On the Integral of Diffusion Process on Interval with Unattainable Edges Boundaries: Semi-Markov Approach
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ON THE INTEGRAL OF DIFFUSION PROCESS ON INTERVAL WITH UNATTAINABLE EDGES BOUNDARIES: SEMI-MARKOV APPROACH B. P. Harlamov∗
UDC 519.217.4
A one-dimension semi-Markov process of diffusion type is considered. A range of values of this process is an open finite interval. It is supposed to have unattainable edges. The integral of this process as a function of time is studied. The invariance principle for this integral is proved. Bibliography: 3 titles.
Introduction In the present paper, we consider a diffusion process with values in the symmetric interval (−c, c) (c > 0). It is assumed that the boundaries of the interval are unattainable (see, for example, [3]). The integral of this process is studied as a function of time. The relevance of such a study is connected with known problems of mathematical physics. To solve the above problem, we apply the so-called semi-Markov method. This method allows one to obtain results in terms of semi-Markov transition functions. The application of semi-Markov methods always gives a possibility to extend the results to a wider class than the class of Markov processes. This is the class of so-called semi-Markov processes of general form and, in particular, the class of semi-Markov diffusion processes (see, for example, [2]). Formulation of the problem Semi-Markov processes of general form. We consider a measurable space of elementary events D, where every ξ ∈ D is a function of type [0, ∞) → R, which is continuous on the right at any point t ≥ 0 and has a limit on the left at any point t > 0. Assume that the Skorokhod metric and the Borel σ-algebra of the subsets F are defined on D. The probability measure P on this σ-algebra is interpreted as the distribution of some one-dimensional random process. Let x ∈ R, and let Px be the measure of all ξ ∈ D for which ξ(0) = x. The family (Px ) (x ∈ R) is called the distribution of the random process defined up to initial state. For the Markov and semi-Markov processes, the elements of this family satisfy some consistency conditions. For every set S ∈ F, the probabilities (Px (S)) (x ∈ R) can be interpreted as the values of some function defined on R. We assume that this function is measurable with respect to the Borel σ-algebra on R. It follows that for every numerical-valued, F-measurable function φ : D → R, the superposition Pφ (S) is also F-measurable function. This property is important to define a homogeneous semi-Markov property of the family (Px ). Let Xt : D → R be an Fmeasurable function defined for any ξ ∈ D as Xt (ξ) = ξ(t). Let Ft be the σ-algebra generated by all the functions Xs (s ≤ t), and let (Ft )t≥0 be the natural filtration. We define in a standard way the Markov moments with respect to the natural filtration and, for any Markov moment τ , the σ-algebra Fτ of previous events on the set {τ < ∞}. A semi-Markov process of general form is defined to be a a random process given up to initial state and such that for any interval Δ, the consistency condition of the form Px (θτ−1 A | Fτ ) = PXτ (A) ∗
Institute of Problems b.p.harlamov@gmai
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