extbf {Last Time} extbf {Definition of Brownian Motion} extbf {Brownian Motion Sample Path} extbf {Brownian Motion Sample Paths} extbf {Equivalent Definition of Brownian Motion} extbf {Existence of Brownian Motion} extbf {Continuity of Brownian Sample Paths} extbf {Properties of Brownian Motion} extbf {Relatives of Brownian Motion} extbf {Illustration: Brownian Motion with drift} extbf {Illustration: Geometric Brownian Motion} extbf {Illustration: Brownian Bridge} extbf {Illustration: Ornstein-Uhlenbeck process}Last Time•Random walk•Convergence of scaled random walks•Stationary and stationary increment processes•Continuous or RCLL modifications•Kolmogorov continuity criteriaToday’s lecture: Section 5.1MATH136/STAT219 Lecture 13, October 20, 2008 – p. 1/13Definition of Brownian MotionA stochastic process {Wt, t ≥ 0} is a (standard) Brownianmotion(aka Wiener process) if:•It is a Gaussian process•IE(Wt) = 0 for all t ≥ 0•IE(WtWs) = min(t, s) for all t, s ≥ 0•For almost every ω, the sample path t 7→ Wt(ω) iscontinuousNote that this implies W0= 0 a.s.MATH136/STAT219 Lecture 13, October 20, 2008 – p. 2/13Brownian Motion Sample Path0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2−1.5−1−0.500.511.52First 5000 stepsMATH136/STAT219 Lecture 13, October 20, 2008 – p. 3/13Brownian Motion Sample Paths0 0.5 1 1.5 2 2.5 3−2.5−2−1.5−1−0.500.511.522.5tWtSample paths of Brownian motionMATH136/STAT219 Lecture 13, October 20, 2008 – p. 4/13Equivalent Definition of Brownian MotionA stochastic process {Wt, t ≥ 0} is a Brownian motion if andonly if:•W0= 0 a.s.•Independent increments: for all 0 ≤ s ≤ tWt− Wsis independent of σ(Wu: 0 ≤ u ≤ s)•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 13, October 20, 2008 – p. 5/13Existence of Brownian Motion•Theorem: Brownian motion exists.•Construction 1: limit of scaled symmetric random walks•Construction 2: uses basis functions of Hilbert spaceMATH136/STAT219 Lecture 13, October 20, 2008 – p. 6/13Continuity of Brownian Sample Paths•If Y has a N(0, σ2) distribution, thenIE(Y2n) =(2n)!2nn!σ2n, n = 1, 2, . . .•Suppose {Vt, t ≥ 0} is a Gaussian SP with IE(Vt) = 0 andIE(VtVs) = min(t, s).•Apply Kolmogorov’s continuity criteria with α = 4, β = 1, andC = 3, i.e.IE[(Vt+h− Vt)4] = 3[IE(Vt+h− Vt)2]2= 3h2,to show that {Vt, t ≥ 0} has a continuous modificationMATH136/STAT219 Lecture 13, October 20, 2008 – p. 7/13Properties of Brownian Motion•Symmetry: the process −W is a BM•Time homogeneity: for all fixed s ≥ 0, {Wt+s− Ws, t ≥ 0} isa BM•Time reversal: for all fixed T > 0, {WT− WT −t, t ≥ 0} is aBM•Scaling (or self-similarity): for all fixed a > 0, {1√aWat, t ≥ 0}is a BM•Time inversion: define˜W0= 0 and˜Wt= tW1/t, t > 0. Then{˜Wt, t ≥ 0} is a BMMATH136/STAT219 Lecture 13, October 20, 2008 – p. 8/13Relatives of Brownian Motion•For µ ∈ IR, σ > 0, x ∈ IR the process {x + µt + σWt, t ≥ 0}is aBrownian motion with drift µ and diffusion coefficient σstarting from x•For Yt= eWt, the process {Yt, t ≥ 0} is a GeometricBrownian motion•For Bt= Wt− min(t, 1)W1, the process {Bt, t ≥ 0} is aBrownian bridge•For Ut= e−t/2Wet, the process {Ut, t ≥ 0} is anOrnstein-Uhlenbeck processMATH136/STAT219 Lecture 13, October 20, 2008 – p. 9/13Illustration: Brownian Motion with drift0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20123456789tXtBrownian Motion with drift: µ= 1, σ= 2, x= 1x + µt + σWtMATH136/STAT219 Lecture 13, October 20, 2008 – p. 10/13Illustration: Geometric Brownian Motion0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2024681012141618tYtGeometric Brownian Motion YtYt= eWtMATH136/STAT219 Lecture 13, October 20, 2008 – p. 11/13Illustration: Brownian Bridge0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.6−0.4−0.200.20.40.60.81Brownian Bridge BttBtBt= Wt− min(t, 1)W1MATH136/STAT219 Lecture 13, October 20, 2008 – p. 12/13Illustration: Ornstein-Uhlenbeck process0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.3−0.2−0.100.10.20.30.40.5tUtOrnstein−Uhlenbeck process UtUt= e−t/2WetMATH136/STAT219 Lecture 13, October 20, 2008 – p.