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
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$\begingroup$ 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 EldredgeFeb 11, 2016 at 7:39
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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 SturmFeb 11, 2016 at 20:07
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$\begingroup$ Good point, but does it's isometry types of results exists for L1 processes? $\endgroup$– ABIMFeb 12, 2016 at 14:03
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$\begingroup$ If you do not require that the expectations are finite but allow an equality infinities, this follows also directly from localization. $\endgroup$– Stephan SturmFeb 12, 2016 at 21:37
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$\begingroup$ I just wonder what you intend to achieve with your question? $\endgroup$– Stephan SturmFeb 12, 2016 at 21:37
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1 Answer
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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
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$\begingroup$ I like this, your answering all my old open questions:)))) $\endgroup$– ABIMJan 14, 2023 at 17:25