1,721,027 research outputs found
On a Fractional Stochastic Risk Model with a Random Initial Surplus and a Multi-Layer Strategy
No full textThe paper deals with a fractional time-changed stochastic risk model, including stochastic premiums, dividends and also a stochastic initial surplus as a capital derived from a previous investment. The inverse of a ν-stable subordinator is used for the time-change. The submartingale property is assumed to guarantee the net-profit condition. The long-range dependence behavior is proven. The infinite-horizon ruin probability, a specialized version of the Gerber–Shiu function, is considered and investigated. In particular, we prove that the distribution function of the infinite-horizon ruin time satisfies an integral-differential equation. The case of the dividends paid according to a multi-layer dividend strategy is also considered
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
On the entropy of fractionally integrated gauss–markov processes
No full textThis paper is devoted to the estimation of the entropy of the dynamical system {Xα (t), t ≥ 0}, where the stochastic process Xα (t) consists of the fractional Riemann–Liouville integral of order α ∈ (0, 1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα (t) is a decreasing function of α ∈ (0, 1)
Generalized fractional calculus for gompertz-type models
No full textThis paper focuses on the construction of deterministic and stochastic extensions of the Gompertz curve by means of generalized fractional derivatives induced by complete Bernstein functions. Precisely, we first introduce a class of linear stochastic equations involving a generalized fractional integral and we study the properties of its solutions. This is done by proving the existence and uniqueness of Gaussian solutions of such equations via a fixed point argument and then by show-ing that, under suitable conditions, the expected value of the solution solves a generalized fractional linear equation. Regularity of the absolute p-moment functions is proved by using generalized Grön-wall inequalities. Deterministic generalized fractional Gompertz curves are introduced by means of Caputo-type generalized fractional derivatives, possibly with respect to other functions. Their stochastic counterparts are then constructed by using the previously considered integral equations to define a rate process and a generalization of lognormal distributions to ensure that the median of the newly constructed process coincides with the deterministic curve
Asymptotics of Two-boundary First-exit-time Densities for Gauss-Markov Processes
No full textThe problem of escape times from a region confined by two time-dependent boundaries is considered for a class of Gauss-Markov processes. Asymptotic approximations of the first exit time probability density functions in case of asymptotically constant and asymptotically periodic boundaries are obtained firstly for the Ornstein-Uhlenbeck process and then extended to the class of Gauss-Markov processes that can be obtained by a specified transformation. Some examples of application to stochastic dynamics and estimations of involved parameters by using numerical approximations are provided
On a stochastic neuronal model integrating correlated inputs
No full textA modified LIF-type stochastic model is considered with a non-delta correlated stochastic process in place of the traditional white noise. Two different mechanisms of reset are specified with the aim to model endogenous and exogenous correlated input stimuli. Ornstein-Uhlenbeck processes are used to model the two different cases. An equivalence between different ways to include currents in the model is also shown. The theory of integrated stochastic processes is evoked and the main features of involved processes are obtained, such as mean and covariance functions. Finally, a simulation algorithm of the proposed model is described; simulations are performed to provide estimations of firing densities and related comparisons are given
Two-boundary first exit time of Gauss-Markov processes for stochastic modeling of acto-myosin dynamics
No full textWe consider a stochastic differential equation in a strip, with coefficients suitably chosen to describe the acto-myosin interaction subject to time-varying forces. By simulating trajectories of the stochastic dynamics via an Euler discretization-based algorithm, we fit experimental data and determine the values of involved parameters. The steps of the myosin are represented by the exit events from the strip. Motivated by these results, we propose a specific stochastic model based on the corresponding time-inhomogeneous Gauss-Markov and diffusion process evolving between two absorbing boundaries. We specify the mean and covariance functions of the stochastic modeling process taking into account time-dependent forces including the effect of an external load. We accurately determine the probability density function (pdf) of the first exit time (FET) from the strip by solving a system of two non singular second-type Volterra integral equations via a numerical quadrature. We provide numerical estimations of the mean of FET as approximations of the dwell-time of the proteins dynamics. The percentage of backward steps is given in agreement to experimental data. Numerical and simulation results are compared and discussed
Fractionally integrated Gauss-Markov processes and applications
No full textWe investigate the stochastic processes obtained as the fractional Riemann-Liouville inte-gral of order alpha is an element of (0 , 1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central role, for the fractional inte-gral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter alpha affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordi-nary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractionally integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed
On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations
No full textFor a fractionally integrated Brownian motion (FIBM) of order (Formula presented.) (Formula presented.) we investigate the decaying rate of (Formula presented.) as (Formula presented.) where (Formula presented.) is the first-passage time (FPT) of (Formula presented.) through the barrier (Formula presented.) Precisely, we study the so-called persistent exponent (Formula presented.) of the FPT tail, such that (Formula presented.) as (Formula presented.) and by means of numerical simulation of long enough trajectories of the process (Formula presented.) we are able to estimate (Formula presented.) and to show that it is a non-increasing function of (Formula presented.) with (Formula presented.) In particular, we are able to validate numerically a new conjecture about the analytical expression of the function (Formula presented.) for (Formula presented.) Such a numerical validation is carried out in two ways: in the first one, we estimate (Formula presented.) by using the simulated FPT density, obtained for any (Formula presented.) in the second one, we estimate the persistent exponent by directly calculating (Formula presented.) Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of (Formula presented.) and we find the upper bound of its covariance function
Successive spike times predicted by a stochastic neuronal model with a variable input signal
Full text linkTwo different stochastic processes are used to model the evolution of the membrane voltage of a neuron exposed to a time-varying input signal. The first process is an inhomogeneous Ornstein-Uhlenbeck process and its first passage time through a constant threshold is used to model the first spike time after the signal onset. The second process is a Gauss-Markov process identified by a particular mean function dependent on the first passage time of the first process. It is shown that the second process is also of a diffusion type. The probability density function of the maximum between the first passage time of the first and the second process is considered to approximate the distribution of the second spike time. Results obtained by simulations are compared with those following the numerical and asymptotic approximations. A general equation to model successive spike times is given. Finally, examples with specific input signals are provided
- …
