Nonparametric Estimation of the Trend in Reflected Fractional SDE

The nonparametric LAN expansion for discretely observed diffusions

Consider a scalar reflected diffusion $(X_t:t\geq 0)$, where the unknown drift function $b$ is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consists of $(X_0,X_Δ,...,X_{nΔ})$ for some fixed sampling distance $Δ>0$, the model satisfies the local asymptotic normality (LAN) property, assuming that $b$ satisfies some mild regularity assumptions. This is established by using the connections of diffusion processes to elliptic and parabolic PDEs. The key tools will be regularity estimates from the theory of parabolic PDEs as well as a detailed analysis …

Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling

We study random series priors for estimating a functional parameter (f\in L^2[0,1]). We show that with a series prior with random truncation, Gaussian coefficients, and inverse gamma multiplicative scaling, it is possible to achieve posterior contraction at optimal rates and adaptation to arbitrary degrees of smoothness. We present general results that can be combined with existing rate of contraction results for various nonparametric estimation problems. We give concrete examples for signal estimation in white noise and drift estimation for a one-dimensional SDE.

关于反射麦肯-沃拉斯方程的适定性及其近似方法的应用

Well-posedness and approximation of reflected McKean-Vlasov SDEs with applications

本文研究了反射麦肯-沃拉斯随机微分方程的适定性及其粒子近似，并应用于受限域中的采样和优化问题。

一种从 M/G/1 工作负载进行非参数推断的筛选方法

Nonparametric inference from possibly unstable M/G/1 workload observations

研究 M/G/1 工作负载模型中服务时间分布的非参数推断问题，提出基于工作负载轨迹的估计器 $B_n(w)$，通过筛选机制简化问题，在适度正则性假设下建立误差界，给出首个达参数化 $L^1$ 风险率且无需相关假设的解决方案。

Jump filtering and efficient drift estimation for Lévy-driven SDE's

The problem of drift estimation for the solution $X$ of a stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density these conditions reduce to $n Δ_n^{3-ε}\to 0,$ where $n$ is the number of observations and $Δ_n$ is the maximal sampling step. This result relaxes the condition $nΔ_n^2 \to 0$ usually require…

基于薛定谔桥的深度条件生成学习

Schrödinger bridge based deep conditional generative learning

提出一种基于薛定谔桥的深度生成方法，通过随机微分方程和深度神经网络学习条件分布，生成的样本质量优于现有方法，并可用于估计条件密度及统计量。

Consistency of Bayesian nonparametric inference for discretely observed jump diffusions

We introduce verifiable criteria for weak posterior consistency of identifiable Bayesian nonparametric inference for jump diffusions with unit diffusion coefficient and uniformly Lipschitz drift and jump coefficients in arbitrary dimension. The criteria are expressed in terms of coefficients of the SDEs describing the process, and do not depend on intractable quantities such as transition densities. We also show that products of discrete net and Dirichlet mixture model priors satisfy our conditions, again under an identifiability assumption. This generalises known results by incorporating jump…

时间齐次随机微分方程路径二分类过程的最小最大收敛速率

Minimax convergence rates of a binary classification procedure for time-homogeneous SDE paths

本文研究了时间齐次随机微分方程路径二分类中插件分类器的最小最大收敛速率，基于非参数估计器推导了漂移系数的收敛速率，并提出新方法研究下界。

Mixing it up: A general framework for Markovian statistics

Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes and in particular hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical sta…

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + σ(X_t) dW_t,$ with discretely sampled observations $X_0, X_Δ, \dots, X_{nΔ}$. We analyse the nonlinear inverse problem corresponding to the `low frequency sampling' regime where $Δ>0$ is fixed and $n \to \infty$. A general theorem is proved that gives conditions for prior distributions $Π$ on the diffusion coefficient $σ$ and the drift function $b$ that ensure minimax optimal contraction rates of the posterior distribution over Hölder-Sobolev smoothness classes. These conditions are verified for nat…

Rates of Convergence for Nearest Neighbor Classification

Nearest neighbor methods are a popular class of nonparametric estimators with several desirable properties, such as adaptivity to different distance scales in different regions of space. Prior work on convergence rates for nearest neighbor classification has not fully reflected these subtle properties. We analyze the behavior of these estimators in metric spaces and provide finite-sample, distribution-dependent rates of convergence under minimal assumptions. As a by-product, we are able to establish the universal consistency of nearest neighbor in a broader range of data spaces than was previo…