Two-dimensional Stochastic Process Research Articles

The continuous-time hidden Markov model based on discretization. Properties of estimators and applications

In this paper we consider continuous-time hidden Markov processes (CTHMM). The model considered is a two-dimensional stochastic process (X_t,Y_t), with X_t an unobserved (hidden) Markov chain defined by its generating matrix and Y_t an observed process whose distribution law depends on X_t and is called the emission function. In general, we allow the process Y_t to take values in a subset of the q-dimensional real space, for some q. The coupled process (X_t,Y_t) is a continuous-time Markov chain whose generator is constructed from the generating matrix of X and the emission distribution. We study the theoretical properties of this two-dimensional process using a formulation based on semi-Markov processes. Observations of the CTHMM are obtained by discretization considering two different scenarii. In the first case we consider that observations of the process Y are registered regularly in time, while in the second one, observations arrive at random. Maximum-likelihood estimators of the characteristics of the coupled process are obtained in both scenarii and the asymptotic properties of these estimators are shown, such as consistency and normality. To illustrate the model a real-data example and a simulation study are considered.

On the random search for a randomly moving particle

ABSTRACT This paper discusses some methods based on the random search problem over the bound domain. The particle moves randomly with a bounded two-dimensional stochastic process in a bounded area. The physical phenomenon determines the boundaries of this area. We have a smart nanosensor tracks randomly for this particle. The necessary condition that maximizes the probability of the first meeting time between the nanosensor and the particle is derived.

Big Data Processing in Geo-Distributed Data Centers Using Cost Minimization Methods

Abstract: The explosive growth of demands on huge process imposes a crucial burden on computation, storage, and communication in data centers, that so incurs wide operational expenditure to data center suppliers. Therefore, Price decrease has become Associate in Nursing rising issue for the longer term huge data era. All totally different from customary cloud services, one in each of the foremost choices of huge data services is that the tight coupling between data and computation as computation tasks is also conducted provided that the corresponding data is obtainable. As a result, 3 factors, i.e., task assignment, data placement and data movement, deeply the operational expenditure of data centers. Throughout this paper, we tend to tend to unit of measurement meant to see the worth decrease downside via a joint improvement of these three factors for big data services in geo-distributed data centers. To elucidate the task completion time with the thought of every data transmission and computation, we tend to tend to propose a two-dimensional stochastic process and derive the standard task completion time in closed-form. Moreover, we tend to tend to model the matter as a mixed-integer non-linear programming (MINLP) and propose economical resolution to correct it. The high efficiency of our proposal is valid by thorough simulation primarily based studies Keywords— Nonlinear Programming, Big Data, Computer Centres, Integer Programming, Minimisation, Mixed Integer Nonlinear Programming, Cost Minimization, Data Centers, Big Data Services, Operational Expenditure, Task Assignment, Big Data

A Bayesian 2D functional linear model for gray-level co-occurrence matrices in texture analysis of lower grade gliomas

In cancer radiomics, textural features evaluated from image intensity-derived gray-level co-occurrence matrices (GLCMs) have been studied to evaluate gray-level spatial dependence within the regions of interest in the brain. Most of these analysis work with summary statistics (or texture-based features) constructed using the GLCM entries, and potentially overlook other structural properties in the GLCM. In our proposed Bayesian framework, we treat each GLCM as a realization of a two-dimensional stochastic functional process observed with error at discrete time points. The latent process is then combined with the outcome model to evaluate the prediction performance. We use simulation studies to assess the performance of our method and apply it to data collected from individuals with lower grade gliomas. We found our approach to outperform competing methods that use only summary statistics to predict isocitrate dehydrogenase (IDH) mutation status.

The value of gas-fired power plants in markets with high shares of renewable energy: A real options application

Using a real options model, this paper quantifies a gas-fired power plant's operating value and the value of a new investment against the background of a market transition to renewable electricity. The model is run with recent data for Germany's power sector and for different types of gas-fired power plants.The result is twofold. First, the paper achieves a more realistic value by improving on existing models: it models electricity and gas prices as a two-dimensional stochastic process, each component consisting of the sum of a seasonal pattern and a mean-reverting process; it uses high granularity by modelling hourly time-steps; and it incorporates power plant ramping times and costs. Second, it compares two types of power plant models, one with daily and one with hourly operating decisions, and thereby quantifies the value of a plant's intraday flexibility. The hourly model replicates operators' and investors' decision making accurately. This is evidenced by the fact that the results trace current major developments like the recent decline and come-back of gas-fired generation in Germany.The paper contributes to a better understanding of the choices operators and investors face in current electricity markets. In the absence of large scale storage solutions flexible supply of electricity, as provided by gas, is important in the transition to renewable energies in Germany and across Europe.

Pricing contingent convertible bonds: An analytical approach based on two-dimensional stochastic processes

In this paper, we study fast pricing methods for contingent convertible bonds (CoCos). Based on two-dimensional stochastic processes, we propose two pricing models for CoCos; a dynamic capital-ratio model and a dynamic debt–equity model. Under these frameworks, we derive analytic-form formulae for CoCos with fixed and floored conversion prices. Many practical implications for analyzing CoCos are observed through numerical tests by choosing the plausible model parameters obtained from empirical results.

Estimating Large Delay Probabilities in Two Correlated Queues

This article focuses on evaluating the probability that both components of a two-dimensional stochastic process will ever, but not necessarily at the same time, exceed some large level u . An important application is in determining the probability of large delays occurring in two correlated queues. Since exact analysis of this probability seems prohibitive, we focus on deriving asymptotics and on developing efficient simulations techniques. Large deviations theory is used to characterise logarithmic asymptotics. The second part of this article focuses on efficient simulation techniques. Using “nearest-neighbour random walk” as an example, we first show that a “naive” implementation of importance sampling, based on the decay rate, is not asymptotically efficient. A different approach, which we call partitioned importance sampling, is developed and shown to be asymptotically efficient. The results are illustrated through various simulation experiments.

A Monitoring and Diagnostic Approach for Stochastic Textured Surfaces

ABSTRACTWe develop a supervised-learning-based approach for monitoring and diagnosing texture-related defects in manufactured products characterized by stochastic textured surfaces that satisfy the locality and stationarity properties of Markov random fields. Examples of stochastic textured surface data include images of woven textiles; image or surface metrology data for machined, cast, or formed metal parts; microscopy images of material microstructure samples; etc. To characterize the complex spatial statistical dependencies of in-control samples of the stochastic textured surface, we use rather generic supervised learning methods, which provide an implicit characterization of the joint distribution of the surface texture. We propose two spatial moving statistics, which are computed from residual errors of the fitted supervised learning model, for monitoring and diagnosing local aberrations in the general spatial statistical behavior of newly manufactured stochastic textured surface samples in a statistical process control context. We illustrate the approach using images of textile fabric samples and simulated two-dimensional stochastic processes, for which the algorithm successfully detects local defects of various natures. Supplemental discussions, results, data and computer codes are available online.

Simulation Solution to a Two-Dimensional Mortgage Refinancing Problem

This work studies a mortgage borrower’s optimal refinancing strategy, which is formulated as the solution to a stochastic minimization problem with contingent conditions. The problem is framed in a business economic environment where the underlying discounting factor and mortgage interest rate are assumed to follow a two-dimensional stochastic process of Vasicek type. A complete Monte Carlo algorithm is developed and implemented. This algorithm generates the optimal refinancing surface as a function of time and the risk-free rate. Numerical examples with financial implications are provided.

Zitterbewegung with spin–orbit coupled ultracold atoms in a fluctuating optical lattice

The dynamics of non-interacting ultracold atoms with artificial spin–orbit coupling is considered. Spin–orbit coupling is created using two moving optical lattices with orthogonal polarizations. Our main goal is to study influence of lattice noise on Rabi oscillations. Special attention is paid to the phenomenon of the Zitterbewegung being trembling motion caused by Rabi transitions between states with different velocities. Phase and amplitude fluctuations of lattices are modelled by means of the two-dimensional stochastic Ornstein-Uhlenbeck process, also known as harmonic noise. In the the noiseless case the problem is solved analytically in terms of the momentum representation. It is shown that lattice noise significantly extends duration of the Zitterbewegung as compared to the noiseless case. This effect originates from noise-induced decoherence of Rabi oscillations.

Minimal diffusion formulation of Markov chain ensembles and its application to ion channel clusters.

We study ensembles of continuous-time Markov chains evolving independently under a common transition rate matrix in some finite state space. A diffusion approximation, composed of two specifically coupled Ornstein-Uhlenbeck processes in stochastic differential equation representation, is formulated to deduce how the number of chains in a given particular state evolves in time. This particular form of the formulation builds upon a theoretical argument adduced here. The formulation is minimal in the sense that it is always a two-dimensional stochastic process regardless of the state space size or the transition matrix density, and that it requires no matrix square root operations. A set of criteria, put forward here as to be necessarily captured by any consistent approximation scheme, is used together with the master equation to determine uniquely the parameter values and noise variances in the formulation. The model is applied to the gating dynamics in ion channel clusters.

On analytic functions of complex Liu process

A complex stochastic process is a fundamental connection between two-dimensional stochastic processes. Similarly, a complex fuzzy process will connect two fuzzy processes to describe the spacial dynamic fuzzy phenomena. In this paper, we first give the formal definitions of complex fuzzy process and complex Liu process. And then the complex differential and complex Liu integral of complex Liu process are also studied in this paper. Especially, some properties of analytic functions of complex Liu process are proved. Finally, complex fuzzy differential equations are proposed as the extension of general fuzzy differential equation. The obtained results provide fundamentals and promote the development of fuzzy calculus.

The Ising Decision Maker: a binary stochastic network for choice response time.

The Ising Decision Maker (IDM) is a new formal model for speeded two-choice decision making derived from the stochastic Hopfield network or dynamic Ising model. On a microscopic level, it consists of 2 pools of binary stochastic neurons with pairwise interactions. Inside each pool, neurons excite each other, whereas between pools, neurons inhibit each other. The perceptual input is represented by an external excitatory field. Using methods from statistical mechanics, the high-dimensional network of neurons (microscopic level) is reduced to a two-dimensional stochastic process, describing the evolution of the mean neural activity per pool (macroscopic level). The IDM can be seen as an abstract, analytically tractable multiple attractor network model of information accumulation. In this article, the properties of the IDM are studied, the relations to existing models are discussed, and it is shown that the most important basic aspects of two-choice response time data can be reproduced. In addition, the IDM is shown to predict a variety of observed psychophysical relations such as Piéron's law, the van der Molen-Keuss effect, and Weber's law. Using Bayesian methods, the model is fitted to both simulated and real data, and its performance is compared to the Ratcliff diffusion model.

Stochastic volatility for interest rate derivatives

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

A CONTINUOUS WAVELET-BASED APPROACH TO DETECT ANISOTROPIC PROPERTIES IN SPATIAL POINT PROCESSES

A two-dimensional stochastic point process can be regarded as a random measure and thus represented as a (countable) sum of Delta Dirac measures concentrated at some points. Integration with respect to the point process itself leads to the concept of the continuous wavelet transform of a point process. Applying then suitable translation, rotation and dilation operations through a non unitary operator, we obtain a transformed point process which highlights main properties of the original point process. The choice of the mother wavelet is relevant and we thus conduct a detailed analysis proposing three two-dimensional mother wavelets. We use this approach to detect main directions present in the point process, and to test for anisotropy.

Distributed delays in a hybrid model of tumor-Immune system interplay

A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect. We here present a hybrid model with a generic delay kernel accounting that, due to many complex phenomena such as chemical transportation and cellular differentiation, the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model with two well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process. Via simulations and parametric sensitivity analysis techniques we (i) relate tumor mass growth with the two kernels, we (ii) measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and (iii) we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication.

On the fractal nature of volcano morphology detected via SAR image analysis: the case of Somma—Vesuvius Volcanic Complex

In this paper an innovative technique for the extraction of natural surfaces geomorphologic parameters from SAR data is presented and applied for volcano monitoring purposes. The observed surface is modeled as a fractal two-dimensional stochastic process. A theoretical framework for the analysis of the SAR imaging process is outlined. An algorithm founded on this imaging model is presented, allowing the retrieving of the point by point fractal dimension of the imaged surface. Significant results regarding the application of the proposed technique to the case study of the Somma-Vesuvius volcanic complex are shown, along with preliminary comments regarding the comparison with ground truth maps of the surface fractal dimension.

The convex hull for a random acceleration process in two dimensions

We compute exactly the mean perimeter 〈L(T)〉 and the mean area 〈A(T)〉 of the convex hull of a random acceleration process of duration T in two dimensions. We use an exact mapping that relates, via Cauchy's formulae, the computation of the perimeter and the area of the convex hull of an arbitrary two-dimensional stochastic process [x(t), y(t)] to the computation of the extreme value statistics of the associated one-dimensional component process x(t). The latter can be computed exactly for the one-dimensional random acceleration process even though the process is non-Markovian. Physically, our results are relevant in describing the average shape of a semi-flexible ideal polymer chain in two dimensions.

Stochastic Volatility for Interest Rate Derivatives

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introduce appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different $\beta$s) and Heston stochastic volatility models informs about what different specifications of the driving SDEs has to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model ($\beta=0$) satisfactorily passes all tests and seems to provide a good match to the market. In contrast we find the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two factors that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified and used to construct a simple terminal Markov-functional type model under a single measure. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

Fractional Klein–Kramers dynamics for subdiffusion and Itô formula

Subdiffusion in the presence of an external force field has been recently described in phasespace by the fractional Klein–Kramers equation. In this paper using a subordinationmethod, we identify a two-dimensional stochastic process (position, velocity) whoseprobability density function is a solution of the fractional Klein–Kramers equation. Thestructure of this process agrees with the two-stage scenario underlying the anomalousdiffusion mechanism, in which trapping events are superimposed on the Langevindynamics. Applying an extension of the celebrated Itô formula for subdiffusion we foundthat the velocity process can be represented explicitly by a corresponding fractionalOrnstein–Uhlenbeck process. A basic feature arising in the context of this stochasticrepresentation is the random change of time of the system made by subordination. For theposition and velocity processes we present a computer visualization of their sample pathsand we derive an explicit expression for the two-point correlation function of thevelocity process. The obtained stochastic representation is crucial in constructing analgorithm to simulate sample paths of the anomalous diffusion, which in turnallows us to detect and examine many relevant properties of the system underconsideration.