A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation

Let X be a fuzzy set-valued random variable (FRV), and \Theta_X the family of all fuzzy sets B for which the Hukuhara difference X-B exists P-almost surely. In this paper, we prove that X can be decomposed as X(\omega)=C+Y(\omega) where the equality holds for P-almost every \omega\ in \Omega, C is the unique deterministic fuzzy set that minimizes E[d_2(X,B)^2] as B is varying in \Theta_X, and Y is a centered FRV (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all FRV translation (i.e. X=M+I_\xi\ for some deterministic fuzzy convex set M and some random element in R^d). In particular, X is an FRV translation if and only if the Aumann expectation EX is equal to C up to a translation. This result includes the well-known case of Gaussian fuzzy random variable for which X=EX+\xi\ with \xi\ being a Gaussian element in R^d, and the fuzzy Brownian motion B_t that can be written as B_t = I_\xi_t where \xi_t is a Brownian process in R^d