Data-driven generative simulation of SDEs using diffusion models

This paper introduces a new approach to generating sample paths of unknown stochastic differential equations (SDEs) using diffusion models, a class of generative AI models 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, our method 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. To demonstrate the effectiveness of our approach, we conduct a simulation experiment to compare our method with alternative benchmark ones including neural SDEs. Furthermore, in an empirical study we leverage these synthetically generated sample paths to enhance the performance of reinforcement learning algorithms for continuous-time mean-variance portfolio selection, hinting promising applications of diffusion models in financial analysis and decision-making.

Reward-Directed Score-Based Diffusion Models via q-Learning

We propose a new reinforcement learning (RL) formulation for training continuous-time score-based diffusion models for generative AI to generate samples that maximize reward functions while keeping the generated distributions close to the unknown target data distributions. Different from most existing studies, our formulation does not involve any pretrained model for the unknown score functions of the noise-perturbed data distributions. We present an entropy-regularized continuous-time RL problem and show that the optimal stochastic policy has a Gaussian distribution with a known covariance matrix. Based on this result, we parameterize the mean of Gaussian policies and develop an actor-critic type (little) q-learning algorithm to solve the RL problem. A key ingredient in our algorithm design is to obtain noisy observations from the unknown score function via a ratio estimator. Numerically, we show the effectiveness of our approach by comparing its performance with two state-of-the-art RL methods that fine-tune pretrained models. Finally, we discuss extensions of our RL formulation to probability flow ODE implementation of diffusion models and to conditional diffusion models.