A concept of fuzzy transition probabilities

Starting from fuzzy real numbers with an arbitrary lattice of belief and following the extension principle, the paper develops concepts of fuzzy probabilities, transition probabilities and random variables and of their combinations, and shows that these concepts are consistent. It derives some results on fuzzy real numbers, on the expectation of fuzzy random variables and on fuzzy stochastic processes. To sketch the range of applications of fuzzy stochastics, the paper gives two examples that show how real-world problems may be modeled by means of fuzzy probabilities and that give small numerical examples. Moreover, it gives a brief outlook for a possible expansion of the present theory to fuzzy Markovian decision processes by means of a partial order on the set of all fuzzy real numbers.