Jan 10, 2006 - (c) Itô integrals with respect to Brownian Motion are martingales ... generated by applying Itô's lemma to unbounded functions like f(x) = x4.
Math 219 Spring 2006 Jonathan C. Mattingly January 10, 2006
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Outline 1. Conditional expectations and σ-algebras. 2. Brownian motion (a.k.a the Wiener process). (a) Definitions of Brownian motion (BM). (standard Brownian Motion refers to a Brownian Motion where the variance divided by t is normalized to 1, in other words EW (t)2 = t.) (b) Forward and Backward Kolmogorov equations for BM. (c) Chapman-Kolmogorov equation 3. Kolmogorov Continuity Theorem (and its application to BM). 4. What is a Markov Process and a Martingale ? (a) Examples of Martingales: W (t) and exp αW (t) −
α2 t 2
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5. Characterization of the law of a process by its finite dimensional distributions. (Kolmogorov Extension Theorem) 6. First and second variation (a.k.a variation and quadratic variation) (see [10, 4]). (a) The Riemann-Stieltjes integral (see [10, 4, 9]) and why it does not work for Brownian Motion. (b) With probability one, Brownian Motion has finite quadratic variation. And hence, it has infinite first variation almost surely. n o (c) In fact, E hW i(t) = t. ( hW i(t) is the quadratic variation of the standard Brownian motion W (t). )
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7. We will treat the stochastic differential equation (SDE) dX(t) = F (X(t), t)dt + σ(X(t), t)dW (t) X(0) = x simply as notation for the integral equation Z t Z t X(t) = x + F (X(s), s)ds + σ(X(s), s)dW (s) . 0
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Rt 8. Building the Itˆo and Stratonovich integrals. (That is, making sense of “ 0 σdW .”) (a) We will first define the integral for simple functions. For these functions the basic definition is rather clear. (b) Then will use that fact that simple functions can be used to approximate, in a sense that will be made clear all of the functions in which we are interested. 9. Doob-Kolmogorov inequality for continuous martingales. 10. Properties Itˆo Integral (a) Standard properties of integrals hold. (b) Itˆo isometry (c) Itˆo integrals with respect to Brownian Motion are martingales adapted to the filtration generated by the Brownian Motion provided that the integrand is square integrable in expectation. 11. Itˆo Processes. 12. Itˆo’s formula: The change of variables formula for Itˆo processes. 13. Connections with PDEs and the Backward Kolmogorov equation. 14. Stochastic differential equations (SDEs) (a) What does it mean to solve an SDE. (b) Gronwall’s inequality. (c) Uniqueness for SDEs. Needs Lipschitz condition to guarantee uniqueness, just like in the ODE setting. (d) Existence. Prove by Picard iteration, just like in the ODE case. Need linear growth restriction on coefficients to insure that the solution exists for all time, just like in the ODE case. 15. Stopping times 2
(a) Definition. σ-algebra associated to stopping time. Bounded stopping times. (b) Doob’s optional stopping theorem (c) (“Localization” of results) Use optional stopping theorem to control processes generated by applying Itˆo’s lemma to unbounded functions like f (x) = x4 . (d) Use optional stopping theorem to control (some) SDEs which do not have linear growth in there coefficients. For example, dX(t) = −X(t)3 dt + X(t)dB(t) . 16. Martingale representation theorem 17. L´evy-Doob theorem (a) How to tell when a continuous martingale is a Brownian Motion. (b) Random time changes to turn a Martingale into a Brownian Motion 18. Hermite Polynomials and the exponential martingale 19. Girsanov’s Theorem, Cameron-Martin formula, and changes of measure. (a) The simple example of i.i.d Gaussian random variables shifted. (b) The shift of a Brownian Motion (c) Changing the drift in a diffusion. 20. General Markov processes (a) Infinitesimal generators and transition semi-groups (b) Forward & backward Kolmogorov equations (c) Dynkin’s formula 21. Exit problems for diffusions, Dirichlet and Poison problems, Feynman-Kac formula 22. One dimensional diffusions. The Feller theory for boundary points. 23. Stochastic averaging: fast and slow time scales 24. Stochastic Calculus of variations
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References [1] Patrick Billingsley, Probability and measure, third ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. [2] Richard Durrett, Probability: Theory and examples, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. [3]
, Stochastic calculus, a practical introduction, CRC Press, 1996.
[4] Gerald B. Folland, Real analysis: Modern techniques and their applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1984. [5] Elliott H. Lieb and Michael Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997. [6] H. P. McKean, Stochastic integrals, Academic Press, New York-London, 1969, Probability and Mathematical Statistics, No. 5. [7] Bernt Oksendal, Stochastic differential equations, 5th ed., Springer-Verlag, 1998. [8] Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 293, Springer-Verlag, Berlin, 1994. [9] Kenneth A. Ross, Elementary analysis: the theory of calculus, Springer-Verlag, New York, 1980, Undergraduate Texts in Mathematics. MR 81a:26001 [10] Karl R. Stromberg, Introduction to classical real analysis, Wadsworth International, Belmont, Calif., 1981, Wadsworth International Mathematics Series. MR 82c:26002
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