Continuous Stochastic Processes Research Articles

A pathwise inference method for the parameters of diffusion terms

ABSTRACTWe consider inference of the parameters of the diffusion term for continuous time stochastic processes with a power-type dependence of the diffusion coefficient from the underlying process such as Cox–Ingersoll–Ross, CKLS, and similar processes. We suggest some original pathwise estimates for this coefficient and for the power index based on an analysis of an auxiliary continuous time complex-valued process generated by the underlying real-valued process. These estimates do not rely on the distribution of the underlying process and on a particular choice of the drift. Some numerical experiments are used to illustrate the feasibility of the suggested method.

Weak measurements, quantum-state collapse, and the Born rule

Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. We describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction can be specific to each system-apparatus pair, a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation-dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. We analyze the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter. This trajectory ensemble distribution can be tested in weak measurement experiments. We also comment on how the required noise may arise in the measuring apparatus.

A domain-theoretic approach to Brownian motion and general continuous stochastic processes

We introduce a domain-theoretic framework for continuous-time, continuous-state stochastic processes. The laws of stochastic processes are embedded into the space of maximal elements of the normalised probabilistic power domain on the space of continuous interval-valued functions endowed with the relative Scott topology. We use the resulting ω-continuous bounded complete dcpo to obtain partially defined stochastic processes and characterise their computability. For a given continuous stochastic process, we show how its domain-theoretic, i.e., finitary, approximations can be constructed, whose least upper bound is the law of the stochastic process. As a main result, we apply our methodology to Brownian motion. We construct a partially defined Wiener measure and show that the Wiener measure is computable within the domain-theoretic framework.

Hybrid colored noise process with space-dependent switching rates.

A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the Itô-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation of stochastic differential equations (SDEs). Recently, we have identified a mechanism for obtaining an Itô SDE based on a form of temporal disorder. Motivated by switching processes in molecular biology, we considered a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state, the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast switching limit one obtains Brownian motion with a space-dependent diffusivity of the Itô form. In this paper, we extend our theory to include colored additive noise. We show that the nature of the effective multiplicative noise process obtained by taking both the white-noise limit (κ→0) and fast switching limit (ε→0) depends on the order the two limits are taken. If the white-noise limit is taken first, then we obtain Itô, and if the fast switching limit is taken first, then we obtain Stratonovich. Moreover, the form of the effective diffusion coefficient differs in the two cases. The latter result holds even in the case of space-independent transition rates, where one obtains additive noise processes with different diffusion coefficients. Finally, we show that yet another form of multiplicative noise is obtained in the simultaneous limit ε,κ→0 with ε/κ^{2} fixed.

Nonlinear Volterra‐type diffusion equations and theory of quantum fields

We study nonlinear Volterra‐type evolution integral equations of the form: in a C∗‐algebra or in a Hilbert algebra of Dixmier‐Segal type, acting on a Hilbert space tensor product , where denotes a Hilbert space and is the Boson‐Einstein (Fermion‐Dirac) Fock space, over a complex Hilbert space . Under suitable Carathéodory‐type conditions on the corresponding Nemytskii operator Φ of f and assuming that k is a quantum dynamical‐type semigroup, we obtain exactly one classical global solution in the space of bounded continuous (operator‐valued) quantum stochastic processes. Moreover, we prove the existence of exactly one positive (respectively completely positive) classical global solution in (respectively in , applying a positivity (respectively completely positivity preserving) quantum stochastic integration process and assuming that k is a quantum dynamical semigroup acting on , where Φ defines a positive (respectively completely positive) quantum stochastic process.

Adaptive Inference for the Bivariate Mean Function in Functional Data

Inference methods are proposed for the bivariate mean function of a continuous stochastic process with a two-dimensional domain. Nonparametric bivariate estimation is facilitated by thresholded projection estimators. Estimators adapt to the sparsity of the bivariate function. Oracle inequality results are developed to describe the adaptive inference methods. The construction of nonparametric bivariate confidence bands is presented. Implementation results show the applicability of the methods in practice.

Temporal disorder as a mechanism for spatially heterogeneous diffusion.

A fundamental issue in analyzing diffusion in heterogeneous media is interpreting the space dependence of the associated diffusion coefficient. This reflects the well-known Ito-Stratonovich dilemma for continuous stochastic processes with multiplicative noise. In order to resolve this dilemma it is necessary to introduce additional constraints regarding the underlying physical system. Here we introduce a mechanism for generating nonlinear Brownian motion based on a form of temporal disorder. Motivated by switching processes in molecular biology, we consider a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast-switching limit one obtains Brownian motion with a space-dependent diffusivity. We show that the resulting multiplicative noise process is of the Ito form. In particular, we solve a first-passage time problem for finite switching rates and show that the mean first-passage time reduces to the Ito version in the fast-switching limit.

The effect of copulas on time-variant reliability involving time-continuous stochastic processes

In structural reliability the dependence structure between random variables is almost exclusively modeled by Gauss (normal or Gaussian) copula; however, this implicit assumption is typically not corroborated. This paper is focusing on time-variant reliability problems with continuous stochastic processes, which are collection of dependent random variables and to our knowledge are not modeled by other than Gauss copula in structural reliability. Therefore, the aim of this contribution is to qualitatively and quantitatively analyze the impact of this copula assumption on failure probability. Three illustrative examples are studied considering bivariate Gauss, t, rotated Clayton, Gumbel, and rotated Gumbel copulas. Time-variant actions are modeled as stationary, ergodic, continuous stochastic processes, and the PHI2 method is adopted for the analyses. The calculations show that the copula function has significant effect on failure probability. In the studied examples, application of Gauss copula can four times underestimate or even 10 times overestimate failure probabilities obtained by other copulas. For normal structures agreement on copula type is recommended, while for safety critical ones inference of copula type from observations is advocated. If data are scare, multiple copula functions and model averaging could be used to explore this uncertainty.

A Poissonian Model of Indel Rate Variation for Phylogenetic Tree Inference.

While indel rate variation has been observed and analyzed in detail, it is not taken into account by current indel-aware phylogenetic reconstruction methods. In this work, we introduce a continuous time stochastic process, the geometric Poisson indel process, that generalizes the Poisson indel process by allowing insertion and deletion rates to vary across sites. We design an efficient algorithm for computing the probability of a given multiple sequence alignment based on our new indel model. We describe a method to construct phylogeny estimates from a fixed alignment using neighbor joining. Using simulation studies, we show that ignoring indel rate variation may have a detrimental effect on the accuracy of the inferred phylogenies, and that our proposed method can sidestep this issue by inferring latent indel rate categories. We also show that our phylogenetic inference method may be more stable to taxa subsampling than methods that either ignore indels or indel rate variation. [evolutionary stochastic process; indel rate variation; Poisson indel process; TKF91.].

Electricity price modeling with stochastic time change

In this paper, we develop a novel approach to electricity price modeling, based on the powerful technique of stochastic time change. This technique allows us to incorporate the characteristic features of electricity prices (such as seasonal volatility, time varying mean reversion and seasonally occurring price spikes) into the model in an elegant and economically justifiable way. The stochastic time change introduces stochastic as well as deterministic (e.g., seasonal) features in the price process' volatility and in the jump component.We specify the base process as a mean reverting jump diffusion and the time change as an absolutely continuous stochastic process with seasonal component. The activity rate of the stochastic time change can be related to the factors that influence supply and demand. Here we use the temperature as a proxy for the demand and hence, as the driving factor of the stochastic time change, and show that this choice leads to realistic price paths. We derive properties of the resulting price process and develop the model calibration procedure. We calibrate the model to the historical EEX power prices and apply it to generating realistic price paths by Monte Carlo simulations. We show that the simulated price process matches the distributional characteristics of the observed electricity prices in periods of both high and low demand.

The quadratic variation of continuous time stochastic processes in vector lattices

We define and study order continuity, topological continuity, γ-Hölder-continuity and Kolmogorov–Čentsov-continuity of continuous-time stochastic processes in vector lattices and show that every such kind of continuous submartingale has a continuous compensator of the same kind. The notion of variation is introduced for continuous time stochastic processes and for a γ-Hölder-continuous martingale with finite variation, we prove that it is a constant martingale. The localization technique for not necessarily bounded martingales is introduced and used to prove our main result which states that the quadratic variation of a continuous-time γ-Hölder continuous martingale X is equal to its compensator 〈X〉.

Regime-switching stochastic volatility model: estimation and calibration to VIX options

ABSTRACTWe develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox–Ingersoll–Ross process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum–Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.

Expectation propagation for continuous time stochastic processes

We consider the inverse problem of reconstructing the posterior measure over the trajectories of a diffusion process from discrete time observations and continuous time constraints. We cast the problem in a Bayesian framework and derive approximations to the posterior distributions of single time marginals using variational approximate inference, giving rise to an expectation propagation type algorithm. For non-linear diffusion processes, this is achieved by leveraging moment closure approximations. We then show how the approximation can be extended to a wide class of discrete-state Markov jump processes by making use of the chemical Langevin equation. Our empirical results show that the proposed method is computationally efficient and provides good approximations for these classes of inverse problems.

Parallel Fokker–Planck-DSMC algorithm for rarefied gas flow simulation in complex domains at all Knudsen numbers

A major challenge for the conventional Direct Simulation Monte Carlo (DSMC) technique lies in the fact that its computational cost becomes prohibitive in the near continuum regime, where the Knudsen number (Kn)—characterizing the degree of rarefaction—becomes small. In contrast, the Fokker–Planck (FP) based particle Monte Carlo scheme allows for computationally efficient simulations of rarefied gas flows in the low and intermediate Kn regime. The Fokker–Planck collision operator—instead of performing binary collisions employed by the DSMC method—integrates continuous stochastic processes for the phase space evolution in time. This allows for time step and grid cell sizes larger than the respective collisional scales required by DSMC. Dynamically switching between the FP and the DSMC collision operators in each computational cell is the basis of the combined FP-DSMC method, which has been proven successful in simulating flows covering the whole Kn range. Until recently, this algorithm had only been applied to two-dimensional test cases. In this contribution, we present the first general purpose implementation of the combined FP-DSMC method. Utilizing both shared- and distributed-memory parallelization, this implementation provides the capability for simulations involving many particles and complex geometries by exploiting state of the art computer cluster technologies.

On the Hausdorff measure of noncompactness for the parameterized Prokhorov metric

We quantify the Prokhorov theorem by establishing an explicit formula for the Hausdorff measure of noncompactness (HMNC) for the parameterized Prokhorov metric on the set of Borel probability measures on a Polish space. Furthermore, we quantify the Arzela-Ascoli theorem by obtaining upper and lower estimates for the HMNC for the uniform norm on the space of continuous maps of a compact interval into Euclidean N-space, using Jung’s theorem on the Chebyshev radius. Finally, we combine the obtained results to quantify the stochastic Arzela-Ascoli theorem by providing upper and lower estimates for the HMNC for the parameterized Prokhorov metric on the set of multivariate continuous stochastic processes.

Stochastic evolution equations with Volterra noise

Volterra processes are continuous stochastic processes whose covariance function can be written in the form R(s,t)=∫0s∧tK(s,r)K(t,r)dr, where K is a suitable square integrable kernel. Examples of such processes are the fractional Brownian motion, multifractional Brownian motion or (in the non-Gaussian case) Rosenblatt process. In the first part, stochastic integral with respect to Volterra processes and cylindrical Volterra process in Hilbert spaces are defined and some of their properties are studied. In the second part, these results are applied to linear stochastic equations in Hilbert spaces driven by cylindrical Volterra processes. Measurability, mean-square continuity and paths continuity of their solutions are proved under various sets of conditions. The general results are illustrated by examples of parabolic and hyperbolic SPDEs.

A Variational EM Algorithm for the Separation of Time-Varying Convolutive Audio Mixtures

This paper addresses the problem of separating audio sources from time-varying convolutive mixtures. We propose a probabilistic framework based on the local complex-Gaussian model combined with non-negative matrix factorization. The time-varying mixing filters are modeled by a continuous temporal stochastic process. We present a variational expectation-maximization (VEM) algorithm that employs a Kalman smoother to estimate the time-varying mixing matrix, and that jointly estimate the source parameters. The sound sources are then separated by Wiener filters constructed with the estimators provided by the VEM algorithm. Extensive experiments on simulated data show that the proposed method outperforms a blockwise version of a state-of-the-art baseline method.

Stochastic modeling to represent wind power generation and demand in electric power system based on real data

A methodology to model two types of random perturbation that affect the operation of electric power systems (EPS) are presented. The first uncertainty is wind power generation and is represented by a one-dimensional and by a multidimensional continuous stochastic process. The second one is power demand, and is modeled by using an hybrid structure based on harmonic regression and the Ornstein–Uhlenbeck (O–U) process. The stochastic models are applied to a real Chilean case, using real data for parametric estimation and validation models.

A new method for estimating the longevity and degradation of photovoltaic systems considering weather states

The power output of solar photovoltaic (PV) systems is affected by solar radiation and ambient temperature. The commonly used evaluation techniques usually overlook the four weather states which are clear, cloudy, foggy, and rainy. In this paper, an ovel analytical model of the four weather conditions based on the Markov chain is proposed. The Markov method is well suited to estimate the reliability and availability of systems based on a continuous stochastic process. The proposed method is generic enough to be applied to reliability evaluation of PV systems and even other applications. Further aspects investigated include the new degradation model for reliability predication of PV modules. The results indicate that the PV module degradation over years, failures, and solar radiation must be considered in choosing an efficient PV system with an optimal design to achieve the maximum benefit of the PV system. For each aspect, a method is proposed, and the complete focusing methodology is expounded and validated using simulated point targets. The results also demonstrate the feasibility and applicability of the proposed method for effective modeling of the chronological aspects and stochastic characteristics of solar cells as well as the optimal configuration and sizing of large PV plants in terms of cost and reliability.

Introduction of a new continuous time and state space stochastic process in condition prediction

Periodic condition assessments of pavements together with condition predictions are the basis for investment decisions in every pavement management system (PMS). Typical approaches include surveys of distress types every 3–6 years with analysis rating and calculation of condition indices for road safety and/or structural health. Furthermore, advanced PMS prediction models allow a comparison of maintenance alternatives and an optimisation of investment strategies. This paper presents an overview of current survey and rating approaches in Germany, Switzerland and Austria, together with an impact analysis of different methods, utilised deterministic performance functions and condition threshold (trigger) values for all major distress types. The core of this paper is a comparison of common deterministic condition prediction models with discrete stochastic approaches and prediction models based on advanced regression techniques mainly from scientific literature and an innovative stochastic continuous time and continuous state space process (HOFFMANN – Process). All prediction models are applied to real-world data from condition surveys in Austria and the long-term pavement performance Database (USA) at single-section and network level. The paper provides evidence why deterministic prediction approaches are leading to substantial bias in condition distribution and remaining service life as they do not account for the stochastic nature of pavements. Classic Markov-chain approaches do not account for censoring of survey data and neglect changes in transition probabilities with increasing age. Applying common bivariate and multiple regression techniques may also lead to certain bias due to collinearity effects and specification bias. The paper provides mathematical evidence on ways to avoid these shortcomings based on the presented innovative stochastic process leading to a higher reliability in condition assessment, rating and accuracy of condition predictions. The aspects of censoring, distress-specific assignment and optimisation of treatments with this new HOFFMANN-process will be covered in forthcoming papers.