Comparing statistical and deep learning techniques for parameter estimation of continuous-time stochastic differentiable equations
Aroon Sankoh, Victor Wickerhauser
TL;DR
The paper investigates parameter estimation for continuous-time stochastic differential equations, focusing on the Ornstein-Uhlenbeck process defined by $dX_t = -\theta X_t \, dt + \sigma \, dW_t$. It directly compares traditional maximum likelihood estimation (MLE) using the OU transition density with a data-driven LSTM estimator trained on trajectory data, employing a composite Huber loss and ELU activations. Results show that MLE is more robust for estimating the volatility $\sigma^2$ across regimes, while the LSTM can yield tighter estimates for the mean-reversion parameter $\theta$ but is less stable for $\sigma^2$ in high- or low-volatility settings, highlighting a complementary role for deep learning in data-rich or model-uncertain contexts. The study suggests using DL as a flexible alternative when model specifications are uncertain or data are plentiful, while maintaining MLE as a strong baseline for volatility estimation; future work should explore broader architectures and real-world, complex distributions.
Abstract
Stochastic differential equations such as the Ornstein-Uhlenbeck process have long been used to model realworld probablistic events such as stock prices and temperature fluctuations. While statistical methods such as Maximum Likelihood Estimation (MLE), Kalman Filtering, Inverse Variable Method, and more have historically been used to estimate the parameters of stochastic differential equations, the recent explosion of deep learning technology suggests that models such as a Recurrent Neural Network (RNN) could produce more precise estimators. We present a series of experiments that compare the estimation accuracy and computational expensiveness of a statistical method (MLE) with a deep learning model (RNN) for the parameters of the Ornstein-Uhlenbeck process.