extbf {Last Time} extbf {Regeneration Properties of Brownian Motion} extbf {Reflection Principle for Brownian Motion} extbf {Illustration: Reflection Principle} extbf {Properties of m $ au _alpha $ubm } extbf {Running Maximum of Brownian Motion} extbf {Limiting Properties of Brownian Motion Paths} extbf {Law of the Iterated Logarithm}Last Time•Continuous time stopping times•Optional stopping theorem•Stopped σ-field•Optional sampling theoremToday’s lecture: Section 5.2MATH136/STAT219 Lecture 19, November 7, 2008 – p. 1/8Regeneration Properties of Brownian Motion•Recall: if {Wt} is a Brownian motion and T is any constant(0 ≤ T < ∞) then{WT +t− WT, t ≥ 0}is a Brownian motion•If {Wt} is a Brownian motion and τ is a stopping time withrespect to its canonical filtration {Ft} then{Wτ+t− Wτ, t ≥ 0}is a Brownian motion, and it is independent of the stoppedσ-field FτMATH136/STAT219 Lecture 19, November 7, 2008 – p. 2/8Reflection Principle for Brownian Motion•Let {Wt} be a BM and {Ft} its canonical filtration•For fixed α > 0 defineτα= inf{t ≥ 0 : Wt= α}•Define a process {˜Wt, t ≥ 0} by˜Wt=(Wtt ≤ τα,2α − Wtt > τα•Then {˜Wt, t ≥ 0} is a Browian motion•The reflection principle implies for w ≤ αIP (τα≤ t, Wt≤ w) = IP (Wt≥ 2α − w)MATH136/STAT219 Lecture 19, November 7, 2008 – p. 3/8Illustration: Reflection Principle0 3−1−0.500.511.522.533.5tWtταReflection principle for Brownian motion with α= 1Wtand˜Wtfor α = 1MATH136/STAT219 Lecture 19, November 7, 2008 – p. 4/8Properties of τα•Let {Wt} be a Brownian motion and {Ft} its canonicalfiltration•For fixed α > 0 define τα= inf{t ≥ 0 : Wt= α}•The distribution function of ταisIP (τα≤ t) =2√2πZ∞α√te−u2/2du,with densityfτα(t) =α√2πt3/2e−α22t, t ≥ 0•In particular, IP (τα< ∞) = 1 and IE(τα) = ∞MATH136/STAT219 Lecture 19, November 7, 2008 – p. 5/8Running Maximum of Brownian Motion•Let {Wt} be a Brownian motion and for fixed α > 0 defineτα= inf{t ≥ 0 : Wt= α}•Sincenmax0≤s≤tWs≥ αo= {τα≤ t},we haveIPmax0≤s≤tWs≥ α= IP (τα≤ t),which can be computed using distribution of ταMATH136/STAT219 Lecture 19, November 7, 2008 – p. 6/8Limiting Properties of Brownian Motion Paths•Let {Wt} be a Brownian motion•For α > 0 define τα= inf{t ≥ 0 : Wt= α} and recall thatIP (τα< ∞) = 1•It follows thatlim supt→∞Wt= +∞ a.s.,lim inft→∞Wt= −∞ a.s.MATH136/STAT219 Lecture 19, November 7, 2008 – p. 7/8Law of the Iterated Logarithm•Let {Wt} be a Brownian motion•Behavior for large t: almost surelylim supt→∞Wtp2t log(log t)= 1, and lim inft→∞Wtp2t log(log t)= −1•Behavior for t near 0: almost surelylim supt→0Wtp2t log(log(1/t))= 1, and lim inft→0Wtp2t log(log(1/t))= −1•Law of large numbers for Browian motion:Wtt→ 0 a.s. as t → ∞MATH136/STAT219 Lecture 19, November 7, 2008 – p.