On a certain class of two-sided continuous local semimartingales: Toward a sample-wise characterization of the Nelson process

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Abstract
Working with the extended framework of stochastic integrals recently discovered by Itô, a complex of stochastic processes inherent in quantum mechanics, the Nelson process, is characterized in terms of sample paths. It is shown that the Nelson process belongs to a certain class of two-sided continuous local semimartingales. Several basics of stochastic calculus in this class are presented. Stochastic calculus of variations is applied in this class to construct the Nelson process and to further illustrate some details of its sample paths. Examples are the bound states, the two-slit interference, and the gravity in quantum mechanics.