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Hi Everyone

I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.

We are given a Brownian Motion $B_t$ and we note $M_t=\sup_{s\le t}B_s$. Moreover we have a smooth real valued function $F(t,x,y)$ (for example a $C^{1,2,1}$) over $\mathbb{R}^+\times \mathbb{R} \times \mathbb{R}^+$ and we are looking for the SDE followed by $F(t,M_t-B_t,M_t)$

I have a hard time trying to express $dF$.

Best regards

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  • $\begingroup$ After two seconds of reflection the answer is simple. It suffices to view $F$ as a function of $B_t$ and of the two increasing functions $M_t$ and $t$ and to apply Itô's lemma (Revuz and Yor bok's version for example). Sorry for this post, I first thought some Itô-Tanaka-Meyer formula were involved because of the fact that the processes $(L_t,|W_t|)$ (L is the local time of W at 0) and $(M_t-B_t,M_t)$ have the same law. Regards $\endgroup$
    – The Bridge
    Apr 22, 2010 at 8:43
  • $\begingroup$ The Bridge: The MO bot bumped your question since in the system it remains "unanswered". Since zhoraster did provide an answer, it would be polite to accept it and award him or her the points. Plus, this will be useful for future generations of students who may ask the same question as you did. $\endgroup$
    – Tom LaGatta
    Sep 8, 2010 at 15:31

1 Answer 1

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Let $F(t,m-b,m)=G(t,m,b)$ (this way it's easier to write). As long as $M$ has bounded variation, we can happily write (I skip arguments, which are $t,M_t,B_t$) $$ dG(t,M_t,B_t) = G'_t dt + G'_m dM_t + G'_b dB_t + \frac12 G''_{bb} dt. $$

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  • $\begingroup$ Sorry but I already answered the question myself in the comment. $\endgroup$
    – The Bridge
    Aug 5, 2010 at 7:50
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    $\begingroup$ Yes, I didn't read it carefully. Still I don't understand why downvote my answer. $\endgroup$
    – zhoraster
    Aug 17, 2010 at 8:47

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