On embedding set functions into covariance functions

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.

A new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loève (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

<p style='text-indent:20px;'>This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.</p>

The relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

In nonparametric Bayesian approaches, Gaussian stochastic processes can serve as priors on real-valued function spaces. Existing literature on the posterior convergence rates under Gaussian process priors shows that it is possible to achieve optimal or near-optimal posterior contraction rates if the smoothness of the Gaussian process matches that of the target function. Among those priors, Gaussian process with a parametric Matérn covariance function is particularly notable in that its degree of smoothness can be determined by a dedicated smoothness parameter. Ma and Bhadra (2023) recently introduced a new family of covariance functions called the Confluent Hypergeometric (CH) class that simultaneously possess two parameters: one controls the tail index of the polynomially decaying covariance function, and the other parameter controls the degree of mean-squared smoothness analogous to the Matérn class. In this paper, we show that with proper choice of rescaling parameters in the Matérn and CH covariance functions, it is possible to obtain the minimax optimal posterior contraction rate for η-regular functions for nonparametric regression model with fixed design. Unlike the previous results for unrescaled cases, the smoothness parameter of the covariance function need not equal η for achieving the optimal minimax rate, for either rescaled Matérn or rescaled CH covariances, illustrating a key benefit of rescaling. We also consider a fully Bayesian treatment of the rescaling parameters and show the resulting posterior distributions still contract at the minimax-optimal rate. The resultant hierarchical Bayesian procedure is fully adaptive to the unknown true smoothness. The theoretical properties of the rescaled and hierarchical Matérn and CH classes are further verified via extensive simulations and an illustration on a geospatial data set is presented.

A note on the product correlation rule

We study the reduction from a bipartite Hermitian matrix H to two single-qubit reduced Hermitian matrices HA and HB and the inverse problem. The inverse problem extends the so-called marginal problem from quantum information theory. We explicitly obtain the expressions of the matrices mentioned above and then extend them to positive semidefinite matrices by showing several inequalities involving a few parameters from the matrices. We also extend our results to multi-quit matrices. Our study constructs the intimate connection between the generalized marginal problems in terms of Hermitian matrices and the traditional marginal problem in terms of positive semidefinite matrices.

A modified LIF-type stochastic model is considered with a non-delta correlated stochastic process in place of the traditional white noise. Two different mechanisms of reset are specified with the aim to model endogenous and exogenous correlated input stimuli. Ornstein-Uhlenbeck processes are used to model the two different cases. An equivalence between different ways to include currents in the model is also shown. The theory of integrated stochastic processes is evoked and the main features of involved processes are obtained, such as mean and covariance functions. Finally, a simulation algorithm of the proposed model is described; simulations are performed to provide estimations of firing densities and related comparisons are given.

The present study devotes to introducing and analyzing the use of the adaptive Fourier decomposition (AFD)‐type methods in the area of stochastic processes and random fields. We involve two types of algorithms, namely, stochastic AFD (SAFD) and stochastic pre‐orthogonal AFD (SPOAFD), for, respectively, the Hardy space format and non‐Hardy space ones, as may be regarded. We provide both their theoretical results and practical algorithms and compare them with the well adopted and, in fact, dominating Karhunen–Loève (KL)‐type expansions. The AFD methods involve a finite or infinite sequence of optimally chosen parameters; they, in contrast, do not rely on and hence not have to compute the eigenpairs of the second type Fredholm integral equation with the covariance function as the kernel. Apart from such computational conveniences, they with the same convergence rate have flexibility of choosing best suitable dictionaries for doing the specific task in the practice. We include a number of experiments showing that in terms of effectiveness, the AFD methods give better approximations before all the positive eigenvalues running out in the case the integral operator being of a finite rank or before the KL iteration step becomes excessively large: The AFD methods normally give better approximations from the very beginning of the iterations.

Least squares prediction with MQ (multiquadric) functions is conceptually different from least squares prediction using covariance functions. MQ kernels are based on geometric or physical considerations rather than stochastic processes, and were found to be superior to covariance functions in topographic applications. This may be true also for gravity anomalies or other phenomena which result from marginally stationary, or non‐stationary random processes. The MQ harmonic kernel is used to develop a formula for estimating the best depth of point mass anomalies as a function of their number and areal extent on a sphere. Functional relationships between geoidal surface parameters are developed which provide linear equation analogs for the solution of Stokes and Vening Meinesz Integral Formulas as well as for the inversion of these classic problems. These relationships are extended to solutions at exterior equipotential surfaces.

In this work, we establish and compute worst case bounds in covariance interpolation for continuous-time stationary stochastic processes, a problem that appears in applications such as broad-band direction-of-arrival estimation and optimal sensor placement. More specifically, for two such stochastic processes whose covariance functions agree on a finite discrete set of time-lags, we would like to compute the maximal possible discrepancy of the covariance functions for real-valued time-lags outside this discrete grid. In array processing, this difference quantifies the inherent uncertainty pertaining to the discrete sampling of space determined by the array geometry. Computing this uncertainty corresponds to solving an infinite-dimensional non-convex problem. However, we herein prove that the maximal objective value may be bounded from above by a finite-dimensional convex optimization problem, allowing for efficient computation by standard methods. Furthermore, we empirically observe that for the case of signals whose spectra are supported on an interval, this upper bound is sharp, i.e., provides an exact quantification of the covariance uncertainty.

Three specific non-linear transformations of Gaussian stochastic processes occurring in signal detection and control theory are considered. The Gaussian stochastic processes are not necessarily stationary. The common feature of the transformed stochastic processes is that the determination of their covariance functions depends upon the evaluation of the orthant probability of four Gaussian variates over the respective correlation matrix. A method for evaluating this orthant probability, when the correlation matrix has certain specific forms, has recently been discussed by Cheng (1969). The application of this method yields closed-form expressions, in terms of tabulated functions, for the output probability distributions and covariance functions of the non-linear transformations investigated.

Stochastic modeling of engineering dynamic excitations for stochastic dynamics of structures

The covariance function of the stochastic component of various soil properties plays an essential role in the estimation and interpolation of spatial phenomena using kriging techniques and in the solution of transport problems in heterogeneous media using stochastic continuum transport modeling. In this study the effect of the sampling network on the estimates of covariance functions of two‐dimensional, isotropic, second‐order stationary processes, characterized by different underlying correlation scales, was analyzed. Estimates of the covariance function based on a standard systematic sampling network were compared with estimates of the covariance functions based on a modified sampling network. Results of these comparisons showed that the modified sampling network was better suited than the systematic network to resolving the statistical properties of the underlying stochastic process at small separation distances. This may result in an improvement in structure identification. Analyses of the effect of the sampling network used for estimating the covariance function on the results of conditional inference procedures such as kriging showed that the kriging estimates were essentially insensitive to the geometric configuration of the sampling network. Uncertainties about these estimates were reduced when using the modified sampling network, however. In the case of unconditional inferences, such as with stochastic analyses of transport problems in heterogeneous media, predictions of the head variance provided by using estimates of the parameters of the covariance function of the log saturated hydraulic conductivity based on the modified sampling network fit the theoretical head variance better than those based on the systematic sampling network. The effect of the sampling network used for estimating the covariance function on the predicted head variance was significant, particularly when the underlying correlation scale was relatively small, when the porous medium texture became finer, and when the hydraulic conductivity variations were treated as being three‐dimensional.

Computation of an ARMA covariance function is a common ingredient in analysis and synthesis of various problems in stochastic control, estimation, and signal processing. Several approaches can be used for this purpose. In this paper, we present an algorithm based on simple polynomial calculations. Compared with alternative strategies, it has small computational load, shows good numerical robustness, and can be extended to handle multivariable ARMA processes, even with complex-valued coefficients.