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(1) Let (B₁) be a Brownian motion started at 0. Show that the process Xt = Bt+1 − B₁ for t≥ 0, is a Brownian motion. Hint: check the definition. (2) Show that a continuous Gaussian process is uniquely determined by its mean function together with its covariance function. Hint: use the result on Gaussian vectors. (3) Let (Bt) be a Brownian motion started at 0. Show that the process X₁ = tB(1/t) for t > 0, and X₁ = 0 is a Brownian motion. Hint: use (2) and the Theorem that Brownian motion is a Gaussian process with zero mean function and covariance function s ^ t.

(1) Let (B₁) be a Brownian motion started at 0. Show that the process Xt = Bt+1 − B₁ for t≥ 0, is a Brownian motion. Hint: check the definition. (2) Show that a continuous Gaussian process is uniquely determined by its mean function together with its covariance function. Hint: use the result on Gaussian vectors. (3) Let (Bt) be a Brownian motion started at 0. Show that the process X₁ = tB(1/t) for t > 0, and X₁ = 0 is a Brownian motion. Hint: use (2) and the Theorem that Brownian motion is a Gaussian process with zero mean function and covariance function s ^ t.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(1) Let (B₁) be a Brownian motion started at 0. Show that the process Xt = Bt+1 − B₁ for t≥ 0, is a Brownian motion. Hint: check the definition. (2) Show that a continuous Gaussian process is uniquely determined by its mean function together with its covariance function. Hint: use the result on Gaussian vectors. (3) Let (B₁) be a Brownian motion started at 0. Show that the process X₁ = tB(1/t) for Xt t> 0, and Xo 0 is a Brownian motion. Hint: use (2) and the Theorem that Brownian motion is a Gaussian process with zero mean function and covariance function s ^ t. = (4) Let Xt = Bt - tB₁ for 0 ≤ t ≤ 1. Show that (Xt) is a Gaussian process. Hint: use the result of Homework 3, the alternative definition of a Gaussian vector. State with reason if (Xt) has independent increments.
Transcribed Image Text:(1) Let (B₁) be a Brownian motion started at 0. Show that the process Xt = Bt+1 − B₁ for t≥ 0, is a Brownian motion. Hint: check the definition. (2) Show that a continuous Gaussian process is uniquely determined by its mean function together with its covariance function. Hint: use the result on Gaussian vectors. (3) Let (B₁) be a Brownian motion started at 0. Show that the process X₁ = tB(1/t) for Xt t> 0, and Xo 0 is a Brownian motion. Hint: use (2) and the Theorem that Brownian motion is a Gaussian process with zero mean function and covariance function s ^ t. = (4) Let Xt = Bt - tB₁ for 0 ≤ t ≤ 1. Show that (Xt) is a Gaussian process. Hint: use the result of Homework 3, the alternative definition of a Gaussian vector. State with reason if (Xt) has independent increments.
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