Stochastic differential equations with fuzzy drift and diffusion

Abstract

A new framework for the fuzzification of stochastic differential equations is presented. It allows for a detailed description of the model uncertainty and the non-predictable stochastic law of natural systems, e.g. in ecosystems even the probability law of the random dynamic changes due to unobservable influences like anthropogenic disturbances or climate variation. The fuzziness of the stochastic system is modelled by a fuzzy set of stochastic differential equations which is identified with a fuzzy set of initial conditions, time-dependent drift and diffusion functions. Using appropriate function spaces the extension principle leads to a consistent theory providing fuzzy solutions in terms of fuzzy sets of processes, fuzzy states, fuzzy moments and fuzzy probabilities.