The moment-generating function of the log-normal distribution, how zero-entropy principle unveils an asymmetry under the reciprocal of an action

The present manuscript is about application of It{^o}'s calculus to the moment-generating function of the lognormal distribution. While Taylor expansion fails when applied to the moments of the lognormal due to divergence, various methods based on saddle-point approximation conjointly employed with integration methods have been proposed. By the Jensen’s inequality, the MGF of the lognormal involves some convexity adjustment, which is one of the aspects under consideration thereof. A method based on zero-entropy principle is proposed part of this study, which deviations from the benchmark by infinitesimal epsilons is attributed to an asymmetry of the reciprocal. As applied to systems carrying vibrating variables, the partial offset by the reciprocal of an action, is a principle meant to explain a variety of phenomena in fields such as quantum physics.