Class Of Stochastic Processes Research Articles

Relation of classical non-equilibrium dynamics and quantum annealing

Non-equilibrium dynamics of the Ising model is a classical stochastic process whereas quantum mechanics has no stochastic elements in the classical sense. Nevertheless, it has been known that there exists a close formal relationship between these two processes. We reformulate this relationship and use it to compare the efficiency of simulated annealing that uses classical stochastic processes and quantum annealing to solve combinatorial optimization problems. It is shown that classical dynamics can be efficiently simulated by quantum- mechanical processes whereas the converse is not necessarily true. This may imply that quantum annealing may be regarded as a more powerful tool than simulated annealing for optimization problems.

Langevin formulation of a subdiffusive continuous-time random walk in physical time.

Systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. One of the most common models to describe such subdiffusive dynamics is the continuous-time random walk (CTRW). Stochastic trajectories of a CTRW can be described in terms of the subordination of a normal diffusive process by an inverse Lévy-stable process. Here, we propose an equivalent Langevin formulation of a force-free CTRW without subordination. By introducing a different type of non-Gaussian noise, we are able to express the CTRW dynamics in terms of a single Langevin equation in physical time with additive noise. We derive the full multipoint statistics of this noise and compare it with the scaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive dynamics. Interestingly, these two noises are identical up to the second order correlation functions, but different in the higher order statistics. We extend our formalism to general waiting time distributions and force fields and compare our results with those of the SBM. In the presence of external forces, our proposed noise generates a different class of stochastic processes, resembling a CTRW but with forces acting at all times.

Large-Dimensional Factor Modeling Based on High-Frequency Observations

This paper develops a statistical theory to estimate an unknown factor structure based on financial high-frequency data. I derive a new estimator for the number of factors and derive consistent and asymptotically mixed-normal estimators of the loadings and factors under the assumption of a large number of cross-sectional and high-frequency observations. The estimation approach can separate factors for normal “continuous” and rare jump risk. The estimators for the loadings and factors are based on the principal component analysis of the quadratic covariation matrix. The estimator for the number of factors uses a perturbed eigenvalue ratio statistic. The results are obtained under general conditions, that allow for a very rich class of stochastic processes and for serial and cross-sectional correlation in the idiosyncratic components.

Sanov and central limit theorems for output statistics of quantum Markov chains

In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not.

Stochastic path summation with memory

Some classes of stochastic processes with memory properties are investigated by evaluating the probability density function as a white noise path integral. The corresponding modified diffusion equation for different types of memory behavior is then discussed.

Gaussian and non-Gaussian processes of zero power variation

This paper considers the class of stochastic processes X defined on [0, T] by X (t) = R T 0 G(t, s) dM (s) where M is a square-integrable martingale and G is a deterministic kernel. When M is Brownian motion, X is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let m be an odd integer. Under the asumption that the quadratic variation [M] of M is differentiable with E[|d[M] /dt| m ] finite, it is shown that the mth power variation

Characterisation of random Gaussian and non-Gaussian stress processes in terms of extreme responses

In the field of military land vehicles, random vibration processes generated by all-terrain wheeled vehicles in motion are not classical stochastic processes with a stationary and Gaussian nature. Non-stationarity of processes induced by the variability of the vehicle speed does not form a major difficulty because the designer can have good control over the vehicle speed by characterising the histogram of instantaneous speed of the vehicle during an operational situation. Beyond this non- stationarity problem, the hard point clearly lies in the fact that the random processes are not Gaussian and are generated mainly by the non-linear behaviour of the undercarriage and the strong occurrence of shocks generated by roughness of the terrain. This non-Gaussian nature is expressed particularly by very high flattening levels that can affect the design of structures under extreme stresses conventionally acquired by spectral approaches, inherent to Gaussian processes and based essentially on spectral moments of stress processes. Due to these technical considerations, techniques for characterisation of random excitation processes generated by this type of carrier need to be changed, by proposing innovative characterisation methods based on time domain approaches as described in the body of the text rather than spectral domain approaches.

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.

Converting a real quantum spin bath to an effective classical noise acting on a central spin

We present a cluster expansion method for approximating quantum spin-bath dynamics in terms of a classical Gaussian stochastic process. The cluster expansion produces the two-point correlation function of the approximate classical bath, permitting rapid evaluation of noise-mitigating quantum control strategies without resorting to computationally intensive dynamical decoupling models. Our approximation is valid for the wide class of models possessing negligible back-action and nearly-Gaussian noise. We study several instances of the central spin decoherence problem in which the central spin and randomly-located bath spins are alike and dipolarly coupled. For various pulse sequences, we compare the coherence echo decay computed explicitly quantum mechanically versus those computed using our approximate classical model, and obtain agreement in most, but not all, cases. We demonstrate the utility of these classical noise models by efficiently searching for the 4-pulse sequences that maximally mitigate decoherence in each of these cases, a computationally expensive task in the explicit quantum model.

Some moment inequalities for fuzzy martingales and their applications

Abstract Martingales are a class of stochastic processes which has had profound influence on the development of probability theory and stochastic processes. Some recent developments are related to mathematical finance. In the real world, some information about these phenomena might be imprecise and represented in the form of vague quantities. In these situations, we need to generalize classical methods to vague environment. Thus, fuzzy martingales have been extended as a vague perception of real-valued martingales. In this paper, some moment inequalities are presented for fuzzy martingales. Several convergence theorems are established based on these inequalities. As an application of convergence theorems, a weak law of large numbers for fuzzy martingales is stated. Furthermore, a few examples are devoted to clarify the main results.

Entropy rate of message sources driven by quantum walks

The amount of information generated by a discrete time stochastic processes in a single step can be quantified by the entropy rate. We investigate the differences between two discrete time walk models, the discrete time quantum walk and the classical random walk in terms of entropy rate. We develop analytical methods to calculate and approximate it. This allows us to draw conclusions about the differences between classical stochastic and quantum processes in terms of the classical information theory.

Validity of Heavy-Traffic Steady-State Approximations in Multiclass Queueing Networks: The Case of Queue-Ratio Disciplines

A class of stochastic processes known as semi-martingale reflecting Brownian motions (SRBMs) is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson [Bramson M (1998) State space collapse with applications to heavy-traffic limits for multiclass queueing networks. Queueing Systems 30:89–148] and Williams [Williams RJ (1998b) Diffusion approximations for open multiclass queueuing networks: Sufficient conditions involving state space collapse. Queueing Systems 30:27–88] laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which an SRBM is obtained as a diffusion limit from a sequence of suitably normalized workload processes. However, for multiclass networks it is still not known in general whether the steady-state distribution of the SRBM provides a valid approximation for the steady-state distribution of the original network. In this paper we study the case of queue-ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative. In addition to standard assumptions made in the literature towards the stability of the pre- and post-limit processes and the existence of diffusion limits, we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate. For the special case of static-priority networks such linear attraction is known to hold under certain conditions on the network primitives. The analysis elucidates interesting connections between stability of the pre- and post-limit processes, their respective fluid models and state-space collapse, and identifies the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.

Stochasticity: A Feature for the Structuring of Large and Heterogeneous Image Databases

The paper addresses image feature characterization and the structuring of large and heterogeneous image databases through the stochasticity or randomness appearance. Measuring stochasticity involves finding suitable representations that can significantly reduce statistical dependencies of any order. Wavelet packet representations provide such a framework for a large class of stochastic processes through an appropriate dictionary of parametric models. From this dictionary and the Kolmogorov stochasticity index, the paper proposes semantic stochasticity templates upon wavelet packet sub-bands in order to provide high level classification and content-based image retrieval. The approach is shown to be relevant for texture images.

A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model

A method to generate first passage times for a class of stochastic processes is proposed. It does not require construction of the trajectories as usually needed in simulation studies, but is based on an integral equation whose unknown quantity is the probability density function of the studied first passage times and on the application of the hazard rate method. The proposed procedure is particularly efficient in the case of the Ornstein-Uhlenbeck process, which is important for modeling spiking neuronal activity.

Algebraic Probability, Classical Stochastic Processes, and Counting Statistics

We study a connection between the algebraic probability and classical stochastic processes described by master equations. Introducing a definition of a state which has not been used for quantum cases, the classical stochastic processes can be reformulated in terms of the algebraic probability. This reformulation immediately gives the Doi-Peliti formalism, which has been frequently used in nonequilibrium physics. As an application of the reformulation, we give a derivation of basic equations for counting statistics, which plays an important role in nonequilibrium physics.

Model selection for long-memory processes in the spectral domain

In this paper, we study a novel empirical criterion for identifying the type of a long-memory time series. The proposed rule for selecting an underlying model suitable to the data is based on the comparison between the normalized prediction errors of the generalized Whittle estimator (a parametric spectral estimator) applied to two or more candidate models. We test this approach by two applications of the procedure: for comparing two distinct statistical models to adjust the data, and for assessing the significance of increasing the number of parameters within a given class of models. Due to the heuristic nature of the method, we test the statistic numerically for several classes of stochastic processes, namely Gaussian processes with long-range dependence (LRD), M/G/∞ (non-Gaussian) processes with LRD, non-stationary processes, and non-linear heteroscedastic models. The numerical results demonstrate that the proposed statistic exhibits good power, is robust and not computationally expensive.

Local linear estimation for stochastic processes driven by $$\alpha $$ α -stable L $$\acute{\mathbf{e}}$$ e ´ vy motion

The \(\alpha \)-stable L\(\acute{\mathrm{e}}\)vy motion together with the Poisson process and Brownian motion are the most important examples of L\(\acute{\mathrm{e}}\)vy processes, which form the first class of stochastic processes being studied in the modern spirit. In this paper, the stochastic processes driven by \(\alpha \)-stable L\(\acute{\mathrm{e}}\)vy motion are considered, local linear estimator of the drift function for these processes is discussed. Under mild conditions, we derive consistency of the local linear estimator of the drift function. The performance of the proposed estimator is assessed by simulation study.

THE QUANTUM DECOMPOSITION OF RANDOM VARIABLES WITHOUT MOMENTS

The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands. Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property. The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like. In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.

Bessel Bridges Decomposition with Varying Dimension: Applications to Finance

We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at t of the process and its integral. We finally give some financial applications to illustrate the panel of applications of our results.

Simulation of stochastic quantum systems using polynomial chaos expansions.

We present an approach to the simulation of quantum systems driven by classical stochastic processes that is based on the polynomial chaos expansion, a well-known technique in the field of uncertainty quantification. The polynomial chaos technique represents the density matrix as an expansion in orthogonal polynomials over the principle components of the stochastic process and yields a sparsely coupled hierarchy of linear differential equations. We provide practical heuristics for truncating this expansion based on results from time-dependent perturbation theory and demonstrate, via an experimentally relevant one-qubit numerical example, that our technique can be significantly more computationally efficient than MonteCarlo simulation.