# A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

## TL;DR

This paper introduces a nonlocal approach to quantum stochastic equations, connecting nonlocal diffusions with quantum processes, and offers new methods for deriving solutions in quantum finance models.

## Contribution

It presents a novel nonlocal framework for deriving quantum Kolmogorov backward and Fokker-Planck equations, linking nonlocal diffusions with quantum stochastic processes.

## Key findings

- Derived quantum equations using nonlocal methods
- Linked nonlocal diffusions with quantum stochastic processes
- Proposed moment matching for solutions

## Abstract

The Accardi-Boukas quantum Black-Scholes equation can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus. In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.