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In Wikipedia a formula is given for the distribution of $$M_t = \max_{0\leq s \leq t} W_s$$ even conditioned on $W_t$.

I wonder if there is also a simple expression for (note the absolute value) $$\tilde M_t = \max_{0\leq s \leq t} |W_s|$$ maybe conditioned on $|W_t|$?

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the derivation of $M_t$ comes from reflection principle. At moment, i cannot see how it would work for $\tilde{M}$ –  Lost1 Apr 10 '13 at 9:32

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up vote 1 down vote accepted

I'd recomment Borodin and Salminen's Handbook of Brownian Motion - Facts and Formulae. There's no simple expression for the distribution, but formula 1.1.8 at page 250 gives for $y > \max(x,z)$:

$$P_x(\max_{0<s<t}|W_s|<y, |W_t| \in dz) = $$

$$\frac{1}{\sqrt{2\pi t}} \sum_{k=-\infty}^\infty(-1)^k \left( e^{-(z-x+2ky)^2/2t} + e^{-(z+x+2ky)^2/2t} \right)\,dz $$

Isn't that a beautiful work of art... You can use standard techniques to obtain all the formulas you want (but they'll still look like this).

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