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Brownian Bridge

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Category: Statistics
Subject: Brownian Bridge
Details: Let B(t) denote a process of Brownian motion.
Let Q(t) be a Brownian Bridge process.
Then, B(t)=(1+t) Q(t/(t+1)).

Using the fact that

P(max((b+B(t))/(1+t))>a)=exp(-2a(a-b))

show that for a Brownian Bridge Q(t)

P(max(Q(u)>a)=exp(-2 a^2)
where 0<=u<=1.

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Solution Summary

Category: Statistics
Subject: Brownian Bridge
Details: Let B(t) denote a process of Brownian motion.
Let Q(t) be a Brownian Bridge process.
Then, B(t)=(1+t) Q(t/(t+1)).

Using the fact that

P(max((b+B(t))/(1+t))>a)=exp(-2a(a-b))

show that for a Brownian Bridge Q(t)

P(max(Q(u)>a)=exp(-2 a^2)
where 0<=u<=1.

Solution Preview

Hi. I've written the solution up in an easier to read format contained in the attached PDF file.
<br>
<br>This question involves making an appropriate substitution to get your expression in a required form. In the question we are told that:
<br>
<br>B(t) = (1+t)Q(t/t+1) (1)
<br>
<br>which, by simply rearranging, may be written ...

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