Stochastic Models Based on Asymptotic Polystochastic Processes

2 Stochastic Models Based on Asymptotic Polystochastic Processes 

For the derivation of one asymptotic variant of a given polystochastic model of a process, we can use the perturbation method. For this transformation, a new time variable is introduced into the stochastic model and then we analyze its behaviour. The new time variable is t = eT, which includes the time evolution t and an arbitrary parameter e, which allows the observation of the model behaviour when its values become very small (e— 0). Here, we study the changes in the operator 0(t, t) when e 0 whilst paying attention to having stable values for t/e or t/e.  

Two different types of asymptotic transformation methods can be used depending on the ratio of t/e used in the first type we operate with fixed values of t/e whereas in the second type we consider t/ e.  

As an example, we show the equation that characterizes a random evolution (see relation (4.90)) written without the arguments for the operator 0(t, t), but developed with the operator V(X(t)). We also consider that, when the random process changes, the operator 0(r,t) will be represented by an identity operator (I = I(T,t))  

Each operator considered in the total operator V(X(t),t) keeps its own mean action when it is applied to one parameter (for example the mean action of operator Vj on the parameter (function) f will be written as follows  

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