论文标题
从低频测量的扩散的一致推断
Consistent inference for diffusions from low frequency measurements
论文作者
论文摘要
令$（x_t）$为$ \ Mathbb r^d $中有界凸的域中的反映扩散过程，求解随机微分方程$$ dx_t = \ nabla f（x_t）dt + \ sqrt 运动。 $ x_0，x_d，\ dots，x_ {nd} $由离散测量和连续观测之间的时间间隔$ d $组成，因此可以固定，以便无法将一个人“缩放”到过程的路径中。目标是推断扩散性$ f $和相关的过渡操作员$ p_ {t，f} $。我们证明了地图$ f \ mapsto p_ {t，f} \ mapsto p_ {d，f}，t <d $的映射定理和稳定不等式。使用这些估计值，我们基于基于高斯工艺先验的无限维度参数$ f $ $ f $的高斯工艺先验建立了一类贝叶斯算法的统计一致性，并显示了获得的某些收敛速率的最佳性。我们讨论了这个反问题的不良程度与光谱几何形状的“热点”猜想之间的潜在关系。
Let $(X_t)$ be a reflected diffusion process in a bounded convex domain in $\mathbb R^d$, solving the stochastic differential equation $$dX_t = \nabla f(X_t) dt + \sqrt{2f (X_t)} dW_t, ~t \ge 0,$$ with $W_t$ a $d$-dimensional Brownian motion. The data $X_0, X_D, \dots, X_{ND}$ consist of discrete measurements and the time interval $D$ between consecutive observations is fixed so that one cannot `zoom' into the observed path of the process. The goal is to infer the diffusivity $f$ and the associated transition operator $P_{t,f}$. We prove injectivity theorems and stability inequalities for the maps $f \mapsto P_{t,f} \mapsto P_{D,f}, t<D$. Using these estimates we establish the statistical consistency of a class of Bayesian algorithms based on Gaussian process priors for the infinite-dimensional parameter $f$, and show optimality of some of the convergence rates obtained. We discuss an underlying relationship between the degree of ill-posedness of this inverse problem and the `hot spots' conjecture from spectral geometry.
