# Conditional probabilities in multiplicative noise processes

## TL;DR

This paper develops a path integral method to compute transition probabilities in multiplicative noise stochastic differential equations, transforming the problem into an additive noise case for easier analysis.

## Contribution

It introduces a time reparametrization technique that simplifies multiplicative noise problems by converting them into additive noise problems, with explicit solutions for harmonic oscillators.

## Key findings

- Established the equivalence between conditional probability and quantum propagator.
- Demonstrated the method with an explicit harmonic oscillator example.
- Provided a new analytical approach for multiplicative noise processes.

## Abstract

We address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach. We show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass. Introducing a {\em time reparametrization}, we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one. We illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment.