Why Are Fuzzy and Stochastic Calculus Different?

Abstract

The theory of fuzzy sets, through possibilistic measures, has always been compared to the theory of probabilities. Much has been said in regards to how they describe different philosophical phenomena and hence how they should be employed in different models. What about their matehamtical development? In this essay, we intend to argue that the challenges found in the development of stochastic analysis are completely different from those found in fuzzy-set analysis. While the former derive from intrinsic irregularity in the paths of stochastic processes, the latter inherits, from the set-valued analysis of Banks and Jacobs [BJ70], the difficulties of finding the appropriate definitions. By stressing these fundamental issues, this work aims at (i) contributing to the discussion about the differences of fuzzy and stochastic, concluding once more that the theories are complementary rather than competing, and (ii) encouraging the fuzzy analyst to attempt to use some techniques often ignored by the probabilist in his pursuit of the construction of a theory of fuzzy calculus, for what does not work for one area could be useful for the other. In particular, while derivatives built using convergence in the sense of distribution don’t usually work in the stochastic realm, the analogous approach – so-called interactive fuzzy calculus – might prove to be a fruitful area for future researches, [dBP17].