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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), tɛ [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) = Бакже which is related to the area under the Brownian bridge curve. Show that ts² $3 (ts)2 E (Y(t)Y(s)) = 2 - s

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), tɛ [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) = Бакже which is related to the area under the Brownian bridge curve. Show that ts² $3 (ts)2 E (Y(t)Y(s)) = 2 - s

Linear Algebra: A Modern Introduction
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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Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.

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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), tɛ [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) = Бакже which is related to the area under the Brownian bridge curve. Show that ts² $3 (ts)2 E (Y(t)Y(s)) = 2 - s<t 6 4 2.3 Show that an equivalent definition to Brownian motion is given by W(t) = (t+1)Q t t +1 (Hint: you can use without proof the fact that if X(t) is a Gaussian process then Y(t) = X(f(t)) is Gaussian if f(t) is a strictly increasing function) 2.4 Show that the solution to the eigenvalue problem [R(t,s)øk(s)ds = køk(t), R(t, s) = min(t, s) — ts, ¢k(0) = ok (1) = 0 [ok(s) be(s) ds = Skl (la) (1b) is given by Hence explain why the following holds 1 Ak = 212 ok(t) = √√2 sin kπt where (k are i.i.d N(0, 1) random variables. Q(t)=√√sin kat -Sk, k=1 Επ (2)
Transcribed Image Text: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), tɛ [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) = Бакже which is related to the area under the Brownian bridge curve. Show that ts² $3 (ts)2 E (Y(t)Y(s)) = 2 - s<t 6 4 2.3 Show that an equivalent definition to Brownian motion is given by W(t) = (t+1)Q t t +1 (Hint: you can use without proof the fact that if X(t) is a Gaussian process then Y(t) = X(f(t)) is Gaussian if f(t) is a strictly increasing function) 2.4 Show that the solution to the eigenvalue problem [R(t,s)øk(s)ds = køk(t), R(t, s) = min(t, s) — ts, ¢k(0) = ok (1) = 0 [ok(s) be(s) ds = Skl (la) (1b) is given by Hence explain why the following holds 1 Ak = 212 ok(t) = √√2 sin kπt where (k are i.i.d N(0, 1) random variables. Q(t)=√√sin kat -Sk, k=1 Επ (2)
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