Continuous Gaussian Process Research Articles

Strict monotonicity of stochastic process extreme distributions

Strict monotonicity is proved for the distributions of extremes of processes consisting of series of bounded function with independent random coefficients, in particular for zero mean continuous Gaussian processes over compact metric space. These results have wide applications to global inference problems on unknown functions.

Estimates for exponential functionals of continuous Gaussian processes with emphasis on fractional Brownian motion

Estimates for exponential functionals of continuous Gaussian processes with emphasis on fractional Brownian motion

The estimations of drift parameters for the Gaussian Vasicek process with time-varying volatility

This article mainly considers the parameter estimation problem for the Gaussian Vasicek process defined as d X t = ( α − β X t ) d t + σ t d G t , t≥0 with unknown parameters α∈ℝ and β < 0, where G t is a continuous centered Gaussian process. The volatility parameter σ t is allowed to vary with time t. Based on the time-continuous observations and least squares method, the Least squares estimations of drift parameters for the non ergodic Gaussian Vasicek model with time-varying volatility are obtained. The strong consistency and asymptotic distributions of proposed estimations are verified under some sufficient conditions. Finally, we verify that the results are still valid under different forms of the volatility function when G t is fractional Brownian motion, sub-fractional Brownian motion, and bi-fractional Brownian motion, respectively. In some sense, our results extend the findings of El Machkouri, Es-Sebaiy, and Ouknine (2016) and Es-Sebaiy and Sebaiy (2021).

A spatially continuous diffusion model of visual working memory

We present results from five visual working memory (VWM) experiments in which participants were briefly shown between 2 and 6 colored squares. They were then cued to recall the color of one of the squares and they responded by choosing the color on a continuous color wheel. The experiments provided response proportions and response time (RT) measures as a function of angle for the choices. Current VWM models for this task include discrete models that assume an item is either within working memory or not and resource models that assume that memory strength varies as a function of the number of items. Because these models do not include processes that allow them to account for RT data, we implemented them within the spatially continuous diffusion model (SCDM, Ratcliff, 2018) and use the experimental data to evaluate these combined models. In the SCDM, evidence retrieved from memory is represented as a spatially continuous normal distribution and this drives the decision process until a criterion (represented as a 1-D line) is reached, which produces a decision. Noise in the accumulation process is represented by continuous Gaussian process noise over spatial position. The models that fit best from the discrete and resource-based classes converged on a common model that had a guessing component and that allowed the height of the normal memory-strength distribution to vary with number of items. The guessing component was implemented as a regular decision process driven by a flat evidence distribution, a zero-drift process. The combination of choice and RT data allows models that were not identifiable based on choice data alone to be discriminated.

On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>\\frac{1}{2}$$\\end{document} with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1-2H}$$\\end{document}, which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

Berry-Esséen bounds and almost sure CLT for the quadratic variation of a class of Gaussian process

We propose a condition which is valid for a class of continuous Gaussian processes that may fail to be self-similar or have stationary increments. Some examples include the sub-fractional Brownian motion and the bi-fractional Brownian motion and the sub-bifractional Brownian motion. Under this assumption, we show an upper bound for the difference between the inner product associated with a class of Gaussian process and that associated with the fractional Brownian motion. This inequality relates a class of Gaussian processes to the well studied fractional Brownian motion, which characterizes their relationship quantitatively. As an application, we obtain the optimal Berry-Esséen bounds for the quadratic variation when H ∈ ( 0 , 2 3 ] and the upper Berry-Esséen bounds when H ∈ ( 2 3 , 3 4 ] . As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when H ∈ ( 0 , 3 4 ] . The results in the present paper extend and improve those in the literature.

Multi-Patch Epidemic Models with General Exposed and Infectious Periods

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as “from distance” from all the other patches. The migration processes among the patches in either of the four states are assumed to be Markovian, and independent of the exposed and infectious periods. These periods have general distributions, and are not affected by the possible migrations of the individuals. The infection “from distance” aspect introduces a new formulation of the infection process, which, together with the migration processes, brings technical challenges in proving the functional limit theorems. Generalizing the methods in Pang and Pardoux [Ann. Appl. Probab. 32 (2022) 1615–1665], we establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.

Formal Safety Envelopes for Provably Accurate State Classification by Data-Driven Flight Models

Aerospace systems are inherently stochastic and increasingly data-driven, thus hard to formally verify. Data-driven statistical models can be used to estimate the state and classify potentially anomalous conditions of aerospace systems from multiple heterogeneous sensors with high accuracy. In this paper, we consider the problem of precisely bounding the regions in the sensor input space of a stochastic system in which safe state classification can be formally proven. As an archetypal application, we consider a statistical model created to detect aerodynamic stall in a prototype wing retrofitted with piezoelectric sensors and used to generate data in a wind tunnel for different flight states. We formally define safety envelopes as regions parameterized by and , to respectively capture how model-predictable observed sensor values are, and given these values, how likely the model’s accurate state classification is. Safety envelopes are formalized in the Agda proof assistant, used to also generate formally verified runtime monitors for sensor data stream analyses in the Haskell programming language. We further propose a new metric for model classification quality, evaluate it on our wing prototype model, and compare it to the model restricted to two different fixed airspeeds, and enhanced to a continuous Gaussian process regression model. Safety envelopes are an important step in formally verifying precise probabilistic properties of data-driven models used in stochastic aerospace systems and could be used by advanced control algorithms to maintain these systems well within safe operation boundaries.

Asymptotic of the running maximum distribution of a Gaussian Bridge

We study the tail behavior of the distribution of the running maximum of a zero mean Gaussian Bridge obtained from a continuous Gaussian process X(t) with by conditioning X(t) to have the value zero at time T. Some explicit examples are shown.

Skorohod and Stratonovich integrals for controlled processes

Given a continuous Gaussian process x which gives rise to a p-geometric rough path for p∈(2,3), and a general continuous process y controlled by x, under proper conditions we establish the relationship between the Skorohod integral ∫0tysd♢xs and the Stratonovich integral ∫0tysdxs. Our strategy is to employ the tools from rough paths theory and Malliavin calculus to analyze discrete sums of the integrals.

Local and uniform moduli of continuity of chi–square processes

Let {ηi(t),t∈[0,1]}i=1k be independent copies of η={η(t),t∈[0,1]}, a mean zero continuous Gaussian process. Let Yk:=Yk(t)= ∑i=1kηi2(t),t∈[0,1]. This paper shows how exact local (at 0) and uniform moduli of continuity (on [0,1]) of Yk can be obtained from the exact local and uniform moduli of continuity of η.

Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

AbstractWe study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime

We study marked Hawkes processes with an intensity process which has a non-stationary baseline intensity, a general self-exciting function of event “ages” at each time and marks. The marks are assumed to be conditionally independent given the event times, while the distribution of each mark depends on the event time, that is, time-varying. We first observe an immigration–birth (branching) representation of such a non-stationary marked Hawkes process, and then derive an equivalent representation of the process using the associated conditional inhomogeneous Poisson processes with stochastic intensities. We consider such a Hawkes process in the high intensity regime, where the baseline intensity gets large, while the self-exciting function and distributions of the marks are unscaled, and there is no time-scaling in the scaled Hawkes process. We prove functional law of large numbers and functional central limit theorems (FCLTs) for the scaled Hawkes processes in this asymptotic regime. The limits in the FCLTs are characterized by continuous Gaussian processes with covariance structures expressed with convolution functionals resulting from the branching representation. We also consider the special cases of multiplicative self-exciting functions, and an indicator type of non-decomposable self-exciting functions (including the cases of “ceasing” and “delayed” reproductions as well as their extensions with varying reproduction rates), and study the properties of the limiting Gaussian processes in these special cases.

The Riemann–Liouville field and its GMC as [formula omitted], and skew flattening for the rough Bergomi model

We consider a re-scaled Riemann–Liouville (RL) process ZtH=∫0t(t−s)H−12dWs, and using Lévy’s continuity theorem for random fields we show that ZH tends weakly to an almost log-correlated Gaussian field Z as H→0. Away from zero, this field differs from a standard Bacry–Muzy field by an a.s.Hölder continuous Gaussian process, and we show that ξγH(dt)=eγZtH−12γ2V ar(ZtH)dt tends to a Gaussian multiplicative chaos (GMC) random measure ξγ for γ∈(0,1) as H→0. We also show convergence in law for ξγH as H→0 for γ∈[0,2) using tightness arguments, and ξγ is non-atomic and locally multifractal away from zero. In the final section, we discuss applications to the Rough Bergomi model; specifically, using Jacod’s stable convergence theorem, we prove the surprising result that (with an appropriate re-scaling) the martingale component Xt of the log stock price tends weakly to Bξγ([0,t]) as H→0, where B is a Brownian motion independent of everything else. This implies that the implied volatility smile for the full rough Bergomi model with ρ≤0 is symmetric in the H→0 limit, and without re-scaling the model tends weakly to the Black–Scholes model as H→0. We also derive a closed-form expression for the conditional third moment E((Xt+h−Xt)3|Ft) (for H>0) given a finite history, and E(XT3) tends to zero (or blows up) exponentially fast as H→0 depending on whether γ is less than or greater than a critical γ≈1.61711 which is the root of 14+12logγ−316γ2. We also briefly discuss the pros and cons of a H=0 model with non-zero skew for which Xt/t tends weakly to a non-Gaussian random variable X1 with non-zero skewness as t→0.

Modeling pedestrian behavior in pedestrian-vehicle near misses: A continuous Gaussian Process Inverse Reinforcement Learning (GP-IRL) approach

Using simulation models to conduct safety assessments can have several advantages as it enables the evaluation of the safety of various design and traffic management options before actually making changes. However, limited studies have developed microsimulation models for the safety evaluation of active road users such as pedestrians. This can be attributed to the limited ability of simulation models to capture the heterogeneity in pedestrian behavior and their complex collision avoidance mechanisms. Therefore, the objective of this study is to develop an agent-based framework to realistically model pedestrian behavior in near misses and to improve the understanding of pedestrian evasive action mechanisms in interactions with vehicles. Pedestrian-vehicle conflicts are modeled using the Markov Decision Process (MDP) framework. A continuous Gaussian Process Inverse Reinforcement Learning (GP-IRL) approach is implemented to retrieve pedestrians’ reward functions and infer their collision avoidance mechanisms in conflict situations. Video data from a congested intersection in Shanghai, China is used as a case study. Trajectories of pedestrians and vehicles involved in traffic conflicts were extracted with computer vision algorithms. A Deep Reinforcement Learning (DRL) model is used to estimate optimal pedestrian policies in traffic conflicts. Results show that the developed model predicted pedestrian trajectories and their evasive action mechanisms (i.e., swerving maneuver and speed changing) in conflict situations with high accuracy. As well, the model provided predictions of the post encroachment time (PET) conflict indicator that strongly correlated with the corresponding values of the field-measured conflicts. This study is a crucial step in developing a safety-oriented microsimulation tool for pedestrians in mixed traffic conditions.

Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging

In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with $H>1/2$, and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model.

Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness

We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a nonnegative function of a Volterra type continuous Gaussian process that may have very rough sample paths. The main results obtained in the paper are sample path and small-noise large deviation principles for the log-price process in a time-inhomogeneous super rough Gaussian model under very mild restrictions. We use these results to study the asymptotic behavior of binary barrier options, exit time probability functions, and call options.

District-level estimation of vaccination coverage: Discrete vs continuous spatial models.

Health and development indicators (HDIs) such as vaccination coverage are regularly measured in many low‐ and middle‐income countries using household surveys, often due to the unreliability or incompleteness of routine data collection systems. Recently, the development of model‐based approaches for producing subnational estimates of HDIs using survey data, particularly cluster‐level data, has been an active area of research. This is mostly driven by the increasing demand for estimates at certain administrative levels, for example, districts, at which many development goals are set and evaluated. In this study, we explore spatial modeling approaches for producing district‐level estimates of vaccination coverage. Specifically, we compare discrete spatial smoothing models which directly model district‐level data with continuous Gaussian process (GP) models that utilize geolocated cluster‐level data. We adopt a fully Bayesian framework, implemented using the INLA and SPDE approaches. We compare the predictive performance of the models by analyzing vaccination coverage using data from two Demographic and Health Surveys (DHS), namely the 2014 Kenya DHS and the 2015‐16 Malawi DHS. We find that the continuous GP models performed well, offering a credible alternative to traditional discrete spatial smoothing models. Our analysis also revealed that accounting for between‐cluster variation in the continuous GP models did not have any real effect on the district‐level estimates. Our results provide guidance to practitioners on the reliability of these model‐based approaches for producing estimates of vaccination coverage and other HDIs.

Parameter Estimation for an Ornstein-Uhlenbeck Process Driven by a General Gaussian Noise

In this paper, we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process (Gt)t≥0. The second order mixed partial derivative of the covariance function $$R(t,s) = \mathbb{E}\left[ {{G_t}{G_s}} \right]$$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by (ts)β−1 up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments; some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise (Gt)t≥0. For the least squares estimator and the second moment estimator constructed from the continuous observations, we prove the strong consistency and the asympotic normality, and obtain the Berry-Esseen bounds. The proof is based on the inner product’s representation of the Hilbert space $$\mathfrak{h}$$ associated with the Gaussian noise (Gt)t≥0, and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.

Functional approximations via Stein’s method of exchangeable pairs

We combine the method of exchangeable pairs with Stein's method for functional approximation. As a result, we give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply this approach to estimate the distance of a sum of random variables, chosen from an array according to a random permutation, from a Gaussian mixture process. This result lets us prove a functional combinatorial central limit theorem. We also consider a graph-valued process and bound the speed of convergence of the distribution of its rescaled edge counts to a continuous Gaussian process.