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

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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