Cylindrical Projections of Occupied Diffusions

TL;DR

This paper introduces cylindrical projections to approximate infinite-dimensional occupation flows in occupied diffusions, enabling practical simulation and analysis for path-dependent stochastic processes.

Contribution

The authors propose a novel finite-dimensional approximation method for occupied diffusions, with proven convergence and applications in finance and Monte Carlo methods.

Findings

Proved strong convergence of cylindrical projections to the original process.

Validated the method through Euler--Maruyama simulations of self-interacting diffusions.

Applied the approach to the Local Occupied Volatility (LOV) model in finance.

Abstract

Occupied diffusions offer a Markovian framework for path-dependent dynamics by lifting the state space with a flow of occupation measures. Because this additional feature is infinite-dimensional, the simulation of these processes remains computationally intractable. We address this by introducing \textit{cylindrical projections}, which approximate the occupation flow via a finite-dimensional system. We establish the strong convergence of this approximation to the initial process and derive corresponding convergence rates. The method is validated through Euler--Maruyama simulations of self-interacting diffusions and an application to the Local Occupied Volatility (LOV) model in finance. Finally, we provide a weak error analysis and explore its consequences for Monte Carlo methods and derivatives pricing.