Fuzzy stochastic differential equations of decreasing fuzziness: Approximate solutions

Abstract

We consider recently introduced fuzzy stochastic differential equations with solutions of decreasing fuzziness. In general, such the equations do not have solutions that could be written in explicit, closed form. Therefore some methods of construction of approximate solutions are proposed in this paper. In considered framework, approximate solutions are some measurable and adapted fuzzy stochastic processes. We analyze two kinds of sequences of approximate solutions. It is showed that each sequence of approximate solutions can be used to prove existence and uniqueness of solution to fuzzy stochastic differential equations of decreasing fuzziness. In fact, both the sequences converge to a unique solution. The rates of convergence related to both the sequences are investigated. All the results apply immediately to set-valued stochastic differential equations with solutions of decreasing diameter.