Before revision:  We prove that a shallow neural network with the logistic activation is a universal approximator in L^{2(R \times } 0, 1 ). As a corollary V_t/K but not C_t/K can be approximated by superpositions of logistic functions on \mathbb{R^+ \times } 0, 1 . This justifies the benefit of time value oriented decision functions in option pricing models .
After revision:  We prove that a shallow neural network with the logistic activation is a universal approximator in L^{2(R \times } 0, 1 By proving a novel universal approximation theorem, we show that our decision function rather than Hutchinson-Lo-Poggio's can be approximated by superpositions of logistic functions on \mathbb{R^+ \times } 0, 1 . This justifies the benefit of time value oriented decision functions in option pricing models .