Path probability of stochastic motion: A functional approach

• The concept of path probability of a random walker is studied.
• The general formula for the path probability distribution functional is derived.
• The overdamped limit of the formula is evaluated.
• The probability of finding paths inside a given tube is calculated.
• The theory developed here is applied to the problem of stock price in finance.

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.