Question 2 Let W(t) be a standard one dimensional Brownian motion and let Q(t) be the Brownian Bridge on [0, 1], i.e. Q(t)=W(t)tW(1), tЄ [0, 1]. 2.1 Show that Q(t) is a Gaussian process with EQ(t) = 0, E(Q(t)Q(s)) = min(t, s) — ts. 2.2 Now consider the stochastic process Y(t) = Q(l)dl which is related to the area under the Brownian bridge curve. Show that ts² $3 E (Y(t)Y(s)) = (ts)2 st 2 6 4

Question 2 Let W(t) be a standard one dimensional Brownian motion and let Q(t) be the Brownian Bridge on [0, 1], i.e. Q(t)=W(t)tW(1), tЄ [0, 1]. 2.1 Show that Q(t) is a Gaussian process with EQ(t) = 0, E(Q(t)Q(s)) = min(t, s) — ts. 2.2 Now consider the stochastic process Y(t) = Q(l)dl which is related to the area under the Brownian bridge curve. Show that ts² $3 E (Y(t)Y(s)) = (ts)2 st 2 6 4

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

See similar textbooks