For the stochastic integrals w.r.t. stable processes of a determinated (non-random) function, does there exist a corresponding integration by parts formula?

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Thank you very much!

Ps. It is my first time to post a question on the web. Very Sorry!

  • $\begingroup$ Define your terms: which "stochastic integrals", "determined function"? $\endgroup$ – T. Amdeberhan Oct 17 '16 at 17:18

These notes from an MIT course discuss Itô processes. Here's the integration by parts formula they give:

$$ \int_0^t f_s \, dB_s = f_t B_t - \int_0^t B_s \, df_s $$

where $f_s$ is a (deterministic) continuous function on $[0,t]$. The notes are from the business school and so the details are skipped.

As long as $f(s)$ is deterministic we have some back-of the envelope computations:

\begin{eqnarray} \Delta \big(f(s) B_s\big) &=& f(s + \Delta s)B_{s + \Delta s} - f(s )B_{s} \\ \\ &=& \big[f(s + \Delta s) - f(s)\big]B_{s + \Delta s} - f(s) \big[ B_{s+\Delta s} - B_s\big] \\ \\ &=& \Delta f(s)\, B_{s + \Delta s} - f(s) \,\Delta B_s\end{eqnarray}

Itô's formula is not goint to tell us anything new. It tells us very familiar things such as: $$ \int_o^t s \, dB_t = t\, B_t - \int_0^t B_s \, ds$$ More in Øksendal's book.


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