Kloeden and Platen  described a method based on the stochastic Taylor series expansion but the major difficulty with this approach is that the double stochastic integrals cannot be easily expressed in terms of simpler stochastic integrals when the Wiener process
The paper "A New Residual Life Prediction Method for Complex Systems Based on Wiener Process
and Evidential Reasoning" by X.
Such models treat uncertainty as stochastic, time-dependent volatility subject to "a standard random Wiener process
" [Silva & Yakovenko, 2003, p.
The first one considers Z(t) as a stochastic process, like Wiener process
, Ornstein-Uhlenbeck process, Poisson process, and so on.
To achieve degradation modeling and RUL estimation for rotating machinery, the optimal degradation indicator is modeled as a Wiener process
. The Wiener-process-based degradation is utilized to describe the equipment degradation process.
or the Brownian movement is a mathematical description of the random motion of a large particle immersed in the fluid and is not subject to any other interaction shocks with small molecules of the surrounding fluid it results a very irregular movement of the large particle the phenomenon was observed for the first time by the English botanist Robert Brown in 1828 , another interpretation was given in notes published between 1877-1880 "The Brownian movements would be in my way of looking at the phenomenon, the result of the calorific movements of molecular surrounding liquid" .
The generalized Wiener process
results in an ending stock price [S.sub.T] that is log-normally distributed
Merton, 'On the Role of the Wiener Process
in Finance Theory and Practice: The Case of Replicating Portfolios', Proceedings of Symposia in Pure Mathematics, 60.
where [Y.sub.N] = 1 - [[summation].sup.N-1.sub.[beta]=1] [Y.sub.[beta]], and [b.sub.[alpha]][S.sub.[alpha]] are parameters, while d[W.sub.[alpha]](t) is an isotropic Wiener process
with independent increments.
genocide; group aspects in the physical interpretation of general relativity theory; the periodic system for understanding group processes and work; and the law of the Wiener process
and path groups.
We model future revenue growth as a continuous-time random walk with drift (called a Wiener process
or a Brownian motion with drift) and use this model to estimate the maximum rate of spending growth consistent with present-value balance, given a jurisdiction's existing revenue structure and volatility.
The result of the Wiener process
control by suggested adaptive NMPC is shown in Fig.