Asymptotic Behavior of the Stochastic Rayleigh-van der Pol Equations with Jumps by ZaiTang Huang, ChunTao Chen

“…Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.…”
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On a Perturbed Risk Model with Time-Dependent Claim Sizes by Longfei Wei, Jia Hao, Shiyu Song, Zhenhua Bao

“…We consider a risk model perturbed by a Brownian motion, where the individual claim sizes are dependent on the inter-claim times. …”
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Least Squares Estimation for α-Fractional Bridge with Discrete Observations by Guangjun Shen, Xiuwei Yin

“…We consider a fractional bridge defined as dXt=-α(Xt/(T-t))dt+dBtH,  0≤t<T, where BH is a fractional Brownian motion of Hurst parameter H>1/2 and parameter α>0 is unknown. …”
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Convergence of numerical solution of stochastic differential equation for the self-thinning process by Petras Rupšys

“… For theoretical and practical analysis of the self-thinning process we use stochastic differential equation, which take the form: dN (t) = N (t) (α - β ln N (t))dt + μN (t)dW (t), N(t0) = N0, t0 ≤ t ≤ T, where N – tree per hectare (stem/ha), t – stand age, W(t) – scalar standard Brownian motion, N0 – not random, α, β and μ are parameters – real constants. …”
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Drift and the Risk-Free Rate by Anda Gadidov, M. C. Spruill

“…It is proven, under a set of assumptions differing from the usual ones in the unboundedness of the time interval, that, in an economy in equilibrium consisting of a risk-free cash account and an equity whose price process is a geometric Brownian motion on [0,∞), the drift rate must be close to the risk-free rate; if the drift rate 𝜇 and the risk-free rate 𝑟 are constants, then 𝑟=𝜇 and the price process is the same under both empirical and risk neutral measures. …”
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Rough Paths above Weierstrass Functions by Cellarosi, Francesco, Selk, Zachary

“…The fundamental observation of rough paths theory is that if one can define “iterated integrals” above a signal, then one can construct solutions to differential equations driven by the signal.The typical examples of the signals of interest are stochastic processes such as (fractional) Brownian motion. However, rough paths theory is not inherently random and therefore can treat irregular deterministic driving signals such as a (vector-valued) Weierstrass function. …”
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About brownian movement in liquids by E. I. Marukovich, V. Yu. Stetsenko, A. V. Stetsenko

“…In metallic liquids, Brownian motion refers to microscopic non-metallic particles and intermetallides that have densities comparable to melt densities. …”
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Holderian functional central limit theorem for linear processes by Mindaugas Juodis

“…The weak convergence of polygonal line process build from sums of (Xt)t ≥ 1 to the standard Brownian motion W in the Hölder space (Hα), 0 < α < 1/2 - 1/τ holds provided the proper noise behavior is satisfied: E|ε1|τ < ∞, τ > 2. …”
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Student Models for a Risky Asset with Dependence: Option Pricing and Greeks by Nikolai Leonenko, Anqi Liu, Nataliya Schestyuk

“… We propose several new models in finance known as the Fractal Activity Time Geometric Brownian Motion (FATGBM) models with Student marginals. …”
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Polar Functions for Anisotropic Gaussian Random Fields by Zhenlong Chen

“…The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise, and the operator-scaling Gaussian random field with stationary increments.…”
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Quasilinear Hyperbolic Systems Applied to Describe the Magnetohydrodynamic Nanofluid Flow by Dayong Nie

“…The results suggest that Brownian motion has a negligible impact on the heat transfer rate. …”
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On multiple-particle continuous-time random walks by Peter Becker-Kern, Hans-Peter Scheffler

“…Scaling limits of continuous-time random walks are used in physics to model anomalous diffusion in which particles spread at a different rate than the classical Brownian motion. In this paper, we characterize the scaling limit of the average of multiple particles, independently moving as a continuous-time random walk. …”
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Convergence Rate of Numerical Solutions for Nonlinear Stochastic Pantograph Equations with Markovian Switching and Jumps by Zhenyu Lu, Tingya Yang, Yanhan Hu, Junhao Hu

“…It is proved that Euler-Maruyama scheme for nonlinear stochastic pantograph equations with Markovian switching and Brownian motion is of convergence with strong order 1/2. …”
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Solution theory of fractional SDEs in complete subcritical regimes by Lucio Galeati, Máté Gerencsér

“…We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. …”
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Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale by Caishi Wang, Jinshu Chen

“…Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.…”
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Reaction-diffusion for fish populations with realistic mobility by Philip Broadbridge

“…However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. …”
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Insider Trading with Memory under Random Deadline by Kai Xiao, Yonghui Zhou

“…By a filtering theory on fractional Brownian motion and the stochastic maximum principle, we obtain a necessary condition of the insider’s optimal strategy, an equation satisfied. …”
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Asymptotically Sufficient Statistics in Nonparametric Regression Experiments with Correlated Noise by Andrew V. Carter

“…We show that the NPR experiment with correlated noise is asymptotically equivalent to an experiment that observes the mean function in the presence of a continuous Gaussian process that is similar to a fractional Brownian motion. These results provide a theoretical motivation for some commonly proposed wavelet estimation techniques.…”
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The Dynamic Spread of the Forward CDS with General Random Loss by Kun Tian, Dewen Xiong, Zhongxing Ye

“…We assume that the filtration F is generated by a d-dimensional Brownian motion W=(W1,…,Wd)′ as well as an integer-valued random measure μ(du,dy). …”
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Liquid Phase Separation Mechanism of Cu-40 wt.% Pb Hypermonotectic Alloys by Xiaosi Sun, Weixin Hao, Teng Ma, Junting Zhang, Guihong Geng

“…The liquid phase separation mechanism in the systems with stable miscibility gaps mainly involved Ostwald ripening, Brownian motion, Marangoni migration, and Stokes motion. …”
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