Transitional probability densities of the diffusion processes.

The theory and practice of the analysis of independent samples from time series is developedenough well and full also. It it is impossible to tell about dependent samples. That is because the multidimensional distributions of probabilities for samples are a little known for the processeswhich are not the normal. Generally speaking, till now even there are no clear approaches to a problem of designing of multidimensional distributions. If marginal density of probabilities of some process is known, whether a structure of joint density will be invariant with respect to typeof dependence what is characteristic for normal distribution- Probably, it is not so. Usually jointdistributions for same marginal density accept various forms depending on statement of a problem, a method of designing and given properties. A situation more or less clear in a case whensample is observation of Markov process. In this case the multidimensional density is designeduniquely by multiplication on initial marginal density of the subsequent conditional density of probabilities (usually named transitional). Thus, for construction of multidimensional density of sample of Markov process it is necessary to have only marginal and transitive probability densities.Until recently an interest to the diffusion processes, which are widely used as mathematicalmodels of real processes, grows, in particular, at the analysis of financial market indicators. In present paper the attention is concentrated to a problem of determination marginal and transitionalprobability densities for diffusion processes when the diffusion function is described by polynomialsof the second order, and the drift function - polynomials of the first order. In this case probabilitydistributions of processes belong to the class of the distributions named as the Pearson distributions.For this class of processes the majority turning out marginal densities appear widelyknown, therefore it was interesting to find out, what will be transitional densities. Such densitiesfor some special cases are known from the literature. In paper all possible versions of a consideredclass are considered. Transitional density of probabilities of Markov processes of diffusiontype are received in the explicit form and the form of expansion on eigenfunctions. It is shown, that the form of density essentially depends on properties of diffusion function and is classified on properties of its roots on six basic types: 1) diffusion is constant-normal distribution; 2) diffusionis linear - the gamma distribution shifted; 3) diffusion is concave and has two various real roots -the beta distribution; 4) diffusion convex and has two various real roots - the beta distribution of the second kind; 5) diffusion has a double real root - the Pearson distribution of type V; 6) diffusionhas a pair of complex conjugate roots - the Pearson distribution of type IV. Other distributionsare special cases of the listed distributions. The received densities cover all types of densitiesof family of Pearson distributions and are the probability densities that often used in practice. In paper transitional densities for following distributions are considered: Normal, Gamma, Exponential, Erlang, Ґц2-distribution, Beta, Rectangular, Beta of second kind, F-distribution, Pareto, Levy, Studentst-distribution, Cauchy, and also Pearson distributions of IV and V types. Results can form a basis for creation of algorithms for modeling of processes with given marginal and transitional densities.On the other hand the knowledge of marginal and transitional densities allows to construct the joint densities of a demanded order for Markov processes. That allows to design the likelihood functionsfor estimation of parameters of observable processes by their sampling.

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