török
• Office: 672 PGH
• Phone: (713) 743-3478
• E-mail: see the home-page
• Office hours: see the home-page

Text: In the first part of the course we will mainly follow the notes by L. C. Evans (UC Berkeley), available on his web-page.
A copy of the notes we will use, version 1.2, is here.

Additional material will be handed out or placed on reserve in the library.

Books on reserve (for the introduction):

• Real and Complex Analysis by Walter Rudin
• Elementary Probability Theory with Stochastic Processes by Kai Lai Chung
• Probability by Leo Breiman

References

The Teacher Evaluation form is available at www.casa.uh.edu/teacherevaluation.

Homeworks

• HW 1: due Tuesday, Sep. 1
  • HAND IN: problem 4 (page 130) from Evans; see page 12 for the definition of the σ-algebra generated by X
  • DO NOT HAND IN:
    • problem 5
    • review sections A, B and C from Chapter 2 of Evans
• HW 2: due Tuesday, Sep. 8
  • HAND IN:
    • problems 9, 12
    • about the hint for 12: show that certain identity involving integrals holds for all nice functions g, e.g. bounded Borel; from here one can conclude that the integrands are equal a.e., which is the desired formula
    • for discrete RV's, the formula - an infinite sum - is easier to obtain
    • note that 12 gives another way to check that the sum of independent normal variables is normal; this is not as elegant as the one using characteristic functions
  • DO NOT HAND IN:
    • problems 6(i), 8, 10
    • prove Lemma 18.5 (last statement on page 18) for k=1, n=2, m=1
    • read sections D (w/o indep of Rademacher functions, we'll discuss that in class), F and G (proofs optional) from Chapter 2 of Evans
• HW 3: due Tuesday, Sep. 15
  • HAND IN:
    • problems 14, 19
    • hint for 19: since W(t) is a normal variable, the question is actually about the moments of a normal RV. Can use the characteristic function to get the answer.
  • DO NOT HAND IN:
    • problems 16, 17
    • take a look to section A of Chapter 3 (motivation for BM)
    • read from section B of Chapter 3 the "Building a one-dimensional Wiener process" part (pages 40-42) which talks about white noise vs. BM
• HW 4: due Tuesday, Sep. 22
  • HAND IN:
    • problems 20, 25 (here the limit is over integers m)
  • DO NOT HAND IN:
    • problems 21, 24
    • read Ch. 3, section 3 (page 49)
• HW 5: due Tuesday, Sep. 29
  • HAND IN:
    • problem 27
  • DO NOT HAND IN:
    • problem 26
    • read the Discussion on page 59
• HW 6: due Tuesday, Oct. 6
  • HAND IN:
    • problem 29
    • prove the claim made in class: If F_t is a non-anticipating family of σ-algebras as in Defn. 64.2, then W_t is a martingale with respect to it (that is, E(W_t | F_s)=W_s for 0 ≤ s ≤ t).
  • DO NOT HAND IN:
    • problem 28
• HW 7: due Tuesday, Oct. 13
  • HAND IN:
    • problems 30, 32 (hint: find d Y and use Thm. 69.2)
    • prove the 1st formula, for ∫ t d W, in the proof of Lemma 72.1. Use the defn of the Ito integral.
  • DO NOT HAND IN:
    • read the proof of Thm. 71.3, about Ito integration of Hermite polynomials
    • in connection with the proof of Lemma 72.1: check that W is in L¹(0,T); it is defined in Defn 65.1 (ii).
• HW 8: due Tuesday, Oct. 20
  • HAND IN:
    • 33, 34
    • from last time: prove the 1st formula, for ∫ t d W, in the proof of Lemma 72.1. Use the defn of the Ito integral.
  • DO NOT HAND IN:
    • problem 4.7 from Øksendal (it is on the page handed out today; work it out in the function spaces introduced in Evans)
• HW 9: due Tuesday, Nov. 10
  • HAND IN:
    • 40 (additional question: Why the Existence Thm does not appply?), 41, 43
  • DO NOT HAND IN:
    • 35, 37, 38, 39, 42, 46
• HW 10: due Tuesday, Nov. 17
  • HAND IN:
    • 44, 45, and the problem I mentioned in class (Find the probability that the BM starting at 0 leaves the interval [-n,1] throught 1; here n is positive.)
    • Hint for 45: this is a linear SDE, formulas are described in the handout with notes from Arnold. You can also solve it by combining the equations in a convenient way (this is equivalent to those formulas, because the linear part has constant coefficients).
  • DO NOT HAND IN:
    • 36, 47, 48

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