Generating solution paths of Markovian stochastic differential equations using diffusion models

Abstract

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.

Figures (6)

• Figure 1: One-dimensional OU process: comparison of mean and standard deviation (std) of solutions obtained by three generative models with 100 synthetic paths. Theoretical and training mean and std are also plotted.
• Figure 2: One-dimensional time-inhomogeneous GBM: comparison of mean and standard deviation (std) of solutions obtained by two generative models with 100 synthetic paths. Theoretical and training mean and std are also plotted.
• Figure 3: 100-dimensional time-homogeneous GBM: comparison of the mean and the standard deviation (std) of solutions (across 4 different dimensions) obtained by three generative models with 10,000 synthetic paths. Theoretical and training mean and std are also plotted.
• Figure 4: Wealth trajectories of two plug-in policies and three RL policies in the out-of-sample test when the target level $z=1.1$. The initial wealth is assume to be one. The "buy and hold" trajectory is the S&P 500 index normalized by its price at the beginning of 2010.
• Figure 5: Wealth trajectories of two plug-in policies and three RL policies in the out-of-sample test when the target level $z=1.2$. The initial wealth is assume to be one. The "buy and hold" trajectory is the S&P 500 index normalized by its price at the beginning of 2010.

Theorems & Definitions (29)

• Remark 1
• Proposition 1
• Proposition 2
• Theorem 1
• proof : Proof of Proposition \ref{['prop: path_KL_decomposition']}
• proof : Overview of the Proof of Proposition \ref{['prop: condition_diffusion_KL_bound']}
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4