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Assume that X(t) and Y(t) are two independent standard Brownian motion processes satisfying X(0) = 0 and Y(0) = 0. Let M(t) = max0 z, X(t) ≤ x) = 1 − Þ ·0 (22-72), z > 0, x 0. [Hint: Apply Reflection Principle for (i).]

Assume that X(t) and Y(t) are two independent standard Brownian motion processes satisfying X(0) = 0 and Y(0) = 0. Let M(t) = max0 z, X(t) ≤ x) = 1 − Þ ·0 (22-72), z > 0, x 0. [Hint: Apply Reflection Principle for (i).]

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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Assume that X(t) and Y(t) are two independent standard Brownian motion processes satisfying X(0) = 0 and Y(0) = 0. Let M(t) = max0<u<t X(u). (i) Show that P(M(t) > z, X(t) ≤ x) = 1 − Þ ·0 (22-72), z > 0, x <z, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fм(t), x(t) (z, x), of M(t) and X(t) fort > 0. [Hint: Apply Reflection Principle for (i).]
Transcribed Image Text:Assume that X(t) and Y(t) are two independent standard Brownian motion processes satisfying X(0) = 0 and Y(0) = 0. Let M(t) = max0<u<t X(u). (i) Show that P(M(t) > z, X(t) ≤ x) = 1 − Þ ·0 (22-72), z > 0, x <z, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fм(t), x(t) (z, x), of M(t) and X(t) fort > 0. [Hint: Apply Reflection Principle for (i).]
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