American Options with Uncertainty of the Stock Prices : The Discrete-Time Model (Mathematical Decision Making under Uncertainty)

2. Fuzzy stochastic processes First we give some mathematical notations regarding fuzzy numbers. Let (Ω,M, P ) be a probability space, where M is a σ-field and P is a non-atomic probability measure. R denotes the set of all real numbers, and let C(R) be the set of all non-empty bounded closed intervals. A ‘fuzzy number’ is denoted by its membership function ã : R → [0, 1] which is normal, upper-semicontinuous, fuzzy convex and has a compact support. Refer to Zadeh [12] regarding fuzzy set theory. R denotes the set of all fuzzy numbers. In this paper, we identify fuzzy numbers with its corresponding membership functions. The α-cut of a fuzzy number ã(∈ R) is given by ãα := {x ∈ R | ã(x) ≥ α} (α ∈ (0, 1]) and ã0 := cl{x ∈ R | ã(x) > 0},