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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