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BerndSchmitz
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### Distribution of max BM - BM

Hi,Can anybody help me with deriving the distribution of $max_{0 \leq s \leq t} W_s - W_t$?Thanks,Bernd

LocalVolatility
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### Distribution of max BM - BM

Let $M_t = \max_{0 \leq u \leq t} W_u$. You know that $M_t$ and $W_t$ have the joint density$f_{M_t, W_t}(m, w) = \frac{2 (2m - w)}{t \sqrt{2 \pi t}} \exp \left\{ -\frac{(2m - w)^2}{2 t} \right\}$. Then I'd just take it from there..$\mathbb{P} \left\{ M_t - W_t \geq x \right\} = \int_0^\infty \int_{-\infty}^{m - x} f_{M_t, W_t}(m, w) \mathrm{d}w \mathrm{d}m$

BerndSchmitz
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Joined: August 16th, 2011, 9:48 am

### Distribution of max BM - BM

Thanks. I had hoped for some elegant trick though

BerndSchmitz
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Posts: 242
Joined: August 16th, 2011, 9:48 am

### Distribution of max BM - BM

QuotePr( max(ws) - wt > a) = Pr( max(ws) > a) How do you justify that?
Last edited by BerndSchmitz on March 16th, 2015, 11:00 pm, edited 1 time in total.

Alan
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### Distribution of max BM - BM

QuoteOriginally posted by: BerndSchmitzThanks. I had hoped for some elegant trick though :-)One elegant trick would note that it is well-known that $M_t - W_t \stackrel{d}{=} |W_t|$, where M is the BM maximum process and |W| is reflected BM. The pdf for the latter is $p(t,x) = \frac{2 \, e^{-x^2/(2 t)}}{\sqrt{2 \pi t}}$ on $(0,\infty)$ and so is also the density of the distribution you seek.
Last edited by Alan on March 16th, 2015, 11:00 pm, edited 1 time in total.

BerndSchmitz
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Joined: August 16th, 2011, 9:48 am

### Distribution of max BM - BM

Thanks a lot Alan. This was the kind of answer I was looking for.Is it intuitive why $M_t-W_t=^d|W_t|$? Or can you recommend any good (online available?) ressources for that?

Alan
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### Distribution of max BM - BM

You're welcome. To me, it is certainly not "obvious", merely plausible. Certainly both sides have non-negative support andhit zero a lot. You'll have to google -- I looked it up in Bertoin's book on Levy processes.

savr
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Joined: January 21st, 2013, 3:28 pm

### Distribution of max BM - BM

The equality in law P(Mt) = P(Mt-Wt) (is this what you mean?) is what is true.The equality almost surely Mt - Wt = |Wt| could not be true, for instance with >0 prob Wt can be <0 and Mt >0.

savr
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Joined: January 21st, 2013, 3:28 pm

### Distribution of max BM - BM

Terminology is the same whether r.v. s or processes. To sum up, in continuous time:for processes: Mt - Wt ~ |Wt| , and for fixed t ~ Mtin discrete time (binomial tree)for fixed t: Mt - Wt ~ Mt !~ |Wt|In a trinomial tree at least |Wt| and Mt have the same support, so there is some hope of redefining things to make them have the same law.
Last edited by savr on March 24th, 2015, 11:00 pm, edited 1 time in total.