Statistical Spatially Inhomogeneous Diffusion Inference

• URL: http://arxiv.org/abs/2312.05793v1
• Date: Sun, 10 Dec 2023 06:52:50 GMT
• Title: Statistical Spatially Inhomogeneous Diffusion Inference
• Abstract summary: Inferring a diffusion equation from discretely-observed measurements is a statistical challenge. We propose neural network-based estimators of both the drift $boldsymbolb$ and the spatially-inhomogeneous diffusion tensor $D = SigmaSigmaT$.
• Score: 15.167120574781153
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Inferring a diffusion equation from discretely-observed measurements is a statistical challenge of significant importance in a variety of fields, from single-molecule tracking in biophysical systems to modeling financial instruments. Assuming that the underlying dynamical process obeys a $d$-dimensional stochastic differential equation of the form $$\mathrm{d}\boldsymbol{x}_t=\boldsymbol{b}(\boldsymbol{x}_t)\mathrm{d} t+\Sigma(\boldsymbol{x}_t)\mathrm{d}\boldsymbol{w}_t,$$ we propose neural network-based estimators of both the drift $\boldsymbol{b}$ and the spatially-inhomogeneous diffusion tensor $D = \Sigma\Sigma^{T}$ and provide statistical convergence guarantees when $\boldsymbol{b}$ and $D$ are $s$-H\"older continuous. Notably, our bound aligns with the minimax optimal rate $N^{-\frac{2s}{2s+d}}$ for nonparametric function estimation even in the presence of correlation within observational data, which necessitates careful handling when establishing fast-rate generalization bounds. Our theoretical results are bolstered by numerical experiments demonstrating accurate inference of spatially-inhomogeneous diffusion tensors.

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