1,721,009 research outputs found
The time-changed stochastic approach and fractionally integrated processes to model the actin-myosin interaction and dwell times
No full textWe propose two stochastic models for the interaction between the myosin head and the actin filament, the physio-chemical mechanism triggering muscle contraction and that is not yet completely understood. We make use of the fractional calculus approach with the purpose of constructing non-Markov processes for models with memory. A time-changed process and a fractionally integrated process are proposed for the two models. Each of these includes memory effects in a different way. We describe such features from a theoretical point of view and with simulations of sample paths. Mean functions and covariances are provided, considering constant and time-dependent tilting forces by which effects of external loads are included. The investigation of the dwell time of such phenomenon is carried out by means of density estimations of the first exit time (FET) of the processes from a strip; this mimics the times of the Steps of the myosin head during the sliding movement outside a potential well due to the interaction with actin. For the case of time-changed diffusion process, we specify an equation for the probability density function of the FET from a strip. The schemes of two simulation algorithms are provided and performed. Some numerical and simulation results are given and discussed
First passage times for some classes of fractional time-changed diffusions
No full textWe consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a time-changed Brownian motion. The time-change is obtained via the inverse of an α-stable subordinator. These processes are specified in terms of time-changed Gauss-Markov processes and fractional time-changed diffusions. A fractional pseudo-Fokker-Planck equation for such processes is given. We investigate their first passage time densities providing a generalized integral equation they satisfy and some transformation rules. First passage time densities for time-changed Brownian motion and Ornstein-Uhlenbeck processes are provided in several forms. Connections with closed form results and numerical evaluations through the level zero are given
Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues
No full textSeveral queueing systems in heavy traffic regimes are shown to admit a diffusive approximation in terms of the Reflected Brownian Motion. The latter is defined by solving the Skorokhod reflection problem on the trajectories of a standard Brownian motion. In recent years, fractional queueing systems have been introduced to model a class of queueing systems with heavy-tailed interarrival and service times. In this paper, we consider a subdiffusive approximation for such processes in the heavy traffic regime. To do this, we introduce the Delayed Reflected Brownian Motion by either solving the Skorohod reflection problem on the trajectories of the delayed Brownian motion or by composing the Reflected Brownian Motion with an inverse stable subordinator. The heavy traffic limit is achieved via the continuous mapping theorem. As a further interesting consequence, we obtain a simulation algorithm for the Delayed Reflected Brownian Motion via a continuous-time random walk approximation
Time-Non-Local Pearson Diffusions
No full textIn this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence
Fractional immigration-death processes
No full textIn this paper we study explicit strong solutions for two difference-differential fractional equations, defined via the generator of an immigration-death process, by using spectral methods. Moreover, we give a stochastic representation of the solutions of such difference-differential equations by means of a stable time-changed immigration-death process and we use this stochastic representation to show boundedness and then uniqueness of these strong solutions. Finally, we study the limit distribution of the time-changed process
Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order and beyond
Full text linkWe introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property
Limit theorems for the fractional nonhomogeneous Poisson process
No full textThe fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse -stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe's theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation
Large deviations for a class of tempered subordinators and their inverse processes
No full textWe consider a class of tempered subordinators, namely a class of subordinators with
one-dimensional marginal tempered distributions which belong to a family studied in
[3]. The main contribution in this paper is a non-central moderate deviations result.
More precisely we mean a class of large deviation principles that fill the gap between
the (trivial) weak convergence of some non-Gaussian identically distributed random
variables to their common law, and the convergence of some other related random
variables to a constant. Some other minor results concern large deviations for the
inverse of the tempered subordinators considered in this paper; actually, in some
results, these inverse processes appear as random time-changes of other independent
processes
Non-local Solvable Birth–Death Processes
No full textIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes
On the Transient Behaviour of Fractional M/ M/ ∞ Queues
No full textWe study some features of the transient probability distribution of a fractional M/ M/ ∞ queueing system. Such model is constructed as a suitable time-changed birth-death process. The fractional differential-difference problem is studied for the corresponding probability distribution and a fractional partial differential equation is obtained for the generating function. Finally, the interpretation of the system as an actual M/ M/ ∞ queue and as a M/M/1 queue with responsive server is given and some conditioned virtual waiting times are studied
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