Singleton Sets Random Attractors for Lattice Dynamical Systems Driven by a Fractional Brownian Motion Revisited

Exponential attractors for first-order lattice dynamical systems

The aim of this paper is to investigate the existence of exponential attractors for lattice reaction-diffusion systems in weighted spaces $l_{\sigma}^{2}$ and for partly dissipative lattice reaction-diffusion systems in weighted spaces $l_{\mu}^{2}\times l_{\mu}^{2}$ , respectively. In contrast to the previous work by Abdallah in J. Math. Anal. Appl. 339, 217---224 (2008) and Commun. Pure Appl. Anal. 8, 803---818 (2009), we get the existence of exponential attractors for lattice dynamical systems in the weak topology spaces.

A lot of processes coming from Physics, Chemistry, Biology, Economy, and other sciences can be described using systems of reaction-diffusion equations. In this chapter, we study the asymptotic behavior of the solutions of a system of infinite ordinary differential equations (a lattice dynamical system) obtained after the spacial discretization of a system of reaction-diffusion equations in an unbounded domain. This kind of dynamical systems is then of importance in the numerical approximations of physical problems.

We first present some sufficient conditions for the existence of exponential attractors for locally coupled lattice dynamical systems in weighted spaces of infinite sequences. Then we apply this result to discuss the existence of exponential attractors for first order lattice systems, partly dissipative lattice systems, and second order lattice systems in weighted spaces of infinite sequences.

We investigate the existence of a global attractor and its upper semicontinuity for the infinite‐dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert space l2 × l2. Such a system is similar to the discretized FitzHugh‐Nagumo system in neurobiology, which is an adequate justification for its study.

The existence of numerical attractors for lattice dynamical systems is established, where the implicit Euler scheme is used for time discretisation. Infinite dimensional discrete lattice systems as well as their finite dimensional truncations are considered. It is shown that the finite dimensional numerical attractors converge upper semicontinuously to the global attractor of the original lattice model as the discretisation step size tends to zero.

This paper is devoted to consider stochastic lattice dynamical systems (SLDS) driven by fractional Brownian motions with Hurst parameter bigger than 1/2. Under usual dissipativity conditions these SLDS are shown to generate a random dynamical system for which the existence and uniqueness of a random attractor are established. Furthermore, the random attractor is, in fact, a singleton sets random attractor.

This paper is concerned with the random attractors for a class of second-order stochastic lattice dynamical systems. We first prove the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the spaceF=lλ2×l2. Then, by proving the asymptotic compactness of the random dynamical systems, we establish the existence of the global random attractor. The system under consideration is quite general, and many existing results can be regarded as the special case of our results.

Solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian Motions

Spatio-temporal chaos

In this paper, we study the asymptotic behavior of the stochastic discrete Klein-Gordon-Schrodinger equations driven by fractional Brownian motions. With the properties of fractional Brownian motions, we prove the existence of a singleton sets random attractor.

Fractional Brownian Motions, Fractional Noises and Applications

Stochastic curtailment has been considered for the interim monitoring of group sequential trials (Davis and Hardy, 1994). Statistical boundaries in Davis and Hardy (1994) were derived using theory of Brownian motion. In some clinical trials, the conditions of forming a Brownian motion may not be satisfied. In this paper, we extend the computations of Brownian motion based boundaries, expected stopping times, and type I and type II error rates to fractional Brownian motion (FBM). FBM includes Brownian motion as a special case. Designs under FBM are compared to those under Brownian motion and to those of O’Brien–Fleming type tests. One- and two-sided boundaries for efficacy and futility monitoring are also discussed. Results show that boundary values decrease and error rates deviate from design levels when the Hurst parameter increases from 0.1 to 0.9, these changes should be considered when designing a study under FBM.

This paper investigates whether the assumption of Brownian motion often used to describe commodity price movements is satisfied. Using historical data from 17 commodity futures contracts specific tests of fractional and ordinary Brownian motion are conducted. The analyses are conducted under the null hypothesis of ordinary Brownian motion against the alternative of persistent or ergodic fractional Brownian motion. Tests for fractional Brownian motion are based on a variance ratio test and compared with conventional R-S analyses. However, standard errors based on Monte Carlo simulations are quite high, meaning that the acceptance region for the null hypothesis is large. The results indicate that for the most part, the null hypothesis of ordinary Brownian motion cannot be rejected for 14 of 17 series. The three series that did not satisfy the tests were rejected because they violated the stationarity property of the random walk hypothesis.

Choosing the ideal number of rotations of planted forests under a silvicultural management regime results in uncertainties in the cash flows of forest investment projects. We verified if there is parity in the Eucalyptus wood price modeling through fractional Brownian motion and geometric Brownian motion to incorporate managerial flexibilities into investment projects in planted forests. We use empirical data from three production cycles of forests planted with Eucalyptus grandis × E. urophylla in the projection of discounted cash flows. The Eucalyptus wood price, assumed as uncertainty, was modeled using fractional and geometric Brownian motion. The discrete-time pricing of European options was obtained using the Monte Carlo method. The root mean square error of fractional and geometric Brownian motions was USD 1.4 and USD 2.2, respectively. The real options approach gave the investment projects, with fractional and geometric Brownian motion, an expanded present value of USD 8,157,706 and USD 9,162,202, respectively. Furthermore, in both models, the optimal harvest ages execution was three rotations. Thus, with an indication of overvaluation of 4.9% when assimilating the geometric Brownian motion, there is no parity between stochastic processes, and three production cycles of Eucalyptus planted forests are economically viable.