Smearing Distributions and their use in Financial Markets

TL;DR

This paper demonstrates that certain superpositions of path integrals with varying parameters maintain Markovian properties if their smearing distributions have a specific form, simplifying complex financial models with stochastic volatility.

Contribution

It introduces a specific functional form for smearing distributions that preserve Markovian properties in superposed path integrals, aiding financial modeling.

Findings

Smearing distributions obey the Chapman-Kolmogorov relation under specific conditions.

Simplification of coupled Fokker-Planck equations for financial models.

Application to stochastic volatility models in finance.

Abstract

It is shown that superpositions of path integrals with arbitrary Hamiltonians and different scaling parameters v ("variances") obey the Chapman-Kolmogorov relation for Markovian processes if and only if the corresponding smearing distributions for v have a specific functional form. Ensuing "smearing" distributions substantially simplify the coupled system of Fokker-Planck equations for smeared and un-smeared conditional probabilities. Simple application in financial models with stochastic volatility is presented.