Generalised Wiener process

The standard Wiener process is a close approximation of the behaviour of asset prices but does not account for some specific aspects of market behaviour. In the first instance, the prices of financial assets do not start at zero, and their price increments have positive mean. The variance of asset price moves is also not always unity. Therefore, the standard Wiener process is replaced by the generalised Wiener process, which describes a variable that may start at something other than zero, and also has incremental changes that have a mean other than zero as well as variances that are not unity. The mean and variance are still constant in a generalised process, which is the same as the standard process, and a different description must be used to describe processes that have variances that differ over time these are known as stochastic integrals (Figure 2.3). 

We now denote the variable as X and for this variable a generalised Wiener process is given by Equation (2.15) ... 

Another type of stochastic process is an Ito process. This a generalised Wiener process where the parameters a and b are functions of the value of the variable X and time t. An ltd process forX can be written as Equation (2.17) ... 

The expected drift rate and variance of an ltd process are liable to change over time indeed the dependence of the expected drift rate and variance on X and t is the main difference between it and a generalised Wiener process. The derivation of ltd s formula is given in Appendix C. 

The spectral density function of the fluctuation can be calculated from the autocorrelation function by the Wiener-Khintchine relation (Wiener, 1930 Khintchine, 1934). The original formulation of the theorem refers to stationary stochastic processes for a possible generalisation see, for example, Lampard, 1954. The relationship connects the autocorrelation function to the spectrum ... 

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... 

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