Some properties of strong solutions to stochastic fuzzy differential equations

Abstract

We consider stochastic fuzzy differential equations driven by m-dimensional Brownian motion. Such equations can be useful in modeling of hybrid dynamic systems, where the phenomena are subjected to two kinds of uncertainties: randomness and fuzziness, simultaneously. Under a boundedness condition, which is weaker than linear growth condition, and the Lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations. Solutions, which are fuzzy stochastic processes, and their uniqueness are considered to be in a strong sense. An estimation of error of the Picard approximate solution is established. We give a boundedness type result for the solution defined on finite time interval. Also the stabilities of solution on initial condition and coefficients of the equation are shown. The existence and uniqueness of a solution defined on infinite time interval is proven. Finally, some applications of fuzzy stochastic differential equations are considered. All the results presented in this paper apply to set-valued stochastic differential equations.