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).]
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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).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe4bc73f-0219-4a5a-be2a-5fbe85a452fa%2F086ecdb2-14d5-407a-a48c-1d0ac484323d%2Fokfyvq_processed.jpeg&w=3840&q=75)
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