Drift Estimation for Stochastic Differential Equations with Denoising Diffusion Models

TL;DR

Across different drift classes, the proposed estimator was found to match classical methods in low dimensions and remained consistently competitive in higher dimensions, with gains that cannot be attributed to architectural design choices alone.

Abstract

We study the estimation of time-homogeneous drift functions in multivariate stochastic differential equations with known diffusion coefficient, from multiple trajectories observed at high frequency over a fixed time horizon. We formulate drift estimation as a denoising problem conditional on previous observations, and propose an estimator of the drift function which is a by-product of training a conditional diffusion model capable of simulating new trajectories dynamically. Across different drift classes, the proposed estimator was found to match classical methods in low dimensions and remained consistently competitive in higher dimensions, with gains that cannot be attributed to architectural design choices alone.

Figures (18)

• Figure 1: The top-left panel shows the drift error $e^{2}(\tau)$ when estimating $\mu^{(1)}$; the top-right panel restricts the range of $\tau$ to $\tau \in[0.15,1]$. The bottom row compares the true drift with $\mathcolor{blue}{\bar{\mu}}(\tau, y)$ for $\tau=1$ (left) and optimal $\tau\approx0.2$ (right) for $\mathcal{K}=100$. Positions $Y$ are uniformly spaced between $[-1.5, 1.5]$, covering $99\%$ of the training distribution.
• Figure 2: The left panels show the drift error $e^{2}(\tau) \forall \tau \in[0,1]$; the right panels restrict the range of $\tau$ to $\tau \in[0.15,1]$. The top row shows the drift error $e^{2}(\tau)$$\mu^{(2)}$ with $D=8$; the bottom row shows the same for $D=12$.
• Figure 3: Panels (left to right) correspond to $\mu_{1}, \mu_{2}, \mu_{3}$. True drift against $\mathrm{DN}$ drift. Shaded region shows $10-90\%$ quantile envelope over $\mathcal{K}$. State values cover $99\%$ of the training distribution.
• Figure 4: True drift against the $\mathrm{DN}$ drift for $\mu_{4}, c=0$. State values cover $99\%$ of the training distribution in each dimension. Top panels show dimensions $3$ (left) and $8$ (right) for $D=8$. Bottom panels show dimensions $3$ (left) and $12$ (right) for $D=12$. Shaded region denotes $10-90\%$ quantile envelope over $\mathcal{K}$.
• Figure 5: OOS $E^{(\mu)}_{j\Delta}$ for $\mu_{4}, c=20$, for $\mathrm{DN}$, and $\mathrm{FC}^{+}$-Conv. Shaded region shows $10-90\%$ quantile envelope over test trajectories $\mathcal{I}$.