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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) = maxoz, X(t) ≤ x) = 1 - ( 2z - √t 2 > 0, x < 2, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fM(t),x(t) (z, x), of M(t) and X(t) for t> 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) = maxoz, X(t) ≤ x) = 1 - ( 2z - √t 2 > 0, x < 2, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fM(t),x(t) (z, x), of M(t) and X(t) for t> 0. [Hint: Apply Reflection Principle for (i).]

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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) = maxo<u<t X(u). (i) Show that P(M(t)>z, X(t) ≤ x) = 1 - ( 2z - √t 2 > 0, x < 2, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fM(t),x(t) (z, x), of M(t) and X(t) for t> 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) = maxo<u<t X(u). (i) Show that P(M(t)>z, X(t) ≤ x) = 1 - ( 2z - √t 2 > 0, x < 2, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fM(t),x(t) (z, x), of M(t) and X(t) for t> 0. [Hint: Apply Reflection Principle for (i).]
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