# Duality in stochastic processes from the viewpoint of basis expansions

## TL;DR

This paper introduces a novel basis expansion approach to derive duality relations in stochastic processes, particularly between stochastic differential equations and birth-death processes, simplifying the derivation process.

## Contribution

It presents a new derivation method based on basis expansions, expanding the types of dual stochastic processes that can be obtained.

## Key findings

- Derived dual processes based on Taylor-type polynomials
- Derived dual processes based on Hermite polynomials
- Simplified derivation method using integration by parts

## Abstract

A new derivation method of duality relations in stochastic processes is proposed. The current focus is on the duality between stochastic differential equations and birth-death processes. Although previous derivation methods have been based on the viewpoint of time-evolution operators, the current derivation is based on basis expansions. In addition, only the tool needed for the derivation is the integration by parts, which is rather simple and understandable. The viewpoint of basis expansions enables us to obtain various dual stochastic processes. As a demonstration, dual processes based on Taylor-type and Hermite polynomials are derived.