Stability of Euler Methods for Fuzzy Differential Equation

The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of alpha and lambda.