10th World Congress in Probability and Statistics

Contributed Session (live Q&A at Track 2, 9:30PM KST)

Contributed 01

Stochastic Partial Differential Equations

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

A sobolev space theory for SPDEs with space-time nonlocal operotors

Junhee Ryu (Korea University)

7

Improved stability for linear SPDEs using mixed boundary/internal controls

Dan Goreac (University Shandong Weiha, University Gustave Eiffel)

3
This talk based on a joint work with I. Munteanu ("Al. I Cuza" Univ., Iasi) is motivated by the asymptotic stabilization of abstract Stochastic PDEs of linear type. As a first step, we exhibit an abstract contribution to the exact controllability (in a general Lp-sense, p > 1) of a class of linear SDEs with general (but time-invariant) rank control coefficient in the noise term. Second, we illustrate on relevant frameworks of SPDEs, a way to drive exactly to 0 their unstable part (of dimension $n \ge 1$) by using M internal (respectively N boundary) controls such that max {M, N} < n. Some examples are presented as is the minimal gain for judicious control dimensions.

Law of the large numbers and Central limit theorems for stochastic heat equations

Kunwoo Kim (Pohang University of Science and Technology)

7

The stochastic heat equation with Lévy noise: existence, moments and intermittency

Carsten Chong (Columbia University)

4
In this talk, we present results about existence, moments and large-time asymptotics of the solution to the stochastic heat equation driven by a Lévy space-time white noise, proved using a combination of decoupling techniques, point process methods and change-of-measure techniques. As one of the more surprising results, we show that the solution exhibits the phenomenon of intermittency for all exponents in all dimensions and for all non-Gaussian Lévy noises, which is fundamentally different to what is known in the Gaussian case. Moreover, we demonstrate that the behavior of the intermittency exponents in terms of a coupling constant depends critically on whether the Lévy noise is light- or heavy-tailed.

This is based on joint work with Quentin Berger (Sorbonne) and Hubert Lacoin (IMPA).

Q&A for Contributed Session 01

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This talk does not have an abstract.

Session Chair

Kunwoo Kim (Pohang University of Science and Technology)

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Contributed 06

Various Aspects of Diffusion Processes

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

On nonlinear filtering of jump diffusions

Fabian Germ (University of Edinburgh)

5
We consider a multi-dimensional signal and observation model, $Z_t=(X_t,Y_t)$, which is a jump diffusion, i.e., the solution of an SDE driven by Wiener processes and Poisson martingale measures. The multi-dimensional “signal” $X_t$ is not observable, and we are interested in classic questions about the mean square estimate of $X_t$ for each time t, given the “observations” $Y_s$ for s in [0,t]. These questions were intensively studied for partially observable diffusion processes $Z_t$ in various generality in the past, and a quite complete filtering theory for diffusion processes was developed. Our aim is to contribute to recent studies in extending the nonlinear filtering theory of diffusion processes to that of jump diffusions. We allow the signal and observation noises to be correlated and the infinitesimal generator of $Z_t$ to be degenerate in the coordinate directions of the signal. First we present the filtering equations: the equations for the time evolution of the conditional distribution $P_t$ and of an unnormalised conditional distribution $Q_t$ of $X_t$, given the observations $Y_s$ for s in [0,t]. These equations are (possibly) degenerate stochastic integro-differential equations. They are stochastic PDEs, often referred as the Kushner-Shiryayev and Zakai equations, respectively, in the special case when Z_t is a diffusion process. Next we present new results on existence and uniqueness of the solutions to the filtering equations in $L_p$-spaces. Finally, using our results on the solutions to the filtering equations, we give conditions ensuring that the conditional density $p_t:=dP_t/dx$, with respect to Lebesgue measure exists and belongs to an $L_p$ space for each $t>0$. Moreover, under quite general regularity conditions on the initial conditional density $p_0$ and on the coefficients of the SDE for $Z_t$, we prove that the process $p_t$ is a cadlag process with values in Bessel potential spaces $H^s_p$ and in Slobodeckij spaces $W^s_p$.

Uniqueness and superposition of the distribution-dependent Zakai equations

Huijie Qiao (Southeast University)

4
The work concerns the Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First, we establish the Kushner-Stratonovich equations, the Zakai equations and the distribution-dependent Zakai equations. And then, the pathwise uniqueness, uniqueness in joint law and uniqueness in law of weak solutions for the distribution-dependent Zakai equations are shown. Finally, we prove a superposition principle between the distribution-dependent Zakai equations and distribution-dependent Fokker-Planck equations. As a by-product, we give some conditions under which distribution-dependent Fokker-Planck equations have a weak solutions.

Quadratic variation and quadratic roughness

Purba Das (University of Oxford)

4
We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the concept of quadratic roughness of a path along a partition sequence and show that, for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Typical paths of Brownian motion are shown to satisfy this quadratic roughness property almost-surely along any partition with a required step size condition. Using these results we derive a formulation of Föllmer's pathwise integration along paths with finite quadratic variation which is invariant with respect to the partition sequence.

Eyring-Kramers formula for non-reversible metastable diffusion processes

Jungkyoung Lee (Seoul National University)

9
In this talk, we consider diffusion processes that admit a Gibbs invariant measure but are non-reversible. Such diffusion processes exhibit metastable behavior if the associated potential function owns multiple local minima. For this model, we provide a proof of the Eyring-Kramers formula which provides sharp asymptotics of the mean of the transition time from a local minimum to a deeper one. In particular, our work indicates that the metastable transitions of non-reversible processes are faster than that of reversible ones.

Q&A for Contributed Session 06

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Session Chair

Insuk Seo (Seoul National University)

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Contributed 18

Inference on Dependence

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

Covariance networks for functional data on multidimensional domains

Soham Sarkar (Ecole Polytechnique Federale de Lausanne)

4
Covariance estimation is ubiquitous in functional data analysis. Yet, the case of functional observations over multidimensional domains introduces computational and statistical challenges, rendering the standard methods effectively inapplicable. To address this problem, we introduce Covariance Networks (CovNet) as a modeling and estimation tool. The CovNet model is universal — it can be used to approximate any covariance up to desired precision. Moreover, the model can be fitted efficiently to the data and its neural network architecture allows us to employ modern computational tools in the implementation. The CovNet model also admits a closed-form eigen-decomposition, which can be computed efficiently, without constructing the covariance itself. This facilitates easy storage and subsequent manipulation in the context of the CovNet. Moreover, we establish consistency of the proposed estimator and derive its rate of convergence. The usefulness of the proposed method is demonstrated using an extensive simulation study.

Large dimensional sample covariance matrices with independent columns and diagonalizable simultaneously population covariance matrices

Tianxing Mei (The University of Hong Kong)

6
We consider the limiting behavior of empirical spectral distribution (ESD) of a sample covariance matrix for n independent but not necessary identical distributed samples with their corresponding population covariance matrices diagonalizable simultaneously asymptotically, when dimension of samples grows proportionally with the sample size. The existing works of different types of sample covariance matrices, including the weighted sample covariance matrix, the centered Gram matrix model and that of linear times series models with diagonalizable simultaneous coefficient matrices, can be covered by our approach. As applications, we obtain the existence and uniqueness of the limiting spectral distribution (LSD) of realized covariance matrix for a multidimensional diffusion process with its co-volatility process equipped with an anisotropic time-varying spectrum. Meanwhile, for a matrix-valued autoregressive model, we derive the common limiting spectral distribution of the sample covariance matrix for each matrix-valued observation when both row and column dimension are large.

Random surface covariance estimation by shifted partial tracing

Tomas Masak (École polytechnique fédérale de Lausanne)

4
The problem of covariance estimation for replicated surface-valued processes is examined from the functional data analysis perspective. Considerations of statistical and computational efficiency often compel the use of separability of the covariance, even though the assumption may fail in practice. We consider a setting where the covariance structure may fail to be separable locally - either due to noise contamination or due to the presence of a non-separable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a non-separable but banded component. We introduce non-parametric estimators hinging on the novel concept of shifted partial tracing, enabling computationally efficient estimation of the model under dense observation. Due to the denoising properties of shifted partial tracing, our methods are shown to yield consistent estimators even under noisy discrete observation, without the need for smoothing. Further to deriving the convergence rates and limit theorems, we also show that the implementation of our estimators, including for the purpose of prediction, comes at no computational overhead relative to a separable model. Finally, we demonstrate empirical performance and computational feasibility of our methods in an extensive simulation study and on a real data set.

This is a joint work with Victor M. Panaretos.

Q&A for Contributed Session 18

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Session Chair

Minsun Song (Sookmyung Women’s University)

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Contributed 24

Time Series Analysis I

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

Statistical modelling of rainfall time series using ensemble empirical mode decomposition and generalised extreme value distribution

Willard Zvarevashe (North West University)

3
The extreme rainfall patterns have direct and indirect effect on all earths spheres particularly the hydrosphere, biosphere and lithosphere. Therefore, an understanding of the extreme rainfall patterns is very important for future planning and management. In this study, using Western Cape (South African province) as a case study, the rainfall time series is decomposed into intrinsic mode functions (IMFs) using a data adaptive method, ensemble empirical mode decomposition. The IMFs are modelled using generalised extreme value distribution (GEVD). The model diagnosis and selection using QQ-Plot, PP-Plot and Akaike information criterion shows that the decomposed IMFs have better models than the original rainfall time series. The rainfall modelling using decomposed data may assist in future planning and further research by providing better predictions.

Regularity of multifractional moving average processes with random Hurst exponent

Fabian Mies (RWTH Aachen University)

3
A recently proposed alternative to multifractional Brownian motion (mBm) with random Hurst exponent is studied, which we refer to as Itô-mBm. It is shown that Itô-mBm is locally self-similar. In contrast to mBm, its pathwise regularity is almost unaffected by the roughness of the functional Hurst parameter. The pathwise properties are established via a new polynomial moment condition similar to the Kolmogorov-Centsov theorem, allowing for random local Hölder exponents. Our results are applicable to a broad class of moving average processes where pathwise regularity and long memory properties may be decoupled, e.g. to a multifractional generalization of the Matérn process.

High-frequency instruments and identification-robust inference for stochastic volatility models

Md. Nazmul Ahsan (Concodia University)

3
We introduce a novel class of generalized stochastic volatility (GSV) models, which can utilize and relate many high-frequency realized volatility (RV) measures to the latent volatility. Instrumental variable methods are employed to provide a unified framework for GSV models' analysis (estimation and inference). We study parameter inference problems in GSV models with nonstationary volatility and exogenous predictors in the latent volatility process. We develop identification-robust methods for joint hypotheses involving the volatility persistence parameter and the composite error's autocorrelation parameter (or the noise ratio) and apply projection techniques for inference about the persistence parameter. The proposed tests include Anderson-Rubin-type tests, dynamic versions of the split-sample procedure, and point-optimal versions of these tests. For distributional theory, three sets of assumptions are considered: we provide exact tests and confidence sets for Gaussian errors, establish exact Monte Carlo test procedures for non-Gaussian errors (possibly heavy-tailed), and show asymptotic validity under weaker distributional assumptions. Simulation results show that the proposed tests outperform the asymptotic test regarding size and exhibit excellent power in empirically realistic settings. We apply our inference methods to IBM's price and option data (2009-2013). We consider 175 different instruments (IV's) spanning 22 classes and analyze their ability to describe the low-frequency volatility. The IV's are compared based on the average length of confidence intervals, which are produced by the proposed tests. The superior instrument set mostly consists of 5-minute HF realized measures, and these IV's produce confidence sets where the volatility persistence parameter lies roughly between 0.85 and 1.0. We find RVs with higher frequency produce wider confidence intervals compared to RVs with slightly lower frequency, showing that these confidence intervals adjust to absorb market microstructure noise or discretization error. Further, when we consider irrelevant or weak IV's (jumps and signed jumps), the proposed tests produce unbounded confidence intervals.

Q&A for Contributed Session 24

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Session Chair

Kyongwon Kim (Ewha Womans University)

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Contributed 30

Spatio-temporal Data Analysis

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

Statistical inference for mean function of longitudinal imaging data over complicated domains

Jie Li (Tsinghua University)

5
Motivated by longitudinal imaging data which possesses inherent spatial and temporal correlation, we propose a novel procedure to estimate its mean function. Functional moving average is applied to depict the dependence among temporally ordered images and flexible bivariate splines over triangulations are used to handle the irregular domain of images which is common in imaging studies. Both global and local asymptotic properties of the bivariate spline estimator for mean function are established with simultaneous confidence corridors (SCCs) as a theoretical byproduct. Under some mild conditions, the proposed estimator and its accompanying SCCs are shown to be consistent and oracle efficient as if all images were entirely observed without errors. The finite sample performance of the proposed method through Monte Carlo simulation experiments strongly corroborates the asymptotic theory. The proposed method is illustrated by analyzing two sea water potential temperature data sets.

Gaussian linear dynamic spatio-temporal models and time asymptotics

Suman Guha (Presidency University, Kolkata)

5
Gaussian linear dynamic spatio-temporal models (LDSTMs) are linear gaussian state-space models for spatio-temporal data which contains deterministic or (and) stochastic spatio-temporal covariates besides the spatio-temporal response. They are extensively used to model discrete-time spatial time series data. The model fitting is carried out either by classical maximum likelihood approach or by calculating Bayesian maximum a posteriori estimate of the unknown parameters. While their finite sample behaviour is well studied, literature on their asymptotic properties is relatively scarce. Classical theory on asymptotic properties of maximum likelihood estimator for linear state-space models is not applicable as it hinges on the assumption of asymptotic stationarity of covariate processes, which is seldom satisfied by discrete-time spatial time series data. In this article, we consider a very general Gaussian LDSTM that can accommodate arbitrary spatio-temporal covariate processeses which grow like power functions wrt. time in deterministic or (and) suitable stochastic sense. We show that under very minimal assumptions, any approximate MLE and Bayesian approximate MAPE of some of the unknown parameters and parametric functions are strongly consistent. Furthermore, building upon the strong consistency theorems we also establish rate of convergence results for both approximate MLE and approximate MAPE.

High-dimensional spectral analysis

Jonas Krampe (University of Mannheim)

7
An important part of multivariate time series analysis is the spectral domain and here key quantities are here the spectral density matrix and the partial coherence. Under a high-dimensional set up, we present how inference can be derived for the partial coherence. Furthermore, we present a valid statistical test for the statistical hypothesis that the partial coherence is almost everywhere smaller than a given bound that includes testing whether the partial coherence is zero or not. Applications of this are among others the construction of graphical interaction models helpful for analyzing functional connectivity among brain regions. We illustrate our procedure by means of simulations and a real data application.

Extreme value analysis for mixture models with heavy-tailed impurity

Ekaterina Morozova (National Research University Higher School of Economics)

6
While there exists a well-established theory for the asymptotic behaviour of maxima of the i.i.d. sequences, very few results are available for the triangular arrays, when the distribution can change over time. Typically, the papers on this issue deal with convergence to the Gumbel law or twice-differentiable distribution. It contributes to the aforementioned problem by providing the extreme value analysis for mixture models with varying parameters, which can be viewed as triangular arrays. In particular, we consider the case of the heavy-tailed impurity, which appears when one of the components has a heavy-tailed distribution, and the corresponding mixing parameter tends to zero as the number of observations grows. We analyse two ways of modelling this impurity, namely, by the non-truncated regularly varying law and its upper-truncated version with an increasing truncation level. The set of possible limit distributions for maxima turns out to be much more diverse than in the classical setting, especially for a mixture with the truncated component, where it includes four discontinuous laws. In the latter case, the resulting limit depends on the asymptotic behaviour of the truncation point, which is shown to be related to the truncation regimes introduced in [1]. For practical purposes we describe the procedure of the application of the considered model to the analysis of financial returns.

The current research is a joint work with Vladimir Panov, available as a preprint on arXiv.org [2].

References:
[1] Chakrabarty, A. and Samorodnitsky, G. (2012). Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not? Stochastic models, 28(1), 109–143.
[2] Panov, V. and Morozova, E. (2021). Extreme value analysis for mixture models with heavy-tailed impurity. arXiv preprint arXiv:2103.07689.

Q&A for Contributed Session 30

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Session Chair

Seoncheol Park (Chungbuk National University)

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Contributed 34

Stochastic Process / Modeling

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

Ruin probabilities in the presence of risky investments and random switching

Konstantin Borovkov (The University of Melbourne)

6
We consider a reserve process where claim times form a renewal process, while between the claim times the process has the dynamics of geometric Brownian motion-type Itô processes with time-dependent random coefficients that are “reset” after each jump. Following the approach of Pergamenshchikov and Zeitoni (2006), we use the implicit renewal theory to obtain power-function bounds for the eventual ruin probability. In the special case of the gamma-distributed claim inter-arrival times and geometric Brownian motions with random coefficients, we obtain necessary and sufficient conditions for existence of Lundberg’s exponent (ensuring the power function behaviour for the ruin probability). [Joint work with Roxanne He.]

Wasserstein convergence rates for random bit approximations of continuous Markov processes

Thomas Kruse (University of Giessen)

3
We determine the convergence speed of the EMCEL scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of certain Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process’ speed measure we prove that the approximating Markov chains converge at fixed times at the rate of $1/4$ with respect to every $p$-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than $1/4$. These results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points. Moreover, we present several further properties of the EMCEL scheme and discuss its differences from the Euler scheme.

The talk is based on joint works with Stefan Ankirchner, Wolfgang Löhr and Mikhail Urusov.

The discrete membrane model on trees

Biltu Dan (Indian Institute of Science)

4
The discrete membrane model (MM) is a random interface model for separating surfaces that tend to preserve curvature. It is similar to the discrete Gaussian free field (DGFF) for which the most likely interfaces are those preserving mean height. However working with the two models presents some key differences. In particular, a lot of tools (electrical networks, random walk representation for the covariance) are available for the DGFF and lack in the MM. In this talk we will investigate a random walk representation for the covariance of the MM and by means of it will define and study the MM on regular trees. In particular, we will study the scaling limit of the maxima of the MM on regular trees.

On a two-server queue with consultation by main server with protected phases of service

Resmi Thekkiniyedath (KKTM Government College)

3
This paper analyses a two-server queueing model with consultations given by the main server to the regular server. The main server not only serves customers but also provides consultation to the regular server with a pre-emptive priority over customers. The customers at the main server undergo interruptions during their service. The interruption is not allowed to a customer at the main server if the service is in any one of the protected phases of service. There are upper bounds for the number of interruptions to a customer at the main server and the number of consultations to the regular server during the service of a customer. A super clock also determines whether to allow further interruption to a customer at the main server or not. A threshold clock decides the restart or resumption of the services at the main and regular servers after each consultation. The arrival process and requirement of consultation follow mutually independent Poisson processes. The service times at the main server and the regular server are assumed to follow mutually independent phase type distributions. The stability condition is established and some performance measures are studied numerically.

Q&A for Contributed Session 34

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Session Chair

Jaehong Jeong (Hanyang University)

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