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The present manuscript is about application of It{\^o}'s calculus to themoment-generating function of the lognormal distribution. While Taylorexpansion fails when applied to the moments of the lognormal due to divergence,various methods based on saddle-point approximation conjointly employed withintegration methods have been proposed. By the Jensen's inequality, the MGF ofthe lognormal involves some convexity adjustment, which is one of the aspectsunder consideration thereof. A method based on zero-entropy principle isproposed part of this study, which deviations from the benchmark byinfinitesimal epsilons is attributed to an asymmetry of the reciprocal. Asapplied to systems carrying vibrating variables, the partial offset by thereciprocal of an action, is a principle meant to explain a variety of phenomenain fields such as quantum physics.

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