MATH136/STAT219 Lecture 21, November 12, 2008 p. 1/11 Last Time Martingale inequalities Martingale convergence theorem Uniformly integrable martingales Today s lecture: Sections 4.4.1, 5.3
MATH136/STAT219 Lecture 21, November 12, 2008 p. 2/11 Doob s Decomposition in Discrete Time Let {X n } be a discrete time SP with IE X n < that is adapted to some filtration {F n } Then there exists a unique decomposition X n = M n + A n such that {M n, F n } is a martingale {A n } is a previsible SP; i.e. A n+1 is F n -measurable A 0 = 0 Proof: define M n = X n A n where A n is defined via the recursive equation A n+1 = A n + IE(X n+1 X n F n ) If {X n } is a submartingale, then {A n } is a nondecreasing process, i.e. A n+1 A n a.s. for all n
MATH136/STAT219 Lecture 21, November 12, 2008 p. 3/11 Doob-Meyer Decomposition Let {M t, F t } be a continuous, square integrable martingale (i.e. IE(M 2 t ) < for all t 0 and {M t } has continuous paths a.s.) Then there exists a unique SP {A t } such that A 0 = 0 {A t } is adapted to {F t } {A t } has continuous sample paths a.s. {A t } is nondecreasing (A t A s a.s. for all t s 0) {(M 2 t A t, F t )} is a martingale {A t } is called the increasing part associated with {M t }, and {A t } is equal to the quadratic variation process of {M t }
MATH136/STAT219 Lecture 21, November 12, 2008 p. 4/11 Illustration: Doob-Meyer decomposition of W 2 t 3 2.5 2 1.5 1 0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 blue: W 2 t, red: W 2 t t
MATH136/STAT219 Lecture 21, November 12, 2008 p. 5/11 Illustration: D-M decomposition in Exercise 4.4.10 2 Exercise 4.4.10 2.5 Exercise 4.4.10 1.5 2 1.5 1 1 0.5 0.5 0 0 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 M t = exp(w t t/2) blue: M 2 t, red: M 2 t A t
MATH136/STAT219 Lecture 21, November 12, 2008 p. 6/11 Variation of a Function For t > 0, let π be a partition of [0, t]: π = {0 = t (π) 0 < t (π) 1 < < t (π) k = t} and define π = max 1 i k (t (π) i t (π) i 1 ) For a function f : [0, t] IR and p 1, the p-th variation of f on [0, t] is defined as V (p) (f) = lim π 0 provided the limit exists k i=1 f(t (π) i ) f(t (π) i 1 ) p Special cases: p = 1 gives total variation and p = 2 gives quadratic variation of f on [0, t]
MATH136/STAT219 Lecture 21, November 12, 2008 p. 7/11 Variation of a Stochastic Process The total variation process of a SP {X t } is the stochastic process {V (1) t, t 0}, where the value at time t 0, V (1) t (ω), is the total variation of the function X s (ω) on the interval [0, t] The quadratic variation process of a SP {X t } is the stochastic process {V (2) t, t 0}, where the value at time t 0, V (2) t (ω), is the quadratic variation of the function X s (ω) on the interval [0, t] (Above definitions apply only in cases where the limits are defined in some sense)
MATH136/STAT219 Lecture 21, November 12, 2008 p. 8/11 Variation of Continuous, L 2 Martingales Let {(X t, F t )} be a continuous, square integrable martingale The quadratic variation process of {X t }, often denoted X t or X, X t, exists and is equal to {A t }, the increasing part of the Doob-Meyer decomposition That is, {(X 2 t X t, F t )} is a martingale Also, the total variation of {X t } on any interval is infinite with probability 1
MATH136/STAT219 Lecture 21, November 12, 2008 p. 9/11 Variation of Brownian Motion Let {W t } be a Brownian Motion Then for all t > 0, k i=1 W(t (π) i ) W(t (π) i 1 ) 2 t in L 2 as π 0 That is, the quadratic variation process of Brownian motion is given by W t = t a.s. Brownian motion accumulates quadratic variation at rate one per unit time Informally, dw t dw t = dt, and also dw t dt = 0 and dtdt = 0 The total variation of Brownian motion on any interval is infinite with probability 1
MATH136/STAT219 Lecture 21, November 12, 2008 p. 10/11 Definition of {F t }-Brownian Motion An {F t }-adapted stochastic process {W t, t 0} is a {F t }-Brownian motion if: W 0 = 0 a.s. Independent increments: for all 0 s t W t W s is independent of F s Stationary increments: for all 0 s t W t W s has a N(0, t s) distribution For almost every ω, the sample path t W t (ω) is continuous
MATH136/STAT219 Lecture 21, November 12, 2008 p. 11/11 Martingale Characterization of Brownian Motion Suppose {(X t, F t )} is a martingale with continuous paths If {(Xt 2 t, F t )} is a martingale Then {X t } is a {F t }-Brownian motion