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This paper deals with the ruin problem of an insurance company investing its capital reserve in a risky asset with the price dynamics given by a conditional geometric Brownian motion whose parameters depend on a Markov process describing random variations in the economic and financial environments. We prove a sufficient condition on the distribution of jumps of the business process ensuring the smoothness of the ruin probability as a function of the initial capital and obtain for this function an integro-differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions and in a Banach space of bounded finitely additive measures. We study sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures. It is proved that the set of all limit measures (points) of such sequences in the weak topology generated by the preconjugate space is non-empty, weakly compact, and all of them are invariant for this operator. We also show that the well-known Doeblin condition (D) for the ergodicity of a Markov chain is equivalent to condition (∗) : all invariant finitely additive measures of the Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures. We give all the proofs for the most general case. [ABSTRACT FROM AUTHOR]

This Special Issue (SI), titled "Probability Theory and Stochastic Modeling with Applications", is concerned with the theory and applications of stochastic models. Footnotes 1 B Disclaimer/Publisher's Note: b The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). It consists of sixteen papers, and we would like to thank all the authors for their positive answers to our call for papers for the SI, as well as for their efforts to provide a high-quality contribution. Using fuzzy variables and robust optimization, the authors provide the electricity transaction policy under different expected costs so that the expectation of the risk-averse distribution system operator is fulfilled. [Extracted from the article]

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system's performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account. [ABSTRACT FROM AUTHOR]

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes. [ABSTRACT FROM AUTHOR]

A ring oscillator is a well-known circuit used for generating random numbers, and interested readers can find many research results concerning the evaluation of the randomness with a packaged test suit. However, the authors think there is room for evaluating the unpredictability of a sequence from another viewpoint. In this paper, the authors focus on Wold's RO-based generator and propose a statistical test to numerically evaluate the randomness of the RO-based generator. The test adopts the state transition probabilities in a Markov process and is designed to check the uniformity of the probabilities based on hypothesis testing. As a result, it is found that the RO-based generator yields a biased output from the viewpoint of the transition probability if the number of ROs is small. More precisely, the transitions 01 → 01 and 11 → 11 happen frequently when the number l of ROs is less than or equal to 10. In this sense, l > 10 is recommended for use in any application, though a packaged test suit is passed. Thus, the authors believe that the proposed test contributes to evaluating the unpredictability of a sequence when used together with available statistical test suits, such as NIST SP800-22. [ABSTRACT FROM AUTHOR]

Content-centric vehicular networks (CCVNs) have considered distributed proactive caching as an attractive approach for the timely provision of emerging services. The naïve caching schemes cache all of the contents to only one selected roadside unit (RSU) for requested vehicles to decrease the data acquisition delay between the data source and the vehicles. Due to the high deployment cost for RSUs and their limited capacity of caching, the vehicular networks could support only a limited number of vehicles and a limited amount of content and thus decrease the cache hit ratio. This paper proposes a mobility-aware distributed proactive caching protocol (MDPC) in CCVNs. MDPC caches contents to the selected RSUs according to the movement of vehicles. To reduce the redundancy and the burden of caching for each RSU, MDPC distributes to cache partial contents by the movement pattern, the probability to predict the next locations (RSUs) on the Markov model based on the current RSU. For recovery of prediction failures, MDPC allows each RSU to request partial missing contents to relatively closer neighbor RSUs with a short delay. Next, we expand the protocol with traffic optimization called MDPC_TO to minimize the amount of traffic for achieving proactive caching in CCVNs. In proportion to the mobility probability of a vehicle toward each of the next RSUs, MDPC_TO controls the amount of pre-cached contents in each of the next RSUs. Then, MDPC_TO has constraints to provide enough content from other next RSUs through backhaul links to remove the delay due to prediction failures. Simulation results verify that MDPC_TO produces less traffic than MDPC. [ABSTRACT FROM AUTHOR]

The stochastic process non-homogeneous Markov system in a stochastic environment in continuous time (S-NHMSC) is introduced in the present paper. The ordinary non-homogeneous Markov process is a very special case of an S-NHMSC. I studied the expected population structure of the S-NHMSC, the first central classical problem of finding the conditions under which the asymptotic behavior of the expected population structure exists and the second central problem of finding which expected relative population structures are possible limiting ones, provided that the limiting vector of input probabilities into the population is controlled. Finally, the rate of convergence was studied. [ABSTRACT FROM AUTHOR]

A practical probabilistic data association filter is proposed for tracking multiple targets in clutter. The number of joint data association events increases combinatorially with the number of measurements and the number of targets, which may become computationally impractical for even small numbers of closely located targets in real target-tracking applications in heavily cluttered environments. In this paper, a Markov chain model is proposed to generate a set of feasible joint events (FJEs) for multiple target tracking that is used to approximate the multi-target data association probabilities and the probabilities of target existence of joint integrated probabilistic data association (JIPDA). A Markov chain with the transition probabilities obtained from the integrated probabilistic data association (IPDA) for single-target tracking is designed to generate a random sequence composed of the predetermined number of FJEs without incurring additional computational cost. The FJEs generated are adjusted for the multi-target tracking environment. A computationally tractable set of these random sequences is utilized to evaluate the track-to-measurement association probabilities such that the computational burden is substantially reduced compared to the JIPDA algorithm. By a series of simulations, the track confirmation rates and target retention statistics of the proposed algorithm are compared with the other existing algorithms including JIPDA to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

To optimally plan maintenance of wind turbine blades, knowledge of the degradation processes and the remaining useful life is essential. In this paper, a method is proposed for calibration of a Markov deterioration model based on past inspection data for a range of blades, and updating of the model for a specific wind turbine blade, whenever information is available from inspections and/or condition monitoring. Dynamic Bayesian networks are used to obtain probabilities of inspection outcomes for a maximum likelihood estimation of the transition probabilities in the Markov model, and are used again when updating the model for a specific blade using observations. The method is illustrated using indicative data from a database containing data from inspections of wind turbine blades. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the notion of the "minus logarithm of stationary probability" as a generalized potential in nonequilibrium systems and attempt to illustrate its central role in an axiomatic approach to stochastic nonequilibrium thermodynamics of complex systems. It is demonstrated that this quantity arises naturally through both monotonicity results of Markov processes and as the rate function when a stochastic process approaches a deterministic limit. We then undertake a more detailed mathematical analysis of the consequences of this quantity, culminating in a necessary and sufficient condition for the criticality of stochastic systems. This condition is then discussed in the context of recent results about criticality in biological systems. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the stability of finite-dimensional distribution of time-inhomogeneous, discrete-time Markov chains on a general state space. The main result of the paper provides an estimate for the absolute difference of finite-dimensional distributions of a given time-inhomogeneous Markov chain and its perturbed version. By perturbation, we mean here small changes in the transition probabilities. Stability estimates are obtained using the coupling method. [ABSTRACT FROM AUTHOR]

In the framework of coding theory, under the assumption of a Markov process (X t) on a finite alphabet A , the compressed representation of the data will be composed of a description of the model used to code the data and the encoded data. Given the model, the Huffman's algorithm is optimal for the number of bits needed to encode the data. On the other hand, modeling (X t) through a Partition Markov Model (PMM) promotes a reduction in the number of transition probabilities needed to define the model. This paper shows how the use of Huffman code with a PMM reduces the number of bits needed in this process. We prove the estimation of a PMM allows for estimating the entropy of (X t) , providing an estimator of the minimum expected codeword length per symbol. We show the efficiency of the new methodology on a simulation study and, through a real problem of compression of DNA sequences of SARS-CoV-2, obtaining in the real data at least a reduction of 10.4%. [ABSTRACT FROM AUTHOR]

The paper is devoted to the investigation of the distributed proof generation process, which makes use of recursive zk-SNARKs. Such distributed proof generation, where recursive zk-SNARK-proofs are organized in perfect Mercle trees, was for the first time proposed in Latus consensus protocol for zk-SNARKs-based sidechains. We consider two models of a such proof generation process: the simplified one, where all proofs are independent (like one level of tree), and its natural generation, where proofs are organized in partially ordered set (poset), according to tree structure. Using discrete Markov chains for modeling of corresponding proof generation process, we obtained the recurrent formulas for the expectation and variance of the number of steps needed to generate a certain number of independent proofs by a given number of provers. We asymptotically represent the expectation as a function of the one variable n / m , where n is the number of provers m is the number of proofs (leaves of tree). Using results obtained, we give numerical recommendation about the number of transactions, which should be included in the current block, idepending on the network parameters, such as time slot duration, number of provers, time needed for proof generation, etc. [ABSTRACT FROM AUTHOR]

This short paper concerns the analysis of the M/M/k queueing system with customer abandonment. In this system, service managers provide a finite buffer space, which is a waiting area that prevents customers from abandoning the system. Abandonment of the system can occur from reneging (exiting from the queue while waiting), and/or balking (leaving the system without waiting). We derive an analytical expression to represent the impact of the buffer space capacity on the delay probability and the abandonment probability for a system with deferred abandonment. The result indicates the provision of the buffer space in a large system could only increase the delay probability while the abandonment probability remains unchanged. Despite the benevolent intentions of service managers, providing a buffer space may exacerbate the performance of larger systems. [ABSTRACT FROM AUTHOR]

Markov chains (MCs) are widely used to model a great deal of financial and actuarial problems. Likewise, they are also used in many other fields ranging from economics, management, agricultural sciences, engineering or informatics to medicine. This paper focuses on the use of MCs for the design of non-life bonus-malus systems (BMSs). It proposes quantifying the uncertainty of transition probabilities in BMSs by using fuzzy numbers (FNs). To do so, Fuzzy MCs (FMCs) as defined by Buckley and Eslami in 2002 are used, thus giving rise to the concept of Fuzzy BMSs (FBMSs). More concretely, we describe in detail the common BMS where the number of claims follows a Poisson distribution under the hypothesis that its characteristic parameter is not a real but a triangular FN (TFN). Moreover, we reflect on how to fit that parameter by using several fuzzy data analysis tools and discuss the goodness of triangular approximates to fuzzy transition probabilities, the fuzzy stationary state, and the fuzzy mean asymptotic premium. The use of FMCs in a BMS allows obtaining not only point estimates of all these variables, but also a structured set of their possible values whose reliability is given by means of a possibility measure. Although our analysis is circumscribed to non-life insurance, all of its findings can easily be extended to any of the abovementioned fields with slight modifications. [ABSTRACT FROM AUTHOR]

As multimedia traffic has been increasing and is expected to grow more sharply, various technologies using caches have been attracting lots of attention. As one breakthrough technology to deal with the explosively growing traffic, exclusive OR (XOR)-based index coding has been widely investigated because it can greatly enhance the efficiency of network resource by reducing the number of transmissions. In this paper, we investigate how to apply XOR-based index coding to large-scaled practical streaming systems for video traffic that accounts for more than 70% of total Internet traffic. Contrary to most previous studies that have focused on theoretical analysis of optimal performance or development of optimal index coding schemes, our study proposes a new XOR coding-based video streaming (XC). We also propose a new grouping algorithm for creating XC groups while guaranteeing the complete backward compatibility of XC with existing streaming schemes such as unicast (UC), multicast (MC), and broadcast (BC). The performance of the proposed scheme is analyzed in two steps. First, the behavior of video contents in caches at clients is modeled as a Markov chain, and the steady-state probabilities and caching probabilities for each piece of video content are derived. Based on the probabilities, the performance of the proposed system is then analyzed in terms of the average number of connections that each client requires in order to receive one video content. Our numerical results show that the proposed video streaming scheme using XC can reduce the average number of transmissions by up to 18%, compared to the conventional scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discrete-time constrained average stochastic games with independent state processes. The state space of each player is denumerable and one-stage cost functions can be unbounded. In these game models, each player chooses an action each time which influences the transition probability of a Markov chain controlled only by this player. Moreover, each player needs to pay some costs which depend on the actions of all the players. First, we give an existence condition of stationary constrained Nash equilibria based on the technique of average occupation measures and the best response linear program. Then, combining the best response linear program and duality program, we present a non-convex mathematic program and prove that each stationary Nash equilibrium is a global minimizer of this mathematic program. Finally, a controlled wireless network is presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In order to improve the dynamic optimization of fleet size and standardized management of dockless bike-sharing, this paper focuses on using the Markov stochastic process and linear programming method to solve the problem of bike-sharing fleet size and rebalancing. Based on the analysis of characters of bike-sharing, which are irreducible, aperiodic and positive-recurrence, we prove that the probability limits the state (steady-state) of bike-sharing Markov chain only exists and is independent of the initial probability distribution. Then a new "Markov chain dockless bike-sharing fleet size solution" algorithm is proposed. The process includes three parts. Firstly, the irreducibility of the bike-sharing transition probability matrix is analyzed. Secondly, the rank-one updating method is used to construct the transition probability random prime matrix. Finally, an iterative method for solving the steady-state probability vector is therefore given and the convergence speed of the method is analyzed. Furthermore, we discuss the dynamic solution of the bike-sharing steady-state fleet size according to the time period, so as improving the practicality of the algorithm. To verify the efficiency of this algorithm, we adopt the linear programming method for bicycle rebalancing analysis. Experiment results show that the algorithm could be used to solve the disordered deployment of dockless bike-sharing. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approach to obtain reduced-order models of Markov chains. Our approach is composed of two information-theoretic processes. The first is a means of comparing pairs of stationary chains on different state spaces, which is done via the negative, modified Kullback–Leibler divergence defined on a model joint space. Model reduction is achieved by solving a value-of-information criterion with respect to this divergence. Optimizing the criterion leads to a probabilistic partitioning of the states in the high-order Markov chain. A single free parameter that emerges through the optimization process dictates both the partition uncertainty and the number of state groups. We provide a data-driven means of choosing the 'optimal' value of this free parameter, which sidesteps needing to a priori know the number of state groups in an arbitrary chain. [ABSTRACT FROM AUTHOR]

The analysis of self-stabilizing algorithms is often limited to the worst case stabilization time starting from an arbitrary state, i.e., a state resulting from a sequence of faults. Considering the fact that these algorithms are intended to provide fault tolerance in the long run, this is not the most relevant metric. A common situation is that a running system is an a legitimate state when hit by a single fault. This event has a much higher probability than multiple concurrent faults. Therefore, the worst case time to recover from a single fault is more relevant than the recovery time from a large number of faults. This paper presents techniques to derive upper bounds for the mean time to recover from a single fault for self-stabilizing algorithms based on Markov chains in combination with lumping. To illustrate the applicability of the techniques they are applied to a new self-stabilizing coloring algorithm. [ABSTRACT FROM AUTHOR]