Multidimensional Stochastic Processes Research Articles

A Bayesian multivariate model using Hamiltonian Monte Carlo inference to estimate total organic carbon content in shale

The prediction of total organic carbon (TOC) content using geophysical logs is one of the key steps in shale reservoir characterization. Various empirical relations have previously been used for the estimation of TOC content from well-logs; however, uncertainty quantification in the model estimation is often ignored while performing TOC estimation in a deterministic framework. We introduce the problem of TOC estimation in a Bayesian setting with the goal of enhancing the TOC content prediction together with the quantification of the uncertainty in the model prediction. To signify the uncertainty, we draw random samples of model parameters from the posterior distribution by realizing multidimensional stochastic processes within the Hamiltonian Monte Carlo algorithm. The posterior model for the variables that influence TOC estimation is conditioned on the available well-log observations and is further defined by a priori and likelihood distributions. We demonstrate examples of applications of this approach to estimate the TOC content on two real field data sets from the well-known Devonian Duvernay shale of Western Canada and the Silurian shale of the Ahnet Basin. The accuracy in the estimation is arbitrated by comparing the prediction results with those obtained using the two most widely used empirical models. Corroborating the results by the laboratory-measured TOC contents demonstrate that the Bayesian approach offers a more reliable and better confidence in predictions when compared with the empirical models, as it provides additional information on the prediction uncertainty. Finally, the implications of the present approach are derived in terms of depositional environments to characterize the high TOC content zone in the studied organic shale formations.

Analyzing and forecasting financial series with singular spectral analysis

Abstract Modern techniques for managing multidimensional stochastic processes that reflect the dynamics of unstable environments are proactive, which refers to decision making based on forecasting the system’s state vector evolution. At the same time, the dynamics of open nonlinear systems are largely determined by their chaotic nature, which leads to a violation of stationarity and ergodicity of the series of observations and, as a result, to a catastrophic decrease in the efficiency of forecasting algorithms based on traditional methods of multivariate statistical data analysis. In this article, we make an attempt to reduce the instability influence by employing singular spectrum analysis (SSA) algorithms. This technique has been employed in a wide class of applied data analysis problems formulated in terms of singular decomposition of data matrices: technologies of immunocomputing and SSA.

Local correlation dimension of multidimensional stochastic process

The computation of the local correlation dimension is a way for estimating the Hausdorff dimension of the image of multidimensional stochastic processes. It can be obtained from the asymptotic behavior of the self-intersection occupation measure around zero. In this paper, we replace the usual indicator function of the occupation measure by a Gaussian kernel. Hence, we obtain the consistency of the local correlation dimension for multivariate fractional Brownian motion. On the other hand, we show that any used norms on Rd give the same asymptotic behavior of the occupation measure. The use of a numerical procedure based on log−log least square estimator and Monte-Carlo experiments confirm the theoretical results and provide an efficient way of estimation of the Hausdorff dimension. In addition, we show that our proposed estimation method performs the univariate one on the estimation of the Hausdorff dimension.

Optimal Singular Control Problem in Infinite Horizon for Stochastic Processes with Regime-Switching

We develop the compactification method to prove the existence of an optimal control for multidimensional stochastic processes with regime-switching. We consider separately the situations with and without state constraints. Three kinds of control policies are studied in this work: the classical controls on the drifts, the singular controls, and the controls on the transition rates of switching processes. Based on the existence of the optimal controls, the dynamic programming principle is also established.

Spatio-temporal modelling of economic phenomena in the context of reducing the dimensions of the random field

Motivation: Clear spatial diversity and high variability in time of economic phenomena and the fact that they show dependencies in space and time dimensions, as well as the spatio-temporal dependencies, lead to the consideration of the phenomena in terms of random fields. On the other hand, applying methods and tools from the field of multidimensional stochastic processes called random fields is difficult due to the specificity of the economic data, in particular to a low number of the observations in space. Hence, there arises the problem of the reduction of the dimensions (especially the space dimension) of the random fields which define economic phenomena. Aim: The aim of the paper is to discuss a model that will reflect the structure of spatial connections and dependence in the spatio-temporal process, while reducing the dimension of space as a non-random argument of the random field. As a result, the set of N time series will be analysed. Therefore, we build a multiple-equation model of autoregressive character with a spatial structure of dependence. In the paper we discuss the advantages and cognitive values of such an approach to the study of economic phenomena in spatio-temporal terms. Results: The empirical example concerns the unemployment rate in Poland across provinces (NUTS2) in the period from January 2011 to April 2019. The data create spatio-temporal series which is the realisation of the two-dimensional random field. The model building strategy is based on a vector autoregressive (VAR) specification, where there are as many time series as the provinces. The study shows that the use of the concept of the conditional, with regard to space dimension, random field simplifies the econometric analysis of the spatio-temporal process under consideration without losing the accuracy of the description of his basic properties.

Some integral inequalities of the Hermite-Hadamard type for s-convex stochastic processes on n-coordinates

In this study, we identified s-convexity of first and second sense for multidimensional stochastic processes. Concordantly, we verified Hermite-Hadamard type inequalities for these processes. Besides, we exemplified these results on two and three-dimensional stochastic processes. Ultimately, we compared our results with multidimensional harmonically convex stochastic processes in the literature. It must be known that the inequalities in our study are especially necessary to compare the maximum and minimum values of s-convex of first and second sense for multidimensional stochastic process with the expected value of stochastic processes. It is used mean-square integrability for the speciality of stochastic processes to obtain these inequalities in this study

The Binarisation and Monitoring of Stochastic Processes in Quality Management

The article examines problems that can be encountered in the monitoring and evaluation of quality control processes. Two types of product/process qualification are distinguished in quality control procedures: quality control/inspection by attributes and quality control/inspection by variables. This division also results in the division of processes. Generally, there are two types of stochastic processes: binary processes and continuous/numerical processes. The first brings the case being described into a zero-one system. The second analyses processes with continuous random variables.This article addresses the following issues: evaluating multidimensional stochastic processes, process binarisation methods, measuring product/process quality and applying statistical procedures to process monitoring. A comprehensive numerical example illustrating these problems is also provided.

A stochastic volatility factor model of heston type. Statistical properties and estimation

In this paper we study the properties of a new multidimensional continuous-time stochastic covariance process, the Stochastic Volatility Factor model. Two helpful conditional characteristic functions, one needed for estimation and the other used in the pricing of financial derivatives are provided, together with the properties of the instantaneous dependence structure. Conditions for stationarity, ergodicity and mixing properties of the increments are studied. The estimation of the model is performed using the Continuum-Generalized Method of Moments (CGMM). A simulation exercise is included showing parameters are recovered, confirming identifiability. The model is also calibrated to exemplary financial data.

Statistical emulators for pricing and hedging longevity risk products

We propose the use of statistical emulators for the purpose of analyzing mortality-linked contracts in stochastic mortality models. Such models typically require (nested) evaluation of expected values of nonlinear functionals of multi-dimensional stochastic processes. Except in the simplest cases, no closed-form expressions are available, necessitating numerical approximation. To complement various analytic approximations, we advocate the use of modern statistical tools from machine learning to generate a flexible, non-parametric surrogate for the true mappings. This method allows performance guarantees regarding approximation accuracy and removes the need for nested simulation. We illustrate our approach with case studies involving (i) a Lee–Carter model with mortality shocks; (ii) index-based static hedging with longevity basis risk; (iii) a Cairns–Blake–Dowd stochastic survival probability model; (iv) variable annuities under stochastic interest rate and mortality.

A Large-Deviation-Based Splitting Estimation of Power Flow Reliability

Given the continued integration of intermittent renewable generators in electrical power grids, connection overloads are of increasing concern for grid operators. The risk of an overload due to injection variability can be described mathematically as a barrier-crossing probability of a function of a multidimensional stochastic process. Crude Monte Carlo is a well-known technique to estimate probabilities, but it may be computationally too intensive in this case as typical modern power grids rarely exhibit connection overloads. In this article, we derive an approximate rate function for the overload probability using results from large deviations theory. Based on this large deviations approximation, we apply a rare event simulation technique called splitting to estimate overload probabilities more efficiently than Crude Monte Carlo simulation. We show on example power grids with up to 11 stochastic power injections that for a fixed accuracy, Crude Monte Carlo would require tens to millions as many samples as the proposed splitting technique required. We investigate the balance between accuracy and workload of three splitting schemes, each based on a different approximation of the rate function. We justify the workload increase of large-deviation-based splitting compared to naive splitting—that is, splitting based on merely the Euclidean distance to the rare event set. For a fixed accuracy, naive splitting requires over 60 times as much CPU time as large-deviation-based splitting, illustrating its computational advantage. In these examples, naive splitting—unlike large-deviation-based splitting—requires even more CPU time than CMC simulation, illustrating its pitfall.

The uniqueness of signature problem in the non-Markov setting

We establish a general framework for a class of multidimensional stochastic processes over [0,1] under which with probability one, the signature (the collection of iterated path integrals in the sense of rough paths) is well-defined and determines the sample paths of the process up to reparametrization. In particular, by using the Malliavin calculus we show that our method applies to a class of Gaussian processes including fractional Brownian motion with Hurst parameter H>1/4, the Ornstein–Uhlenbeck process and the Brownian bridge.

Langevin dynamics for vector variables driven by multiplicative white noise: A functional formalism.

We discuss general multidimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals. We present a functional formalism to build up the generating functional of correlation functions without any type of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables, as well as Grassmann variables. In this representation, time reversal transformation became a linear transformation in the extended variables, simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. We study some examples such as higher order derivative Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.

Estimation of Air Surface Temperature From Remote Sensing Images and Pixelwise Modeling of the Estimation Uncertainty Through Support Vector Machines

The knowledge of air temperature near the Earth's surface plays a relevant role in weather and climate studies as well as in the framework of solar energy management; e.g., for identifying the most suitable locations for a new solar installation or monitoring the performance of existing systems. Remote sensing allows air temperature to be estimated on a spatially distributed basis, thus complementing the spatially sparse observations collected by ground micro-meteorological stations. In this paper, a novel approach to periodic (e.g., daily or monthly) air temperature estimation from satellite images based on support vector machines (SVMs) is proposed. A recently developed SVM-based approach to supervised land and sea surface temperature estimation using satellite images is generalized to the case of air temperature and integrated with case-specific techniques aimed at computing periodic statistics of air temperature using the expectation-maximization algorithm. The method is fully automated and allows the statistics of the estimation error to be modeled on a pixelwise basis. This last result is accomplished by combining nonstationary multidimensional stochastic processes and Clark's variance approximation. The method is experimentally validated with MSG-SEVIRI data acquired over Provence-Alpes-Côte d'Azur (France).

Asymptotically Pointwise Optimal Change Detection in Multiple Channels

A Bayesian multichannel change-point detection problem is studied in the following general setting. A multidimensional stochastic process is observed; some or all of its components may experience changes in distribution, simultaneously or not. The loss function penalizes for false alarms and detection delays, and the penalty increases with each missed change-point. For wide classes of stochastic processes, with or without nuisance parameters and practically any joint prior distribution of change-points, asymptotically pointwise optimal (APO) rules are obtained, translating the classical concept of Bickel and Yahav to the sequential change-point detection. These APO rules are attractive because of their simple analytic form and straightforward computation. An application to a multidimensional autoregressive time series is shown.

Affine Principal-Component-Based Term Structure Model

We present an affine multifactor Gaussian term structure model, where we let factors be principal components of yields. Taking forward the affine-yield idea presented in Duffie and Kan (1996) we derive general expressions for the coefficients of a multidimensional affine Gaussian stochastic process given an exogenous matrix of loadings of selected yields on the state variables.

Approximation of multidimensional stochastic processes from average sampling

The convergence property of sampling series, the estimate of truncation error in the mean square sense and the almost sure results on sampling theorem for multidimensional stochastic processes from average sampling are analyzed. These results are generalization of the classical results which were given by Balakrishnan (IRE Trans. Inf. Theory 3(2):143-146, 1957) and Belyaev (Theory Probab. Appl. 4(4):437-444, 1959) for random signals. Using inequalities in the mean square sense, the results of Pogány and Peruničić (Glas. Mat. 36(1):155-167, 2001) were improved too.MSC:42C15, 60G10, 94A20.

A Fokker–Planck control framework for multidimensional stochastic processes

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model.

Pricing of mountain range derivatives under a principal component stochastic volatility model

In this paper, a multidimensional stochastic volatility process is introduced. This process is simpler than existing ones in terms of number of parameters while keeping practical stylized facts like stochastic correlation and volatility. The pricing of two mountain range derivatives, Altavista and Everest, is analyzed under this framework, showing sensitivities to parameters, number of eigenvalues, and maturity time. Copyright © 2012 John Wiley & Sons, Ltd.

Minimax Prediction Problem for Multidimensional Stationary Stochastic Processes

We consider the problem of estimation of the unknown value of the functionals and which depend on the unknown values of a multidimensional stationary stochastic process based on observations of the process for t < 0 from the class Ξ of processes which satisfy conditions , . The maximum values of the mean-square errors of the optimal estimates of the functionals and are found. It is shown that these maximum values of the errors in the class Ξ give the moving average processes which are determined by eigenvectors of compact operators constructed with the help of the vector-function .

Real Option Pricing of Network Design Investments

A real option solution and method are used to quantify the value of flexibility for deferral and design strategies in investments made in a network. The model is proposed to address managerial flexibility in transportation planning and can be applied to any network design problem under uncertainty. The core framework is based on the assumption that some variable such as travel demand can be characterized by nonstationary, multi-dimensional stochastic processes, such as geometric Brownian motion. By exploiting that characterization, it is possible to determine the value of the flexibility to defer the investment or to redesign the network by solving a dynamic program with network design subproblem using a least squares Monte Carlo simulation algorithm. The option premium is shown to decompose into a basic deferral premium and a flexible network design premium, which quantifies the cost of committing to preferred alternative in transportation planning. The proposed model is tested on the classic Sioux Falls, South Dakota, network, showing that a traditional discounted cash-flow analysis may support an immediate investment of a continuous network design but that under some conditions, deferral with the option to redesign can offer the greatest option value.