Stochastic gradient descent in continuous time for drift identification in multiscale diffusions

TL;DR

Due to the incompatibility between the multiscale data and the homogenized model, the estimator alone is not able to reconstruct the exact drift, so it is proposed to filter the original trajectory through appropriate kernels and include filtered data in the stochastic differential equation for the estimator, which indeed solves the misspecification issue.

Abstract

We consider the setting of multiscale overdamped Langevin stochastic differential equations, and study the problem of learning the drift function of the homogenized dynamics from continuous-time observations of the multiscale system. We decompose the drift term in a truncated series of basis functions, and employ the stochastic gradient descent in continuous time to infer the coefficients of the expansion. Due to the incompatibility between the multiscale data and the homogenized model, the estimator alone is not able to reconstruct the exact drift. We therefore propose to filter the original trajectory through appropriate kernels and include filtered data in the stochastic differential equation for the estimator, which indeed solves the misspecification issue. Several numerical experiments highlight the accuracy of our approach. Moreover, we show theoretically in a simplified framework the asymptotic unbiasedness of our estimator in the limit of infinite data and when the multiscale parameter describing the fastest scale vanishes.

Figures (8)

• Figure 1: Comparison between SGDCT estimator (a) without filtered data and (b) with exponential filtered data, for the example in \ref{['sec:verification']}. In each plot, the dashed orange line is the biased value $\alpha$, the solid orange line is the unbiased value $A$, and the blue line is the estimate over time $t$.
• Figure 2: Effect of filter width $\delta$ dependence on $\varepsilon$, for the example in \ref{['sec:verification']}. The filter width is given by $\delta = \varepsilon^\xi$, where $\xi \in (0,3)$ is the variable on the horizontal axis. The top dashed orange line corresponds to the biased value $\alpha$, while the bottom dashed orange line corresponds to the unbiased value $A$. The blue line gives the value of the exponential filter estimator at the final time $T$.
• Figure 3: Rate of convergence in time of SGDCT estimates $\widehat{A}_T^\varepsilon$ with exponential filtered data, for the example in \ref{['sec:verification']}. A Monte Carlo approximation of the $L^2$ error of the estimates $\widehat{A}_T^\varepsilon$ from the true value $A$ is given in blue, and a reference line giving an $\mathcal{O}(T^{-1/2})$ rate is given in orange.
• Figure 4: Sample estimates using exponential filtered SGDCT, for the multidimensional drift coefficient in \ref{['sec:multidimensional']}. In (a) and (b), the blue line is the estimate over time $t$ for $A_1$ and $A_2$, respectively. The orange lines correspond to $A_1$ and $A_2$, respectively. In (c), the curve is the path of the estimate of $(A_1, A_2)$ with the color of the curve representing the time of the estimate and the red star representing to true value $(A_1, A_2)$.
• Figure 5: Sample estimates using the moving average SGDCT, for the multidimensional drift coefficient in \ref{['sec:multidimensional']}. In (a) and (b), the blue line is the estimate over time $t$ for $A_1$ and $A_2$, respectively. The orange lines correspond to $A_1$ and $A_2$, respectively. In (c), the curve is the path of the estimate of $(A_1, A_2)$ with the color of the curve representing the time of the estimate and the red star representing to true value $(A_1, A_2)$.

Theorems & Definitions (19)

• Remark 2.1
• Proposition 3.1
• proof
• Remark 3.2
• Theorem 4.2
• Remark 4.3
• Lemma 4.4
• proof
• Lemma 4.5
• proof