# Asymptotic Expansion and Weak Approximation for a Stochastic Control Problem on Path Space

## TL;DR

This paper analyzes a stochastic control problem using asymptotic expansion and confirms its accuracy through numerical experiments.

## Contribution

The paper introduces a precise error estimate for asymptotic expansion in stochastic control problems and validates it numerically.

## Key findings

- The expansion error is shown to depend on the regularity of path space functionals.
- A numerical scheme using Monte Carlo simulation effectively implements the expansion in multidimensional cases.
- Numerical experiments confirm the approximation error matches the theoretical convergence rate.

## Abstract

The paper provides a precise error estimate for an asymptotic expansion of a certain stochastic control problem related to relative entropy minimization. In particular, it is shown that the expansion error depends on the regularity of functionals on path space. An efficient numerical scheme based on a weak approximation with Monte Carlo simulation is employed to implement the asymptotic expansion in multidimensional settings. Throughout numerical experiments, it is confirmed that the approximation error of the proposed scheme is consistent with the theoretical rate of convergence.