With this question I try to build up a systematization of different kinds of integrals. The following table differentiates between deterministic and stochastic integrals, the summation processes ("from left" or "average between left and right") and different kinds of variations (bounded and quadratic).

The following integrals are already present:

(A) Riemann integral
(F) Ito integral
(G) Stratonovich integral

Unfortunately I didn't find references for the remaining fields. Is this because it doesn't make any sense to define these? And then, why not?

In any case: I would very much appreciate your help here. Could you please reference the field (X) and give some further information regarding

  • type of integral
  • examples
  • refences
  • links
  • if it doesn't make any sense: why not

Every little bit of information helps - thank you very much and I promise to consolidate the information given here into a proper form.

Addendum: For the cases (D) and (E) [perhaps even (H) and (I)?] there seem to be some possible connections to deterministic fractal functions (like e.g. Weierstrass function) but I can't find any further references on definitions for integrals/integration here.

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I think that you can define for some integrands with restrictive conditions, integrals with respect to q-variational integrator (q>0). You can google for Young Integrals.

You could also have a look at Rough Path theory also where you solve equations with respect to path of infinite variations.

Moreover there is a way to define a path-by-path Itô integral for continuous integrators with Itô's formula also being derived in a path-by-path way. This is done (in french) in an article "Calculd'Itô sans probabilité" by Hans Föllmer in volume XV of the Séminaire of Probablity of Strasbourg LNM 850 . To my knowledge it hasn't been translted in english.

All this is very partial but are leads that you may exlpore yourself



It seems unlikely to me that (J) and (K) have any substantial positive answer. The general theory of stochastic integration is basically about semimartingales, and the quadratic variation exists for these.


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