Path probability of stochastic motion: A functional approach

TL;DR

This paper develops a functional approach to derive the path probability distribution for stochastic motion, with applications in physics and finance, providing analytical evaluations of path probabilities within specified bands.

Contribution

Introduces a general functional formula for path probability distribution and applies it to stochastic processes in physics and finance.

Findings

Derived a general formula for path probability distribution functional.

Analytically evaluated path probabilities within specified tubes or bands.

Applied the formalism to stochastic stock price dynamics.

Abstract

The path probability of a particle undergoing stochastic motion is studied by the use of functional technique, and the general formula is derived for the path probability distribution functional. The probability of finding paths inside a tube/band, the center of which is stipulated by a given path, is analytically evaluated in a way analogous to continuous measurements in quantum mechanics. Then, the formalism developed here is applied to the stochastic dynamics of stock price in finance.