Konstantin Borovkov, Mikhail Zhitlukhin
We show that the distribution of the maximum of the fractional Brownian motion $B^H$ with Hurst parameter $H\to 0$ over an $n$-point set $τ\subset [0,1]$ can be approximated by the normal law with mean $\sqrt{\ln n}$ and variance $1/2$ provided that $n\to \infty$ slowly enough and the points in $τ$ are not too close to each other.
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Fisher's Information for Discretely Sampled Levy Processes
本文研究了递增频率离散采样下Levy过程Fisher信息的渐近性态。我们表明，它是可能的，以区分不仅是连续的过程的一部分，从它的跳跃部分，但也不同类型的跳跃，并推导出有效的估计的收敛速度。
Discretely sampled variance and volatility swaps versus their continuous approximations
离散抽样方差和波动率掉期交易活跃在场外市场。为了给这些互换定价，经常使用连续采样近似来简化计算。本文的目的是研究这种近似有效的条件。我们的第一组定理的特征条件下，离散采样交换值是有限的，给定的连续近似值存在。令人惊讶的是，对于一些其他合理的价格过程，离散抽样掉期价格不存在，从而使近似无效。提供了实例。进一步假设两个交换值都存在，我们研究了离散采样值收敛到连续值的充分条件。由于它在文献中的普及，我们将我们的定理应用到3/2随机波动率模型。虽然我们可以证明所有交换值的有限性，但我们只能证明某些参数值的近似收敛性。
The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian li…
We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead-lag increments. In particular, by sampling a $d$-dimensional continuous semimartingale $X:[0,1] \rightarrow \mathbb{R}^d$ at a set of times $D=(t_i)$, we construct a piecewise linear, axis-directed process $X^D: [0,1] \rightarrow\mathbb{R}^{2d}$ comprised of a past and future component. We call such an object the Hoff process associated with the discrete data $\{X_{t}\}_{t_i\in D}$. The Hoff process can be lifted to its natural rough path enhancement and we consider the…
We derive consistent and asymptotically normal estimators for the drift and volatility parameters of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space and the bounded domain. We establish the exact spatial regularity of the solution, which in turn, using power-variation arguments, allows building the desired estimators. We show that naive approximations of the derivatives appearing in the power-variation based estimators may create nontrivial biases, which we compute explicit…
On filter-type estimation of discretely sampled cyclic long-memory processes
本文扩展广义滤波矩方法至离散采样循环长记忆过程，证明估计量强相合且收敛速度与连续情形一致，并通过数值实验给出所需采样率与分辨率。
Berry-Esséen bounds for the least squares estimator for discretely observed fractional Ornstein-Uhlenbeck processes
令$θ>0$。考虑一个一维分数Ornstein-Uhlenbeck过程，定义为$dX_t= -θ X_t dt+dB_t,\quad t\geq0,$，其中$B$是Hurst参数为$H\in(1/2,1)$的分数布朗运动.我们感兴趣的问题，估计未知参数$θ$。为此，我们处理离散化的轨迹，在$n$等距时间$t_i=iΔ_{n}, i=0,...,n$，和$T_n=nΔ_{n}$表示的“观察窗口”的长度观察。我们假设$Δ_{n} \rightarrow 0$和$T_n\rightarrow \infty$为$n\rightarrow \infty$。作为$θ$的估计量，我们选择最小二乘估计量（LSE）$\hatθ_{n}$。建立了该估计量的相合性。在LSE $\hatθ_{n}$的中心极限定理中，得到了当$H\in(1/2,3/4)$时Kolmogorov距离的显式界.这些结果在过程$X$上没有任何遍历性。
Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of $Δ$-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for $Δ$-power variations of the iterated integrals of a fractional Brownian motion (fBm). These abstract results, besides being of independent interest, in the second part of the paper are applied to e…
We consider a parameter estimation problem for one dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed …
Estimation of several parameters in discretely-observed Stochastic Differential Equations with additive fractional noise
本文针对由加性分数布朗运动驱动的随机微分方程，基于离散观测构造Hurst参数、扩散参数和漂移项的联合估计量，证明一致性并给出收敛速率，且验证了分数Ornstein-Uhlenbeck过程满足可识别性假设。
The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consi…
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…
Error bounds and convergence for American put option pricing based on translation-invariant Markov chains
考虑一个离散的有限维马尔可夫市场模型。在这种情况下，离散抽样美式期权可以使用所谓的"非重组“树算法定价。通过连续增加执行次数，美式期权价格本身就可以计算出来;出于组合的原因，我们将考虑一种递归算法，它在每一步递归中将执行次数增加一倍。首先，我们证明，由初等参数，误差界的一阶差分在这个递归算法。从这一点上，边界的高阶差异，可以得到使用组合参数的粗糙路径理论的动机。我们将得到一个显式的$L^1(C)$收敛估计的递归算法，价格离散采样的美国$\max$ -看跌期权（在一篮子大小$d$）在每个递归步骤，$C$属于某类紧子集的$\RR^d$，在假设足够小的波动率。在$d=1$，$L^1(C)$的情况下，将导出$C$的更自然选择的界限。
Asymptotic and Exact Pricing of Options on Variance
本文研究了以标的证券的离散抽样已实现方差为基础的衍生品定价问题。在文献中，已实现方差通常近似于其连续时间极限，基础对数价格的二次变化。在这里，我们描述了两个对象上的选项的小时间限制。我们发现，它们之间的差异强烈依赖于股票价格过程是否有跳跃。随后，我们提出了两种新的方法来评估离散采样的已实现方差的期权价格。其中一种方法是近似的，它是基于我们的渐近结果的二次变差的期权价格的校正。另一种方法是精确的;它使用一种新的随机化方法，并应用傅立叶-拉普拉斯技术。我们比较的方法，并说明我们的结果通过一些数值例子。
Power law Polya's urn and fractional Brownian motion
我们引入了一个自然的家庭的随机游动的整数集上的比例分数布朗运动。增量X_n具有这样的性质：给定{X_k：k < n}，X_n的条件律是X_{n-k_n}的条件律，其中k_n独立于正整数上的固定律μ进行采样。当μ具有大致幂律衰减时（准确地说，当它位于α稳定从属子吸引区域中时，0 < α< 1/2），行走缩放为具有Hurst参数α+ 1/2的分数布朗运动。行走是很容易模拟和他们的增量满足FKG不等式。从某种意义上说，它们是Z上简单随机游动的自然“分数”模拟。
Estimating the Fractal Dimension of the S&P 500 Index using Wavelet Analysis
标准普尔500指数数据采样在一分钟的时间间隔超过11.5年（1989年1月至2000年5月）的过程中进行了分析，特别是赫斯特参数的平稳段（赫斯特参数几乎是恒定的时间段）估计。使用对数尺度谱的渐近无偏和有效的估计。估计量是渐近高斯的，从$N$点的数据段获得的估计量的方差是$\frac{1}{N}$阶的。小波分析是为高频数据集量身定制的，因为它由于用于计算细节系数的金字塔算法而具有低的计算复杂性。该估计是强大的添加剂非平稳性，在这里，它表现出一定程度的鲁棒性乘法非平稳性，如季节性和波动持续性，以及。这一分析表明，市场在1997-2000年期间变得更有效率。
Predicting path-dependent processes by deep learning
本文提出一种基于离散观测历史信息的路径依赖过程深度学习预测方法，从理论上证明了其L2误差收敛性与近似合理性，并通过数值模拟验证了方法的准确性，分析了影响预测精度的相关因素。
Rough differential equations for volatility
本文提出了一种利用粗糙微分方程建模粗糙波动性的方法，通过联合提升布朗运动和分数布朗运动，构建了新的波动性模型并进行了市场数据校准。
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