Monte Carlo Test for Stochastic Trend in Space State Models for the Location-Scale Family

In space state models for time series, a key point is the decision between modeling the trend of non-stationary processes through a deterministic or a stochastic term. The present paper introduces a Monte Carlo hypothesis test procedure to guide in such a decision. The method works for any time series distribution belonging to the location-scale family. The proposed method provides an alpha-level test for any time series of length greater than 3 and it does not demand assumptions on the distribution of the trend term when it is actually stochastic.

On the distribution of fluxes of gamma-ray blazars: hints for a stochastic process?

ABSTRACT We examine a model for the observed temporal variability of powerful blazars in the γ-ray band in which the dynamics is described in terms of a stochastic differential equation, including the contribution of a deterministic drift and a stochastic term. The form of the equation is motivated by the current astrophysical framework, accepting that jets are powered through the extraction of the rotational energy of the central supermassive black hole mediated by magnetic fields supported by a so-called magnetically arrested accretion disc. We apply the model to the γ-ray light curves of several bright blazars and we infer the parameters suitable to describe them. In particular, we examine the differential distribution of fluxes (dN/dFγ) and we show that the predicted probability density function for the assumed stochastic equation naturally reproduces the observed power-law shape at large fluxes $\mathrm{ d}N/\mathrm{ d}F_{\gamma } \propto F_{\gamma }^{-\alpha }$ with α > 2.

Random Pullback Attractor of a Non-autonomous Local Modified Stochastic Swift-Hohenberg with Multiplicative Noise

In this paper, we study the existence of the random -pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg equation with multiplicative noise in stratonovich sense. It is shown that a random -pullback attractor exists in when its external force has exponential growth. Due to the stochastic term, the estimate are delicate, we overcome this difficulty by using the Ornstein-Uhlenbeck(O-U) transformation and its properties.

Periodic stochastic high-order Degasperis–Procesi equation with cylindrical fBm

This paper studies the periodic stochastic high-order Degasperis–Procesi (DP) equation driven by a cylindrical fractional Brownian motion (fBm) which is white in space. And it has the covariance with Hurst parameter [Formula: see text] in the time variable. The local existence and uniqueness of the solution [Formula: see text] in [Formula: see text] with [Formula: see text] are proved by fixed point theorem combined with the stochastic term estimations in the Besov-type Bourgain space [Formula: see text] and the second iteration of non-linear term.

Nonclassical point of view of the Brownian motion generation via fractional deterministic model

In this paper, we present a dynamical system based on the Langevin equation without stochastic term and using fractional derivatives that exhibit properties of Brownian motion, i.e. a deterministic model to generate Brownian motion is proposed. The stochastic process is replaced by considering an additional degree of freedom in the second-order Langevin equation. Thus, it is transformed into a system of three first-order linear differential equations, additionally [Formula: see text]-fractional derivative are considered which allow us to obtain better statistical properties. Switching surfaces are established as a part of fluctuating acceleration. The final system of three [Formula: see text]-order linear differential equations does not contain a stochastic term, so the system generates motion in a deterministic way. Nevertheless, from the time series analysis, we found that the behavior of the system exhibits statistics properties of Brownian motion, such as, a linear growth in time of mean square displacement, a Gaussian distribution. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.

An impulsive delay discrete stochastic neural network fractional-order model and applications in finance

In this paper, we propose a new tool for modeling and analysis in finance, introducing an impulsive discrete stochastic neural network (NN) fractional-order model. The main advantages of the proposed approach are: (i) Using NNs which can be trained without the restriction of a model to derive parameters and discover relationships, driven and shaped solely by the nature of the data; (ii) using fractional-order differences, whose nonlocal property makes the fractional calculus a suitable tool for modeling actual financial systems; (iii) using impulsive perturbations, which give an opportunity to control the dynamic behavior of the model; (iv) including a stochastic term, which allows to study the effect of noise disturbances generally existing in financial assets; (v) taking into account the existence of time delayed influences. The modeling approach proposed in this paper can be applied to investigate macroeconomic systems.

Absolute continuity of the law for solutions of stochastic differential equations with boundary noise

We study the existence and regularity of the density for the solution [Formula: see text] (with fixed [Formula: see text] and [Formula: see text]) of the heat equation in a bounded domain [Formula: see text] driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in [Formula: see text].

Bose-Einstein Condensation of a Stochastic Liquid

The Bogoliubov-Lee-Huang theory of superfluid 4He is modified by introducing an effective temperature scale (which accounts for the deep well of the interatomic potential) and by incorporating into the Hamiltonian a stochastic term Vl, which simulates liquidity of HeI and liquidity of the normal and superfluid component of HeII. Vl depends on two independent random angles αn, αs ∈ [0, π], which characterize the locally ordered motion of the two fluids (the normal fluid and superfluid) comprising HeII. The resulting thermodynamics improves the thermodynamic functions and excitation spectrum Ep(αn, αs) of the superfluid phase, obtained previously, leaving the heat capacity CV (T) of the normal phase, with a minimum at Tmin > 2.17K, unchanged. The theoretical velocity of sound in HeII, equal to the initial slope of Ep(π, π), agrees with experiment.

Stochastic Bose superfluid

The ideas of Mitus et al. are exploited to define the liquid state as a state of matter, in which particles perform locally ordered motion. The presence of the liquid phase is accounted for by a stochastic term in the Hamiltonian, which simulates this property of a liquid. The Bogoliubov–Lee–Huang theory of He II, recently modified by use of effective temperature scale and more stringent reduction procedure (DHSET theory) is extended, by incorporating this term into the 4 He Hamiltonian. The resulting thermodynamics accounts for effects, which are beyond the scope of other He I and He II theories, e.g., the atomic momentum distribution and excitation spectrum have the form of diffused bands, similarly as in He II; the He I, theoretical heat capacity CV(T) is a convex function, with a minimum at T min > Tλ, which qualitatively simulates experimental He I heat capacity. Other thermodynamic functions are similar to those of DHSET theory.