# Generating solution paths of Markovian stochastic differential equations using diffusion models

arXiv: https://arxiv.org/abs/2509.08731 · HTML: https://mlsanity.com/paper/2509.08731/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 5 of its 50 references that are in the map.
- First posted on arXiv 10 Sep 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 5 of 5)

- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)
- [Sampling, Diffusions, and Stochastic Localization](https://mlsanity.com/paper/2305.10690/) (2023)
- [Contractive Diffusion Probabilistic Models](https://mlsanity.com/paper/2401.13115/) (2024)
- [Diffusion Factor Models: Generating High-Dimensional Returns with Factor Structure](https://mlsanity.com/paper/2504.06566/) (2025)
- [Dynamic data generation and dynamic portfolio selection: an application of a score-based diffusion model](https://mlsanity.com/paper/2507.09916/) (2025)

## Built on by (most cited first, 2 of 2)

- [Factor-Based Conditional Diffusion Model for Contextual Portfolio Optimization](https://mlsanity.com/paper/2509.22088/) (2025)
- [Learning Transition Kernels of Jump-Diffusion Processes with Conditional Diffusion Models](https://mlsanity.com/paper/2610.09045/) (2026)

## Abstract (arXiv)

This paper introduces a new approach to generating sample paths of unknown Markovian stochastic differential equations (SDEs) using diffusion models, a class of generative AI methods commonly employed in image and video applications. Unlike the traditional Monte Carlo methods for simulating SDEs, which require explicit specifications of the drift and diffusion coefficients, ours takes a model-free, data-driven approach. Given a finite set of sample paths from an SDE, we utilize conditional diffusion models to generate new, synthetic paths of the same SDE. Numerical experiments show that our method consistently outperforms two alternative methods in terms of the Kullback--Leibler (KL) divergence between the distributions of the target SDE paths and the generated ones. Moreover, we present a theoretical error analysis deriving an explicit bound on the said KL divergence. Finally, in simulation and empirical studies, we leverage these synthetically generated sample paths to boost the performance of reinforcement learning algorithms for continuous-time mean--variance portfolio selection, hinting promising applications of our study in financial analysis and decision-making.