Quadratic Stochastic Processes Research Articles

The practical calculation method of the high-order moment spectra of quadratic Gaussian stochastic processes

In the frequency-domain analysis of structures under non-Gaussian excitations, the determination of high-order moment spectra of excitations remains a substantial challenge due to the complexity and inaccuracy of the existing estimation method. This paper proposes the practical calculation method of the high-order moment spectra models for quadratic Gaussian stochastic processes. Firstly, the expressions of the first four moment spectra of quadratic Gaussian stochastic processes are derived based on the relation between the high-order cumulant function of Gaussian stochastic processes and high-order moment function. Secondly, a practical calculation method of the high-order moment spectra is presented by the ratio of the corresponding component moment functions. Finally, two numerical examples are investigated to validate the efficiency and accuracy of the practical calculation method of the high-order moment spectra for quadratic Gaussian stochastic processes.

On options pricing under a quadratic stochastic process modulated GBM model

On options pricing under a quadratic stochastic process modulated GBM model

Markov processes of cubic stochastic matrices: Quadratic stochastic processes

We consider Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices we have to fix first a multiplication and then consider the KCE with respect to the fixed multiplication. Moreover, the notion of stochastic cubic matrix also varies depending on the real models of application. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP. In this paper, our aim is to construct QSPs for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices (known as Maksimov's multiplications). We construct a wide class of QSPs and give some time-dependent behavior of such processes. We give an example with applications to the Biology, constructing a QSP which describes the time behavior (dynamics) of a population with the possibility of twin births.

On the stability of matrix-valued Riccati diffusions

The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation. This class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. The analysis developed here applies to filtering problems with unstable signals.

On the stability and the concentration of extended Kalman-Bucy filters

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estimates in terms of the filter forgetting properties with respect to erroneous initial conditions. For uniformly stable and fully observable signals, we also provide explicit non-asymptotic estimates for the exponential forgetting rate of the filters and the associated stochastic Riccati equations w.r.t. Frobenius norms. These non asymptotic exponential concentration and quantitative stability estimates seem to be the first results of this type for this class of nonlinear filters. Our techniques combine $\chi $-square concentration inequalities and Laplace estimates with spectral and random matrices theory, and the non asymptotic stability theory of quadratic type stochastic processes.

The simulation solution of nonlinear problems of ergodic electric-power processes transformation

The algorithm for the simulation solution of the stochastic electric-power processes problems (quadratic inertial smoothing (QIS) and quadratic cumulative averaging (QCA)) has been stated. It includes simulation of initial process realizations, transformation of realizations and ensemble statistical analysis of these processes, as well as simulation quality check by the test problems. Peculiarities of skewness, kurtosis and rated maxima (quantiles) calculations based on the ensemble of the stochastic ergodic process realizations have been considered. Density and cumulative distribution function of the quadratic stochastic processes with normal distribution have been analyzed. Ensemble and single realization errors for rated maxima have been evaluated. It has been proposed to use test check with distribution quantiles when searching for simulation solution of QIS on the stage of quadratic transformation. Approximation of rated maxima using the gamma distribution law is given.

Flow of finite-dimensional algebras

Each finite-dimensional algebra can be identified to the cubic matrix given by structural constants defining the multiplication between the basis elements of the algebra. In this paper we introduce the notion of flow (depending on time) of finite-dimensional algebras. This flow can be considered as a particular case of (continuous-time) dynamical system whose states are finite-dimensional algebras with matrices of structural constants satisfying an analogue of Kolmogorov–Chapman equation. These flows of algebras (FAs) can also be considered as deformations of algebras with the rule (the evolution equation) given by Kolmogorov–Chapman equation. We mainly use the multiplications of cubic matrices which were introduced by Maksimov and consider Kolmogorov–Chapman equation with respect to these multiplications. If all cubic matrices of structural constants are stochastic (there are several kinds of stochasticity) then the corresponding FA is called stochastic FA (SFA). We define SFA generated by known quadratic stochastic processes. For some multiplications of cubic matrices we reduce Kolmogorov–Chapman equation given for cubic matrices to the equation given for square matrices. Using this result many FAs are given (time homogeneous, time non-homogeneous, periodic, etc.). For a periodic FA we construct a continuum set of finite-dimensional algebras and show that the corresponding discrete time FA is dense in the set. Moreover, we give a construction of an FA which contains algebras of arbitrary (finite) dimension. For several FAs we describe the time depending behavior (dynamics) of the properties to be baric, limit algebras, commutative, evolution algebras or associative algebras.

On cubic stochastic operators and processes 1

In this paper analogically as quadratic stochastic operators and processes we define cubic stochastic operator (CSO) and cubic stochastic processes (CSP). These are defined on the set of all probability measures of a measurable space. The measurable space can be given on a finite or continual set. The finite case has been investigated before. So here we mainly work on the continual set. We give a construction of a CSO and show that dynamical systems generated by such a CSO can be studied by studying of the behavior of trajectories of a CSO given on a finite dimensional simplex. We define a CSP and drive differential equations for such CSPs with continuous time.

On Marginal Processes of Quadratic Stochastic Processes

It is known that the theory of Markov processes is a rapidly developing field with numerous applications to many branches of mathematics and physics, biology, and so on. But there are some physical models which cannot be described by such processes. One of such models is related to population genetics. These processes are called quadratic stochastic processes (q.s.p.). In the present paper, we associate to given q.s.p. two kind of processes, which call marginal processes. Note that one of them is Markov process. We prove that such kind of processes uniquely define q.s.p. Moreover, we provide a construction of nontrivial examples of q.s.p. Weak ergodicity of q.s.p. is also studied in terms of the marginal processes.

On quadratic stochastic processes and related differential equations

It is known that the theory of Markov process is a rapidly developing field with numerous applications to many branches of mathematics and physics, biology and so on. But there are some physical models which cannot be described by such processes. One of such models is related to population genetics. These processes are called quadratic stochastic processes (q.s.p.). In this theory it is important to construct nontrivial examples of such processes. In the present paper we are going to provide a construction of q.s.p. by means of two given processes. We should stress that such a construction allows us to produce lots of nontrivial examples of q.s.o. We also associate to given q.s.p. two kind of processes. Note that one of such processes is Markov. It is proved that such kind of processes uniquely define q.s.p. Moreover, we also derive some differential equations for q.s.p.

On L 1-weak ergodicity of nonhomogeneous discrete Markov processes and its applications

In the present paper we investigate the L 1-weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the L 1-weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition for such processes to satisfy the L 1-weak ergodicity. Moreover, we apply the obtained results to establish L 1-weak ergodicity of discrete time quadratic stochastic processes. As an application of the main result, certain concrete examples are also provided.

QUADRATIC STOCHASTIC OPERATORS AND PROCESSES: RESULTS AND OPEN PROBLEMS

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

On the ergodic principle for Markov and quadratic stochastic processes and its relations

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras

We give an expansion of a quantum quadratic stochastic process (q.q.s.p.) into a so-called fibrewise Markov process and prove that, conversely, such an expansion uniquely determines the quantum quadratic stochastic process. As an application, we give a criterion (in terms of this expansion) for the q.q.s.p. to satisfy the ergodic principle. Using this result, we prove that a q.q.s.p. satisfies the ergodic principle if and only if the associated Markov process satisfies this principle. The expansion obtained is used to introduce a new notion of conjugacy of two q.q.s.p.'s and to study the relation between this notion and the ergodic principle.

On One Regularity Condition for Quantum Quadratic Stochastic Processes

We present necessary and sufficient conditions for the validity of a regularity condition for homogeneous quantum quadratic stochastic processes defined on von Neumann algebras.

Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras

The paper contains necessary and sufficient conditions under which the ergodic principle and the regularity condition hold for discrete quantum quadratic stochastic processes defined on von Neumann algebras. A connection between these processes and Markov processes is established.

On some regularity properties of quadratic stochastic processes

On some regularity properties of quadratic stochastic processes

On some regularity properties of quadratic stochastic processes

On some regularity properties of quadratic stochastic processes

On quadratic stochastic processes

On quadratic stochastic processes

On quadratic stochastic processes

On quadratic stochastic processes