Statistical Spatially Inhomogeneous Diffusion Inference
Yinuo Ren Yiping Lu Lexing Ying Grant M. Rotskoff
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.