Wiener process models

Compared with linear degradation model, stochastic process model can reasonably identify the random variation in degradation process. Baussaron (Baussaron 2011) proposed a reliability demonstration method based on ADT using Wiener process model. Generally speaking, the performance degradation of product is usually a... 

In the following, two basic models are presented [2, p.l29 4, p.203] the Wiener process and the Kolmogorov equation. Applications of the resulting equations in Chemical Engineering are also elaborated. The common to all models concerned is that they are one-dimensional and, certainly, obey the fundamental Markov concept - that past is not relevant and thaX future may be predicted from the present and the transition probabilities to the future. 

The reader should note that the microscale model is used to determine the nonzero terms in B, and thus for the following discussion B can be assumed to be known. Using matrix notation and the properties of the Wiener process (Gardiner, 2004), a symmetric N x N diffusion matrix D can be defined by... 

The Brownian force is the well-known force that becomes important in the case of very small particles suspended in a continuous phase. The Brownian force can be defined as the instantaneous momentum exchange due to collisions between the molecules of the continuous phase with a suspended particle. When the particle is small enough to perceive the molecular nature (and motion) of the continuous phase (i.e. when the particle Knudsen number is large enough), it exhibits a random motion, which was observed as early as 2000 years ago by the Roman Lucretius. The Brownian force is typically described as a stochastic process (Gardiner, 2004), and it can be modeled as a Wiener process " ... 

The fluctuations are often modeled as a Wiener process, i.e., i (t)dt = dW t) (Gardiner, 2003) the function is known as white noise. For its precise mathematical definition and that of the Wiener process, the reader is referred to Gardiner (2003). Then Equation (2.44) is replaced by... 

Chemical processes can be modelled in detail as a bunch of equations and differential equations based on chemical and physical laws. These laws rely on static assumptions about the environment where the corresponding processes take place. In practice, all components are infiuenced by stochastic factors that can influence the static process behaviour and/or the dynamic process characteristics. Incorporating continuous stochastic processes to a differential equation leads to stochastic differential equations. Linear stochastic differential equations (SDEs) are usually formulated in the following general form using the Wiener process W t) ... 

For the piuposes of employing option pricing models, the d3mamic behaviour of asset prices is usually described as a function of what is known as a Wiener process, which is also known as Brownian motion. The noise or volatility component is described by an adapted Brownian or Wiener process, and involves introducing a random increment to the standard random process. This is described next. 

Brownian motion or the Wiener process is employed by virtually all option pricing models and we introduce it here with respect to a change in the variable W over an interval of time t. If W represents a variable following a Wiener process and AW is a change in value over a period of time t, the relationship between AW and At is given by Equation (2.7) ... 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

The expression describes a stochastic process composed of n independent Wiener processes, from which the whole forward rate curve, from the initial curve at time 0, is derived. Each individual forward rate maturity is a function of a specific volatility coefficient. The volatility values ( t, t, T, w)) are not specified in the model and are dependent on historical Wiener processes. From Equation (4.28) following the HJM model, the spot rate stochastic process is given by Equation (4.29) ... 

Doksum, K. and A. Hoyland (1992). Models for variable-stress accelerated life testing experiments based on wiener processes and the inverse gaussian distribution. Technometrics 34(1), 74- 2. 

Instead of fitting a fully nonlinear model, another approach to nonlinear system identification is to partition the nonUnearities from the linear component A common application of this approach is the Wiener-Hammerstein model. A Wiener-Hammerstein model is a generalisation of the Hammerstein model, where non-linearities are assumed cmly to be in the input and the Wiener model, where nonlinearities are assumed only to be in the output, which allows nonlinearities to be present in both the input and output The process model is assumed to be linear. Thus, the general form of the model can be written as... 

The drift Brownian motion (Wiener process) is commonly used stochastic processes degradation model for the degradation process. 

The Wiener process represents one possible form of diffusion processes. It is actually the integral of what in practical applications is called a white noise. The Wiener process with drift will be used in our application. The initial mean value (drift) is p and standard deviations for each time increment have been previously calculated—see Table 1. For our model we apply Wiener process with drift given by stochastic differential equation. 

In the article we have introduced possible approaches to modelling indirect diagnostic measures. We were looking for first hitting time distribution which represents a critical limit of Fe particles amount. The modelling is based on the diffusion Wiener process with drift. The observation and the analysis focused on the individual and mean value of Fe particles amount in oil. 

The achieved results complement the set of approaches to the indirect observation of a technical condition. The approaches using purely a regression analysis and fuzzy logic, see e.g. Koucky Valis (2011), Valis et al. (2012), have been applied so far. Following the conclusions of modelling with the Wiener process, the results of previous approaches might be completed when searching for ... 

Doksum K.A. Hoyland A. 1992. Models for Variable-Stress Accelerated Life Testing Experiments Based on Wiener Processes and the Inverse Gaussian Distribution. Technometrics. 1(34) 74-82. 

Linden M. 2000. Modelling Strike Duration Distribution a Controlled Wiener Process Approach. Applied Stochastic Models in Bussiness and Industry 16 35 5. 

In this chapter, discrete linear-state space models will be discussed and their similarity to ARX models will be shown. In addition Wiener models are introduced. They are suitable for non-linear process modeling and consist of a linear time variant model and a non-linear static model. Several examples show how to develop both types of models. 

The mathematical model of a one dimensional diffusion is the Wiener process (W,), which satisfies the following three conditions (1) Wo = 0, (2) Wt is continuous with independent increments and (3) the trajectory of [Wt+st - Wj] can be sampled from a normal distribution with mean (/u.) of zero and variance (a ) of St (strong Markov property). 

In the limit 5t 0, this random walk becomes a Wiener process. A Wiener process has the same long-time behavior as a random walk with steps of /Ji each St time period, but tile steps are taken infinitely close togetiier. This is unphysical however, when modeling Brownian diffusion, we are really only interested in behavior on time scales longer than the velocity antocorrelation time. 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

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