extbf {Last Time} extbf {Doob's Submartingale Inequalities} extbf {Doob's Maximal Inequality} extbf {Martingale Convergence Theorem} extbf {Illustration: Martingale Convergence} extbf {Martingale Convergence and Uniform Integrability} extbf {Uniformly Integrable Martingales} extbf {Levy Martingales}Last Time•Reflection principle for BM•Brownian hitting times•Running maximum of BM•Law of large numbers for BMToday’s lecture: Section 4.4.2, 4.5MATH136/STAT219 Lecture 20, November 10, 2008 – p. 1/8Doob’s Submartingale Inequalities•If {Xt} is a RCLL submartingale then for all x > 0 and t > 0IPsup0≤s≤tXs> x≤IE|Xt|x•If {Xt} is a martingale then for all x > 0 and t > 0IPsup0≤s≤t|Xs| > x≤IE|Xt|x•Analogous versions exist for discrete timeMATH136/STAT219 Lecture 20, November 10, 2008 – p. 2/8Doob’s Maximal Inequality•If either◦{Xt} is a RCLL martingale, or◦{Xt} is a nonnegative RCLL submartingale•Then for all p > 1 and t ≥ 0IEhsup0≤s≤t|Xs|pi≤pp − 1pIE|Xt|p•Analogous version exists for discrete timeMATH136/STAT219 Lecture 20, November 10, 2008 – p. 3/8Martingale Convergence Theorem•If {Xt, Ft} is a RCLL submartingale withsupt≥0IE(X+t) < ∞•Then there exists a RV X∞such that IE|X∞| < ∞ andXt→ X∞almost surely as t → ∞•In addition,IE|X∞| ≤ limt→∞IE|Xt| < ∞MATH136/STAT219 Lecture 20, November 10, 2008 – p. 4/8Illustration: Martingale Convergence0 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.70.8nM(n)Example 4.5.4See Exercise 4.5.4MATH136/STAT219 Lecture 20, November 10, 2008 – p. 5/8Martingale Convergence and Uniform Integrability•If {Xt, Ft} is a uniformly integrable RCLL submartingalewithsupt≥0IE(X+t) < ∞•Then there exists a RV X∞such that IE|X∞| < ∞ andXt→ X∞almost surely as t → ∞•In addition,Xt→ X∞in L1as t → ∞•In this case, X∞is a last element, i.e.IE(X∞|Ft) = Xtfor all t ≥ 0MATH136/STAT219 Lecture 20, November 10, 2008 – p. 6/8Uniformly Integrable MartingalesIf {Xt, Ft} is martingale then the following are equivalent:•{Xt} is uniformly integrable•{Xt} has a last element: i.e. there exists anF∞-measurable RV X∞satisfying:◦IE|X∞| < ∞, and◦IE(X∞|Ft) = Xtfor all t ≥ 0•Xtconverges almost surely as t → ∞•Xtconverges in L1as t → ∞MATH136/STAT219 Lecture 20, November 10, 2008 – p. 7/8Levy Martingales•Let Y be a RV with IE|Y | < ∞ and let {Ft} be somefiltration. Define F∞= σ(∪t≥0Ft)•Then {IE(Y |Ft), t ≥ 0} is a uniformly integrable martingaleand as t → ∞,IE(Y |Ft) → IE(Y |F∞),almost surely and in L1•All uniformly integrable martingales are of this formMATH136/STAT219 Lecture 20, November 10, 2008 – p.