Convergence Martingale convergence theorem Let (Y, F) be a submartingale and suppose that for all n there exist a real value M such that E(Y + n ) M. Then there exist a random variable Y such that Y n Y almost surely as n. Y has a finite mean if E Y 0 < Y n Y in mean if the sequence {Y n : n 0} is uniformly integrable. Any submartingale or supermartingale (Y, F) converges almost surely if it satisfies E Y n <.
Consequences If (Y, F) is either a non-negative supermartingale or a non-positive submartingale then Y = lim n Y n exists almost surely Submartingale and Supermartingale E Y n < E(Y n+1 F n ) Y n submartingale E(Y n+1 F n ) Y n supermartingale
De Moivre s martingale convergence (Ferrary Example) Convergence of Y n = (q/p) Sn (martingale) X 1,..., X n with X i {1, 1} with P(X i = 1) = p P(X i = 1) = q. S n = n i+1 X i fortune at time n.
De Moivre s martingale convergency Y n = (q/p) Sn is a non-negative martingale = Y = lim n Y n exists almost surely. If p = q Y n = 1 for all n. If p q????
Upcrossings Given a sequence of real numbers y = {y n : n 0} and [a, b] a real interval. Upcrossing: a crossing by y of [a, b] in the upwards direction.
Upcrossings definition Let Y = {y n : n 0} be a real sequence and [a, b] a real interval. Upcrossing of [a,b]: Is an up crossing by y of [a, b] T 1 = min{n : y n a} First time that y hits (, a]. T 2 = min{n > T 1 : y n b} next time that y hits [b, ). [T 1, T 2 ] is an upcrossing of [a,b] Upcrossing for k 2 T 2k 1 = min{n > T 2k 2 : y n a} T 2k = min{n > T 2k 1 : y n b} In general [T 2k 1, T 2k ] is an upcrossing of [a,b] for k 1
Number of upcrossings U n (a, b; y) is the number of upcrossings of [a, b] by the sequence y 0, y 1,..., y n. Total number of upcrossings of [a, b] by y U(a, b; y) = lim n U n (a, b; y)
Upper bound for number of upcrossings Upccrossings inequality If a < b then EU n (a, b; Y ) E((Yn a)+ ) b a Bound for the number of upcrossings for a martingale sequence key point to proof the convergence martingale theorem.
Convergence Martingale convergence theorem Let (Y, F) be a submartingale and suppose that for all n there exist a real value M such that E(Y + n ) M. Then there exist a random variable Y such that Y n Y almost surely as n. Y has a finite mean if E Y 0 < Y n Y in mean if the sequence {Y n : n 0} is uniformly integrable.
Doob s martingale convergency Doob s martingale Y n = E(Z F n ) Let Z be a random variable on (Ω, F, P) such that E Z <. Let F, = {F 0, F 1,...} be a filtration with F = lim n F n F to be the smallest σ-algebra containing every F n.
Doob s martingale convergence Doob s martingale Y n = E(Z F n ) The pair (Y, F) is a martingale and since E Y n < then, by the Martingale convergence theorem, there exist Y such that almost surely as n. Y n Y
Doob s martingale convergence In addition, if {Y n } is a uniformly integrable sequence. Y n Y in mean as n by the Convergence Martingale Theorem. sup n E( Y n I { Yn a}}) 0 as a
Doob s martingale convergence Martingale convergence Theorem Y n Y a.s and in mean. What is the limit Y? Y = E(Z F ) Let N positive integer and A F N. {Y n : n 0} uniformly integrable. {Y n I A : n N}uniformly integrable. E(Y n IA) E(Y IA) for all A F N (Th 3 page 351). E(Y n I A ) = E(Y N I A ) = E(ZI A ) for all n N and A F N E(ZI A ) = E(Y I A ) E((Z Y )I A ) = 0 as N for all A F Y = E(Z F ).
Theorem Theorem Let (Y, F) be a martingale then Y n converges in mean if and only if there exist a random variable Z with E(Z) < such that Y n = E(Z F n ) If Y n Y in mean then Y n = E(Y F n ) If such random variable Z exists we say that the martingale (Y, F) is closed.
Zero-One Law Consider X 0, X 1,... independent random variables. T = n H n tail σ-algebra with H n = σ(x n, X n+1,...) Claim: For all A T either P(A) = 0 or P(A) = 1 Let A T and we define Y n = E(I A F n ) with F n = σ(x 1, X 2,... X n ). A T F = lim n F n Y n E(I A F ) = I A a.s and in mean. Y n = E(I A F n ) = P(A) since A T is independent of the events in F n Then P(A) = I A almost surely P(A) = 0 or P(A) = 1.
Dubins s inequality Dubins s inequality Let (Y, F) be a non-negative supermartingale. Then P(U n (a, b; Y ) j) (a/b) j E(min{1, Y 0 /a}) for 0 a b and j 0. Summing over j upccrossing inequality. E(U n (a, b; Y ) a/(b a)e(min{1, Y 0 /a})
Maximal inequality Let Y n = max{y i : 0 i n} Theorem If (Y, F) is a submartingale then P(Yn x) E(Y n + ) for x > 0 x If (Y, F) is a supermartingale and E Y 0 < then P(Y n x) E(Y 0) + E(Y n ) x for x > 0 Doob-Kolmogorov inequality Kolmogorov s inequality
Stopping Times or Markov Times Stopping time A random variable T taking values in {0, 1, 2, } { } is called a stopping time with respect to the filtration F if {T = n} F n for all n 0 {T > n} = {T n} c F n for all n since F is a filtration. T are not require to be finite.
Notation Given a filtraton F and a stopping time T. Then F T the collection of all events A such that A {T n} F n fo all n. F T is a σ-algebra representing the set of events whose occurrence or non-occurrence is known by time T.
Example We toss a fair coin repetedly. T the time of the first head Let X i = number of heads on the i th toss {T = n} = {X n = 1, X j = 0, 1 j < n} F n with F n = σ(x 1, X 2,..., X n ). T is a stopping time. T is finite almost surely.
Stopping and martingales A martingale which is stopped at a random time T remains a martingale as long as T is stopping time! Theorem Let (Y, F) be a submartingale and let T be an stopping time (with respect to F ). Then (Z n, F) defined as is a submartingale. x y = minx, y Z n = Y T n If (Y, F) submartingale submartingale and a supermartingale Y T n is a supermartingale.
Switching time A possible strategy of a gambler in a casino is to change games. If he is playing fair games then there should not gain or lose (on average) at such a change... Let (X, F) and let (Y, F) be two martingales with respect to the filtration F. T be a stopping time with respect to F. T is switching time from X to Y and X T is the capital-fortune which is carried forward.
Optional switching Suppose { that X T = Y T on the event {T < }. Then Xn if n < T Z n = if n T Y n defines a martingale with respect to F. Changing games, if both of them are fair, will not increase our loses or gains!!