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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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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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.

and and Very very grateful!

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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