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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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    https://brainmass.com/math/basic-algebra/brownian-bridge-19304

    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 ...

    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.

    $2.49

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