Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$?
For examples: are there:
- Ito Isometry(-types) of results for L1 processes
$\begingroup$
$\endgroup$
5
-
$\begingroup$ Can you be more specific about what kind of results you are looking for? It isn't really clear to me what you mean by the calculus being "defined on" an $L^1$ space. Certainly there are plenty of results in either area where $L^1$ spaces arise. $\endgroup$– Nate EldredgeCommented Feb 11, 2016 at 7:39
-
1$\begingroup$ Localization is a standard practice written down in nearly every stochastic analysis textbook. As limits in the definition of stochastic integrals are taken in probability, the generic setting for Ito's formula is neither $L^2$ nor $L^1$ but $L^0$. $\endgroup$– Stephan SturmCommented Feb 11, 2016 at 20:07
-
$\begingroup$ Good point, but does it's isometry types of results exists for L1 processes? $\endgroup$– ABIMCommented Feb 12, 2016 at 14:03
-
$\begingroup$ If you do not require that the expectations are finite but allow an equality infinities, this follows also directly from localization. $\endgroup$– Stephan SturmCommented Feb 12, 2016 at 21:37
-
$\begingroup$ I just wonder what you intend to achieve with your question? $\endgroup$– Stephan SturmCommented Feb 12, 2016 at 21:37
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
Note that even for usual Ito calculus wrt to Brownian motion, there is a variation of Ito isometry for processes with infinite variance see. https://nualart.ku.edu/StochasticCalculus.pdf
Now in terms of a calculus in terms of general processes that are not in $L^{2}$, it gets hard because we lose the Hilbert-space structure and thus nice things such as orthonormality/projections (eg. "Stochastic Calculus with Respect to Gaussian Processes"). Even in Rough-paths, one needs some regularity to do calculus.
One reference I found online for processes with infinite variance is "On generalized Malliavin calculus".
answered Jan 13, 2023 at 22:58
-
$\begingroup$ I like this, your answering all my old open questions:)))) $\endgroup$– ABIMCommented Jan 14, 2023 at 17:25