extbf {Last Time} extbf {Doob's Decomposition in Discrete Time} extbf {Doob-Meyer Decomposition} extbf {Illustration: Doob-Meyer decomposition of m $W_t^2$ubm } extbf {Illustration: D-M decomposition in Exercise 4.4.10} extbf {Variation of a Function} extbf {Variation of a Stochastic Process} extbf {Variation of Continuous, m $L^2;$ubm Martingales} extbf {Variation of Brownian Motion} extbf {Definition of m ${clf _t}$ubm -Brownian Motion} extbf {Martingale Characterization of Brownian Motion}Last Time•Martingale inequalities•Martingale convergence theorem•Uniformly integrable martingalesToday’s lecture: Sections 4.4.1, 5.3MATH136/STAT219 Lecture 21, November 12, 2008 – p. 1/11Doob’s Decomposition in Discrete Time•Let {Xn} be a discrete time SP with IE|Xn| < ∞ that isadapted to some filtration {Fn}•Then there exists a unique decomposition Xn= Mn+ Ansuch that◦{Mn, Fn} is a martingale◦{An} is a previsible SP; i.e. An+1is Fn-measurable◦A0= 0•Proof: define Mn= Xn− Anwhere Anis defined via therecursive equationAn+1= An+ IE(Xn+1− Xn|Fn)•If {Xn} is a submartingale, then {An} is a nondecreasingprocess, i.e. An+1≥ Ana.s. for all nMATH136/STAT219 Lecture 21, November 12, 2008 – p. 2/11Doob-Meyer Decomposition•Let {Mt, Ft} be a continuous, square integrable martingale(i.e. IE(M2t) < ∞ for all t ≥ 0 and {Mt} has continuouspaths a.s.)•Then there exists a unique SP {At} such that◦A0= 0◦{At} is adapted to {Ft}◦{At} has continuous sample paths a.s.◦{At} is nondecreasing (At≥ Asa.s. for all t ≥ s ≥ 0)◦{(M2t− At, Ft)} is a martingale•{At} is called the increasing part associated with {Mt}, and{At} is equal to the quadratic variation process of {Mt}MATH136/STAT219 Lecture 21, November 12, 2008 – p. 3/11Illustration: Doob-Meyer decomposition of W2t0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.500.511.522.53blue: W2t, red: W2t− tMATH136/STAT219 Lecture 21, November 12, 2008 – p. 4/11Illustration: D-M decomposition in Exercise 4.4.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.500.511.52Exercise 4.4.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.500.511.522.5Exercise 4.4.10Mt= exp(Wt− t/2)blue: M2t, red: M2t− AtMATH136/STAT219 Lecture 21, November 12, 2008 – p. 5/11Variation 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|| = max1≤i≤k(t(π)i− t(π)i−1)•For a function f : [0, t] 7→ IR and p ≥ 1, the p-th variation off on [0, t] is defined asV(p)(f) = lim||π||→0kXi=1|f(t(π)i) − f(t(π)i−1)|pprovided the limit exists•Special cases: p = 1 gives total variation and p = 2 givesquadratic variation of f on [0, t]MATH136/STAT219 Lecture 21, November 12, 2008 – p. 6/11Variation of a Stochastic Process•The total variation process of a SP {Xt} is the stochasticprocess {V(1)t, t ≥ 0}, where the value at time t ≥ 0,V(1)t(ω), is the total variation of the function Xs(ω) on theinterval [0, t]•The quadratic variation process of a SP {Xt} is thestochastic process {V(2)t, t ≥ 0}, where the value at timet ≥ 0, V(2)t(ω), is the quadratic variation of the functionXs(ω) on the interval [0, t]•(Above definitions apply only in cases where the limits aredefined in some sense)MATH136/STAT219 Lecture 21, November 12, 2008 – p. 7/11Variation of Continuous, L2Martingales•Let {(Xt, Ft)} be a continuous, square integrable martingale•The quadratic variation process of {Xt}, often denoted hXitor hX, Xit, exists and is equal to {At}, the increasing part ofthe Doob-Meyer decomposition•That is, {(X2t− hXit, Ft)} is a martingale•Also, the total variation of {Xt} on any interval is infinite withprobability 1MATH136/STAT219 Lecture 21, November 12, 2008 – p. 8/11Variation of Brownian Motion•Let {Wt} be a Brownian Motion•Then for all t > 0,kXi=1|W (t(π)i) − W (t(π)i−1)|2→ t in L2as ||π|| → 0•That is, the quadratic variation process of Brownian motionis given by hW it= t a.s.•Brownian motion accumulates quadratic variation at rateone per unit time•Informally, dWtdWt= dt, and also dWtdt = 0 and dtdt = 0•The total variation of Brownian motion on any interval isinfinite with probability 1MATH136/STAT219 Lecture 21, November 12, 2008 – p. 9/11Definition of {Ft}-Brownian MotionAn {Ft}-adapted stochastic process {Wt, t ≥ 0} is a{Ft}-Brownian motion if:•W0= 0 a.s.•Independent increments: for all 0 ≤ s ≤ tWt− Wsis independent ofFs•Stationary increments: for all 0 ≤ s ≤ tWt− Wshas a N(0, t − s) distribution•For almost every ω, the sample path t 7→ Wt(ω) iscontinuousMATH136/STAT219 Lecture 21, November 12, 2008 – p. 10/11Martingale Characterization of Brownian Motion•Suppose {(Xt, Ft)} is a martingale with continuous paths•If {(X2t− t, Ft)} is a martingale•Then {Xt} is a {Ft}-Brownian motionMATH136/STAT219 Lecture 21, November 12, 2008 – p.