Random Jumps Research Articles

Fluctuations of Current in Polycrystalline Ferroelectric Ceramics

Experimental data on fluctuations of leakage currents over time in samples of polycrystalline ferroelectric ceramics (SBPZT-1, ceramics based on sodium bismuth lead zirconate–titanate) upon the application of a constant electric field are considered. Variations in the type of leakage current dependence on time with an increase in electric field strength on samples with different values of the loss-angle tangent are determined. Exponential decay is accompanied by random jumps, i.e., variations in the values of the leakage current.

Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

A Novel Multi-Phase Stochastic Model for Lithium-Ion Batteries’ Degradation with Regeneration Phenomena

A lithium-Ion battery is a typical degradation product, and its performance will deteriorate over time. In its degradation process, regeneration phenomena have been frequently encountered, which affect both the degradation state and rate. In this paper, we focus on how to build the degradation model and estimate the lifetime. Toward this end, we first propose a multi-phase stochastic degradation model with random jumps based on the Wiener process, where the multi-phase model and random jumps at the changing point are used to describe the variation of degradation rate and state caused by regeneration phenomena accordingly. Owing to the complex structure and random variables, the traditional Maximum Likelihood Estimation (MLE) is not suitable for the proposed model. In this case, we treat these random variables as latent parameters, and then develop an approach for model identification based on expectation conditional maximum (ECM) algorithm. Moreover, depending on the proposed model, how to estimate the lifetime with fixed changing point is presented via the time-space transformation technique, and the approximate analytical solution is derived. Finally, a numerical simulation and a practical case are provided for illustration.

Random multi-hopper model: super-fast random walks on graphs

We develop a model for a random walker with long-range hops on general graphs. This random multi-hopper jumps from a node to any other node in the graph with a probability that decays as a function of the shortest-path distance between the two nodes. We consider here two decaying functions in the form of the Laplace and Mellin transforms of the shortest-path distances. Remarkably, when the parameters of these transforms approach zero asymptotically, the multi-hopper's hitting times between any two nodes in the graph converge to their minimum possible value, given by the hitting times of a normal random walker on a complete graph. Stated differently, for small parameter values the multi-hopper explores a general graph as fast as possible when compared to a random walker on a full graph. Using computational experiments we show that compared to the normal random walker, the multi-hopper indeed explores graphs with clusters or skewed degree distributions more efficiently for a large parameter range. We provide further computational evidence of the speed-up attained by the random multi-hopper model with respect to the normal random walker by studying deterministic, random and real-world networks.

A new characterization of the jump rate for piecewise-deterministic Markov processes with discrete transitions

ABSTRACTPiecewise-deterministic Markov processes form a general class of non diffusion stochastic models that involve both deterministic trajectories and random jumps at random times. In this paper, we state a new characterization of the jump rate of such a process with discrete transitions. We deduce from this result a non parametric technique for estimating this feature of interest. We state the uniform convergence in probability of the estimator. The methodology is illustrated on a numerical example.

Complex and Entropy of Fluctuations of Agent-Based Interacting Financial Dynamics with Random Jump

This paper investigates the complex behaviors and entropy properties for a novel random complex interacting stock price dynamics, which is established by the combination of stochastic contact process and compound Poisson process, concerning with stock return fluctuations caused by the spread of investors’ attitudes and random jump fluctuations caused by the macroeconomic environment, respectively. To better understand the fluctuation complex behaviors of the proposed price dynamics, the entropy analyses of random logarithmic price returns and corresponding absolute returns of simulation dataset with different parameter set are preformed, including permutation entropy, fractional permutation entropy, sample entropy and fractional sample entropy. We found that a larger λ or γ leads to more complex dynamics, and the absolute return series exhibit lower complex dynamics than the return series. To verify the rationality of the proposed compound price model, the corresponding analyses of actual market datasets are also comparatively preformed. The empirical results verify that the proposed price model can reproduce some important complex dynamics of actual stock markets to some extent.

Continuous-time random walks with reset events

In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.

Linked Gauss-Diffusion processes for modeling a finite-size neuronal network

A Leaky Integrate-and-Fire (LIF) model with stochastic current-based linkages is considered to describe the firing activity of neurons interacting in a (2×2)-size feed-forward network. In the subthreshold regime and under the assumption that no more than one spike is exchanged between coupled neurons, the stochastic evolution of the neuronal membrane voltage is subject to random jumps due to interactions in the network. Linked Gauss-Diffusion processes are proposed to describe this dynamics and to provide estimates of the firing probability density of each neuron. To this end, an iterated integral equation-based approach is applied to evaluate numerically the first passage time density of such processes through the firing threshold. Asymptotic approximations of the firing densities of surrounding neurons are used to obtain closed-form expressions for the mean of the involved processes and to simplify the numerical procedure. An extension of the model to an (N×N)-size network is also given. Histograms of firing times obtained by simulations of the LIF dynamics and numerical firings estimates are compared.

Partially observed time‐inconsistent stochastic linear‐quadratic control with random jumps

SummaryThis paper studies a partially observed time‐inconsistent stochastic linear‐quadratic control system, in which the state follows a stochastic differential equation driven by a Brownian motion and an independent Poisson random measure. The cost functional contains a state‐dependent term and a quadratic term of the conditional expected state process, which will cause the time inconsistency in dynamic systems. By virtue of a classical spike variation approach, we define an equilibrium and derive a sufficient condition for the equilibrium in the fully observed system with stochastic coefficients. Then, we obtain the equilibrium with an explicit feedback form in deterministic coefficients case and discuss the existence and uniqueness of the solution of corresponding Riccati equations. Furthermore, we get filtering equations of the partially observed system and get the explicit equilibrium in some special case.

Mean stability and stabilization of positive linear systems subject to mode-dependent Poisson jumps

This paper deals with the class of positive linear systems subject to Poisson jumps. The random jumps are described by an exponential distribution with mode-dependent parameter. Contrary to classical Markov Jump Linear Systems, no probabilistic information on the sequence of modes is assumed to be available. Mean stability of this class of systems is analyzed, showing that it is equivalent to stability under arbitrary switching of a suitably defined deterministic switching system. Sufficient conditions for mean stability are worked out by relying on common copositive (linear or quadratic) Lyapunov functions. Similar conditions are derived for designing a state-feedback control law guaranteeing closed-loop mean stability and positivity.

Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations

We discuss the stochastic linear-quadratic (LQ) optimal controlproblem with Poisson processes under the indefinite case. Based on thewellposedness of the LQ problem, the main idea is expressed by thedefinition of relax compensator that extends the stochasticHamiltonian system and stochastic Riccati equation with Poissonprocesses (SREP) from the positive definite case to the indefinite case. Wemainly study the existence and uniqueness of the solution for thestochastic Hamiltonian system and obtain the optimal control withopen-loop form. Then, we further investigate the existence anduniqueness of the solution for SREP in some special case and obtainthe optimal control in close-loop form.

Assessing the Impact of Market Microstructure Noise and Random Jumps on the Relative Forecasting Performance of Option-Implied and Returns-Based Volatility

This paper presents a comprehensive empirical evaluation of option-implied and returnsbased forecasts of volatility, in which new developments related to the impact on measured volatility of market microstructure noise and random jumps are explicitly taken into account. The option-based component of the analysis also accommodates the concept of model-free implied volatility, such that the forecasting performance of the options market is separated from the issue of misspecification of the option pricing model. The forecasting assessment is conducted using an extensive set of observations on equity and option trades for News Corporation for the 1992 to 2001 period, yielding certain clear results. According to several different criteria, the model-free implied volatility is the best performing forecast, overall, of future volatility, with this result being robust to the way in which alternative measures of future volatility accommodate microstructure noise and jumps. Of the volatility measures considered, the one which is, in turn, best forecast by the option-implied volatility is that measure which adjusts for microstructure noise, but which retains some information about random jumps.

Lifetime prognostics for deteriorating systems with time-varying random jumps

In this paper, we propose a jump diffusion process with non-homogeneous compound Poisson process to model the degradation process with randomly occurring jumps, which combines two stochastic processes, i.e., traditional diffusion process to describe the continuous degradation and non-homogeneous compound Poisson process to depict random jumps with a time-varying intensity. The approximated analytical lifetime under the concept of the first passage time (FPT) is obtained by a time–space transformation technique. To identify the model parameters, we first present a general method based on Maximum Likelihood Estimation (MLE) for the proposed model, and then specifically provide a two-step approach for linear jump diffusion process via combining MLE and Expectation Conditional Maximization (ECM) algorithm. Finally, a numerical example and a study on the furnace wall are provided to illustrate the effectiveness of the proposed method.

How much is the gap?—Efficient jump risk-adjusted valuation of leveraged certificates

This paper develops a novel and highly efficient numerical algorithm for the gap risk-adjusted valuation of leveraged certificates. The existing literature relies on Monte Carlo simulations, which are not fast enough to be used in a market-making environment. This is because issuers need to compute thousands of price updates per second. By valuing leveraged certificates as multi-window barrier options, we explicitly model random jumps that occur at known times, such as between the exchange closing and re-opening. Our algorithm combines the one-day transition probability with Simpson’s numerical integration rule. This yields a backward induction scheme which requires a significantly coarser spatial and time grid than finite-difference methods. We confirm its robustness and accuracy through Monte Carlo simulations.

How Much Is the Gap? Efficient Jump Risk-Adjusted Valuation of Leveraged Certificates

This paper develops a novel and highly efficient numerical algorithm for the gap risk-adjusted valuation of leveraged certificates. The existing literature relies on Monte Carlo simulations, which are not fast enough to be used in a market making environment. This is because issuers need to compute thousands of price updates per second. By valuing leveraged certificates as multi-window barrier options, we explicitly model random jumps that occur at known times, such as between the exchange closing and re-opening. Our algorithm combines the one-day transition probability with Simpson's numerical integration rule. This yields a backward induction scheme which requires a significantly coarser spacial and time grid than finite difference methods. We confirm its robustness and accuracy through Monte Carlo simulations.

Toward Efficient Hub-Less Real Time Personalized PageRank

In the era of big data, reduced models capable of reducing big data graph to estimate personalized PageRank are limited. Personalized PageRank is a page rank calculation where random jumps are only allowed to a subset of start nodes. The resources of current process of calculation of personalized PageRank are highly prohibitive, thus in this paper we propose a novel fast accurate and less resource intensive algorithm to the personalized PageRank problem. FAST Personalized PageRank is utilized to find the target node set. Using the mentioned target set, the algorithm gives an estimation of the closeness of any pair of nodes in the graph. As the time taken by the estimation of personalized PageRank is directly proportional to the network size, in this paper a node reduction method is used to prune the graph. In this pruning model, most popular nodes also known as hubs are found using personalized page vector. To decrease the entropy and reduce the number of alternate paths to the target nodes, popular nodes are identified and flagged. The flagged nodes are, then, given a lower priority in the computation. This way the redundant path will being ignored in the computation process. After pruning the graph, estimation results achieve an improved time complexity. In our experiment, we compare our result with the benchmark FAST personalized PageRank approach. Our algorithm significantly reduces the computation time and outperforms the benchmark FAST personalized PageRank algorithm in highly dense graphs.

Acceptance of the Dana Medal of the Mineralogical Society of America for 2016

Once you have digested the news, two lists fleet through your mind—those who have received the recognition before you and those who have not. The first list is daunting and inspirational; the second is humbling. Both remind you of the responsibility the award bestows upon you. This award recognizes work done on diffusion. Diffusion is about taking a random walk, exploring neighborhoods, and improbable jumps. It is inefficient, unfocused but omnipresent and about the journey rather than reaching a goal. Along the way, it allows one to reach seemingly impossible spaces with extraneous encounters that are enriching. I have been privileged to experiment with that all my life, including throughout my career, and am now even being recognized for it. With parents in Kolkata, India, who never went to college but supported education despite financial difficulties, it was doubtful at one point whether I would even go to college. From there, to be standing here, defines improbable. I am sorry that they are not here to see this day. My aunt, who saw us through those difficult financial times and much else, has been able to share the surprise. How did it happen? I have learned that “…it is twice blest; It blesseth him that gives and him that takes,” said originally in a different context, applies to teaching as well. I have been multiply blessed all my life to have outstanding teachers and students, who have taught me with patience and compassion in spite of my failings. I am very glad that many of them are right here today. It is well and good to hang out and explore, as long as someone pays you to do that. I had the great fortune to find employers in my early career (Jibamitra Ganguly, Fritz Seifert, Herbert Palme) who did not …

Mobility-Aware Coded Probabilistic Caching Scheme for MEC-Enabled Small Cell Networks

Caching on the edge has been recognized as an effective solution to tackle the backhaul constraint of network densification. However, most related works ignored user mobility in wireless networks, which is unreasonable under the background of network densification. For a more flexible and context-aware caching decision, the concept of caching on the edge can be extended to mobile edge computing (MEC) that enables computation and storage resources at mobile edge networks. With MEC servers deployed on base stations, a huge amount of collected radio access network context data can be analyzed and utilized to render a caching scheme adaptive to user’s context-aware information. In this regard, a novel mobility-aware coded probabilistic caching scheme is proposed for MEC-enabled small cell networks (SCNs). Different from previous mobility-aware caching schemes, user mobility and distributed storage are incorporated into a conventional probabilistic caching scheme, with the aim of throughput maximization. Based on stochastic geometry theory and a modified mobility model of discrete random jumps, the explicit expression of throughput is derived. Due to the complexity of the expression, two light-weight heuristic algorithms are provided to numerically obtain the optimal solutions. Moreover, a significant trade-off among the gains of mobility diversity, content diversity, and channel selection diversity is discussed, and we further numerically analyze how such a trade-off is influenced by user mobility, content popularity, and backhaul capacity, with some fundamental insights into the application of the proposed scheme in MEC-enabled SCNs. The superiority of our proposed scheme is demonstrated by the comparisons with the classical M most popular caching scheme and the conventional probabilistic caching scheme. Numerical results show that our proposed caching scheme achieves higher throughput than those of the other two, especially when users of intense mobility request contents, of which the popularity profile is not skewed, in MEC-enabled SCNs with poor backhaul capacity, indicating that the proposed caching scheme is a promising solution for network densification.

Stochastic Interest Model Based on Compound Poisson Process and Applications in Actuarial Science

Considering stochastic behavior of interest rates in financial market, we construct a new class of interest models based on compound Poisson process. Different from the references, this paper describes the randomness of interest rates by modeling the force of interest with Poisson random jumps directly. To solve the problem in calculation of accumulated interest force function, one important integral technique is employed. And a conception called the critical value is introduced to investigate the validity condition of this new model. We also discuss actuarial present values of several life annuities under this new interest model. Simulations are done to illustrate the theoretical results and the effect of parameters in interest model on actuarial present values is also analyzed.

Multi-objective electric distribution network reconfiguration solution using runner-root algorithm

This paper presents a runner-root algorithm (RRA) for electric distribution network reconfiguration (NR) problem. The considered NR problem in this paper is to minimize real power loss, load balancing among the branches, load balancing among the feeders as well as number of switching operations and node voltage deviation using max-min method for selection of the final compromised solution. RRA is equipped with two explorative tools, which are random jumps with large steps and re-initialization strategy to escape from local optimal. Moreover, RRA is also equipped with an exploitative tool to search around the current best solution with large and small steps to ensure the obtained result of global optimization. The effectiveness of the applied RRA in both single- and multi-objective has been tested on 33-node and 70-node distribution network systems and the obtained test results have been compared to those from other methods in the literature. The simulation results show that the applied RRA can be an efficient method for network reconfiguration problems with single- and multi-objective.