FILTERING OF STATES AND PARAMETERS OF SPECIAL MARKOV JUMP PROCESSES VIA INDIRECT PERFECT OBSERVATIONS

This paper investigates a class of stochastic differential systems with random structure, the transitions of which are generated by the special Markov jump processes. A statement concerning the Markovian property of the couple “jump process–governed diffusion" is presented. An analogue of the Fokker–Plank–Kolmogorov equation in the form of a system of partial integro-differential equations, describing the evolution of the mutual transition probability of this couple, is derived.

The aim of this paper is manufacturing of turbocharger engine jet blades made of nickel superalloys. Processes for producing molds and casting realized in a production line are special processes. It means that the results are known only after inspection of the finished product. There is lack of the methods and techniques of effective and efficient quality control of the work in stock, above all molds. Therefore, the unknown is the state ceramic mold for the precision casting, which resulting in risk of referral to a defective mold of the casting process and thus give the product does not comply, is eliminated in the final inspection. One method of reducing this risk is particularly thorough monitoring of all parameters of each process and keeping them in the desired operating point. Operating point is a set of parameters of processes. Such monitoring is possible with the commitment to the methods and techniques to automatically, without human intervention, data collection and processing methods appropriate for use in operational control. The paper presents results of research on the attitude to the problem of a special process. This change is the introduction to the process efficient and effective form of quality control tools in the course of its preparation. In this case, the method of photogrammetry, thermal imaging and computed tomography were used. With the infrared camera will be possible to determine the temperature field, the disorder in relation to the pattern indicates the type of defect. Computed tomography and will be used to develop patterns of correlated defects associated with thermal imagers. Photogrammetry is the use of a model set of quality control (comparison of the actual state of the model *.CAD). It also allows the designation of a wall thickness of the mold.

This article investigates the zonotope-based state estimation for boost converter system with Markov jump process. DC-DC boost converters are pivotal in modern power electronics, enabling renewable energy integration, electric vehicle charging, and microgrid operations by elevating low input voltages from sources like photovoltaics to stable high outputs. However, their nonlinear dynamics and sensitivity to uncertainties/disturbances degrade control precision, driving research into robust state estimation. To address these challenges, the boost converter is modeled as a Markov jump system to characterize stochastic switching, with time delays, disturbances, and noises integrated for a generalized discrete-time model. An adaptive event-triggered mechanism is adopted to administrate the data transmission to conserve communication resources. A zonotopic set-membership estimation design is proposed, which involves designing an observer for the augmented system to ensure performance and developing an algorithm to construct zonotopes that enclose all system states. Finally, numerical simulations are performed to verify the effectiveness of the proposed approach.

The paper introduces observation systems with special Markov Jump processes as the states arid their indirect observations in presence of Wiener noises. Solution of the optimal filtering problem in the classes of linear, polynomial and all nonlinear estimators is presented, and interrelation of obtained estimates is also demonstrated.

Empirical evidences on corporate bond found e.g. in Frydman (2005) and Frydman and Schuermann (2008) suggest the facts that the attribution of constant intensity in the credit rating dynamics, represented by finite-state jump Markov processes, is not borne out by actual data. On the other hand, it is known facts that continuous-time finite-state jump Markov processes can be represented by means of Poisson process and embedded discrete-time Markov chains, see, e.g., Ch. 7 of Pardoux (2008), Sec. 5.10 of Resnick (2002) and and Jakubowski and Nieweglowski (2010, 2008). Motivated by the above facts, we consider representation of jump Markov processes in terms of doubly stochastic Poisson process (see e.g. Cox (1955), Kingman (1964), Serfozo (1972), and Bremaud (1981)) whose intensity is driven by observed explanatory covariates. However, we impose weaker conditions than that of specified in Kingman (1964) and Jakubowski and Nieweglowski (2010, 2008) in the construction of such jump Markov process in which the conditional rate of jump arrival in the conditional Poisson process follows the landmarking approach. This approach has been recently introduced in event history analysis by van Houwelingen (2007) and van Houwelingen and Putter (2012). By this approach, we do not require the whole trajectory (dynamics) of the observed covariates. Analogous to Cox (1955) and Jakubowski and Nieweglowski (2010), we call such representation as doubly stochastic jump Markov processes. We derive lifetime distributions until its absorption of doubly stochastic finite state absorbing jump Markov process, and propose generalization of the phase-type distribution introduced in Neuts (1981, 1975). Also, we derive conditional forward intensity of future occurrences of the jump Markov process. The new distribution and intensity are given in closed form and have ability to capture explanatory covariates and heterogeneity. The results can be seen as an alternative structural approach to the reduced-form model of credit default discussed in Duffie et al. (2009, 2007), and Duan et al. (2012). Some numerical examples are discussed to motivate the main results.

We are interested in the connection between a metastable continuous state space Markov process (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring-Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.

The paper presents a solution of optimal filtering problem for stochastic differential systems of random structure with switches generated by a special class of Markov jump processes. The equations for both the conditional expectation of some signal process given a noisy observation, and conditional probability density function (pdf) are obtained. Numerical methods for solution of corresponding Fokker-Plank and Zakai equation analogues are given and illustrated by an example.

It is usually difficult to express a family of tri-point transition function (TTF) by a transition matrix as Markov processes with one parameter. In this paper, we define three kinds of connection matrixes on the states of standard tri-point transition function (STTF) and study their essential character, give a constructive method on the constant-value standard tri-point transition function and a general expression of the state-symmetric standard tri-point transition function by a sequence of the transition matrixes of special and simple Markov processes with one parameter.

In this paper we prove the weak consistency and the asymptotic normality of the maximum likelihood estimation based on discrete observations ofn independent Gaussian Markov processes. The Ornstein Uhlenbeck process is a special Gaussian Markov process. We derive asymptotic simultaneous confidence regions for the parameters of the Ornstein Uhlenbeck process as an application.

The effective reproduction number is an important descriptor of an infectious disease epidemic. In small populations, ideally we would estimate the effective reproduction number using a Markov Jump Process (MJP) model of the spread of infectious disease, but in practice this is computationally challenging. We propose a computationally tractable approximation to an MJP which tracks only latent and infectious individuals, the EI model, an MJP where the time-varying immigration rate into the E compartment is equal to the product of the proportion of susceptibles in the population and the transmission rate. We use an analogue of the central limit theorem for MJPs to approximate transition densities as normal, which makes Bayesian computation tractable. Using simulated pathogen RNA concentrations collected from wastewater data, we demonstrate the advantages of our stochastic model over its deterministic counterpart for the purpose of estimating effective reproduction number dynamics, and compare against a state of the art method. We apply our new model to inference of changes in the effective reproduction number of SARS-CoV-2 in several college campus communities that were put under wastewater pathogen surveillance in 2022.

We give sharp sufficient conditions for nonexplosions and explosions in Markov pure jump processes in terms of the holding time parameters and moments of the jump distributions.

This paper studies optimum control of random differential equations of the form \[\dot x = f^{r(t)} (t,x,u)\] in which $r(t)$ is a jump Markov process. Optimality conditions of dynamic programming type and stochastic minimum principles are given. The problems posed involve terminal conditions, and transversality conditions are shown to hold.

Semi-Markov and Jump Markov Controlled Models: Average Cost Criterion

A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

Fixed time output feedback containment for uncertain nonlinear multiagent systems with switching communication topologies