Abstract
The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms.
Highlights
Deterministic fuzzy differential equations have been developed due to investigations of dynamic systems where the information on parameters of such systems is incomplete or vague. They play an important role in an increasing number of system models in biology [1], engineering [2], civil engineering [3], bioinformatics and computational biology [4], quantum optics and gravity [5], and hydraulic [6, 7] and modeling of mechanical systems [8]
A different approach was proposed by Hullermeier in [13] where fuzzy differential equations were interpreted as a family of differential inclusions associated with level sets of their fuzzy right hand sides
The main problem is a concept of a fuzzy stochastic integral which should cover the notion of the classical stochastic Itointegral
Summary
Deterministic fuzzy differential equations have been developed due to investigations of dynamic systems where the information on parameters of such systems is incomplete or vague. The second aim and the novelty of the paper is the analysis of this problem
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