Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes

TL;DR

The paper addresses parameter estimation for nonlinear, high-dimensional SDEs with intractable transition densities by introducing two splitting-based, likelihood-type estimators based on Lie-Trotter and Strang schemes. It proves $L^p$ convergence for both schemes, and establishes consistency and asymptotic normality under a mild one-sided Lipschitz condition, achieving optimal rates for drift and diffusion parameters. In a 3D stochastic Lorenz system, the Strang-splitting estimator demonstrates superior accuracy and substantially faster computation compared with state-of-the-art methods like Euler-Maruyama, Kessler, local linearization, and Hermite expansion. The work delivers practical, scalable tools for likelihood-based inference in nonlinear SDEs, with strong theoretical guarantees and demonstrated efficiency on a challenging chaotic system.

Abstract

The likelihood functions for discretely observed nonlinear continuous-time models based on stochastic differential equations are not available except for a few cases. Various parameter estimation techniques have been proposed, each with advantages, disadvantages, and limitations depending on the application. Most applications still use the Euler-Maruyama discretization, despite many proofs of its bias. More sophisticated methods, such as Kessler's Gaussian approximation, Ozaki's Local Linearization, Aït-Sahalia's Hermite expansions, or MCMC methods, might be complex to implement, do not scale well with increasing model dimension, or can be numerically unstable. We propose two efficient and easy-to-implement likelihood-based estimators based on the Lie-Trotter (LT) and the Strang (S) splitting schemes. We prove that S has $L^p$ convergence rate of order 1, a property already known for LT. We show that the estimators are consistent and asymptotically efficient under the less restrictive one-sided Lipschitz assumption. A numerical study on the 3-dimensional stochastic Lorenz system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on precision and computational speed compared to the state-of-the-art.

Figures (7)

• Figure 1: An example trajectory of the stochastic Lorenz system \ref{['eq:Lorenz']} starting at $(0, 1, 0)$ for $N = 10000$ and $h = 0.005$. The first row shows the evolution of the individual components $X, Y$, and $Z$. The second row shows the evolution of component pairs: $(Y,Z)$, $(X,Z)$ and $(X, Y)$. Parameters are $p = 10$, $r = 28$, $c = 8/3$, $\sigma_1^2 = 1$, $\sigma_2^2 = 2$ and $\sigma_3^2 = 1.5$.
• Figure 2: Comparing the absolute relative error (ARE) as a function of increasing discretization step $h$ for eight estimators in the stochastic Lorenz system. The sample size is $N = 10000$. The $y$-axis is on log scale. The HE estimator (purple dot) converged only for $h = 0.005$, and only for 60% of the simulated data sets.
• Figure 3: Comparing the normalized distributions of $(\hat{\bm{\theta}}_N - \bm{\theta}_0) \oslash \bm{\theta}_0$ (where $\oslash$ is the element-wise division) of the Lorenz system for the $\mathrm{S_{mix}}$, $\mathrm{S_{avg}}$, LL and HE estimators for $N = 10000$. Each column represents one parameter, and each row represents one value of the discretization step $h$. The black dot with a vertical bar in each violin plot represents the mean and the standard deviation. The HE estimator (purple) converged only for $h = 0.005$, and only for 60% of the simulated data sets.
• Figure 4: Running times as a function of $N$ for different estimators of the Lorenz system. Each column shows one value of $h$. On the $x$-axis is the sample size $N$, and on the $y$-axis is the running time in seconds. The HE estimator (purple) achieved convergence only for $h = 0.005$, and only in $43\% - 72\%$ of cases across various sample sizes $N$.
• Figure 5: Comparing distributions of $\hat{\bm{\theta}}_N - \bm{\theta}_0$ for the $\mathrm{S_{mix}}$ estimator with theoretical asymptotic distributions \ref{['eq:asymptoticdist']} for each parameter (columns), for $h = 0.01$ and $N \in \{1000, 5000, 10000\}$ (colors). The black lines correspond to the theoretical asymptotic distributions computed from data and true parameters for $N = 10000$ and $h = 0.01$.

Theorems & Definitions (26)

• Lemma 2.1
• Proposition 2.2
• Definition 2.3
• Definition 3.1: $L^p$ consistency of a numerical scheme
• Definition 3.2: Bounded moments of a numerical scheme
• Theorem 3.3: $L^p$ convergence of a numerical scheme
• Proposition 3.4: One-step prediction of LT splitting
• Theorem 3.5: $L^p$ convergence of the LT splitting
• Proposition 3.6
• Theorem 3.7: $L^p$ convergence of S splitting