Instructor: Dr. Yuri Bakhtin
- (2007-04-25, NEW) The following will be covered on the final exam on Thu, May 3.
- Markov property and Strong Markov property for the Wiener process
- Properties of paths for Wiener process: the Law of the Iterated logarithm(LIL) (with proof),
the Functional LIL(no proof), the local LIL(derives from the usual one), monotonicity intervals, points of increase,
local maxima, non-differentiablility (see the book for the correct version of the Paley-Wiener-Zygmund theorem, I gave you a wrong statement), modulus of continuity (no proof), quadratic variation (with proof)
- Ito's integral: functional-analytic construction and properties (Ito's isometry, martingale property, continuity\...) (with proofs)
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- Ito's formula of change of variables in stochastic integrals(including multidimensional case) (with proof)
- Representation of Wiener functionals via stochastic integrals(with proof's ideas)
- Problems mostly similar to the homework
- Material related to the midterm test
- (2007-04-23)The last homework is posted online (p.6-7 of the updated pdf file)
- (2007-04-11) The notes for the make-up class today are
posted online
- (2007-04-09) The make up class will be at the Conference room, Skiles 114, on Wednesday, April,
11, at 4pm.
- (2007-04-09) The homework distributed to you last Thursday is posted online (p.4-5 of the updated pdf file)
- (2007-03-22) As I have announced to you already, there will be no lecture for our course on Tue, 2007-03-27
and Thu, 2007-03-29. There will be an additional make up class in mid-April.
- (2007-02-27) The following will be covered on the test on March, 6:
- The notion of a stochastic process, cylindric sigma-algebra, finite-dimensional distributions
- The Consistency Theorem (with proof)
- The Consistency Theorem via characteristic functions(with proof)
- Kolmogorov--Chentsov condition for the existence of a continuous modification(with proof)
- Existence of Wiener process via the Consistency theorem and continuous modification(with proof)
- Explicit construction of the Wiener process via Haar functions (with proof)
- Processes with independent increments. Existence criterion via characteristic functions. Poisson process
- Gaussian vectors and processes. Their basic properties(the meaning of parameters; uncorrelatedness implies independence;
conditional expectation). Existence
criterion. Wiener process, Ornstein--Uhlenbeck process as Gaussian processes.
- Weak convergence. Tightness, relative compactness. Prokhorov's theorem (no proof). Weak convergence in C is equivalent
to tightness plus convergence of f.d.d. (with proof). Characterization
of tightness in C. Kolmogorov--Chentsov tightness condition (with proof)
- Donsker's invariance principle in C, or Functional CLT(with proof)
Weak convergence in D and Martingale/Markov property for Wiener process won't be covered on the test.
- (2007-02-20) The test will be on Tue, March 6.
- (2007-02-20) The second assignment is posted online, see p.3 of the file.
It is due on Feb, 27.
- (2007-02-01) Problems 5,8,9 are all used for explicit construction of i.i.d. sequences from the Lebesgue
measure on [0,1]. By mistake they are separated by Problems 6 and 7 in the list.
- (2007-02-01) A misprint in Problem 1 is corrected. The random variables are assumed to be positive.
- (2007-01-30) The first homework assignment is posted online.
It is due on Thursday, Feb 8. Please report misprints.
- (2007-01-05) All updates on the course will be posted on this web page.