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By N. I. Portenko

Diffusion tactics function a mathematical version for the actual phenomenon of diffusion. some of the most vital difficulties within the idea of diffusion procedures is the advance of tools for developing those procedures from a given diffusion matrix and a given waft vector. targeting the research of this challenge, this booklet is meant for experts within the conception of random methods and its applications.

A generalized diffusion strategy (that is, a continuing Markov technique for which the Kolmogorov neighborhood features exist within the generalized feel) can function a version for diffusion in a medium relocating in a nonregular approach. the writer constructs generalized diffusion techniques less than assumptions: first, that the diffusion matrix is adequately average; and moment, that the go with the flow vector is a functionality integrable to a few strength, or is a generalized functionality of the kind of the by-product of a degree.

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Generalized Diffusion Processes

Diffusion approaches function a mathematical version for the actual phenomenon of diffusion. some of the most vital difficulties within the conception of diffusion methods is the advance of tools for developing those procedures from a given diffusion matrix and a given float vector. concentrating on the research of this challenge, this publication is meant for experts within the concept of random strategies and its purposes.

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3 the process (Rs (t) , 4(1', QsX) is a martingale, so that ESX RS (T) - 1 for all x c Rm and 0 < s < T < +oo. Therefore, setting P X(A) = RS(T)QsX(d(o), AE A we get a probability measure PS on Os . /llT coincides with PS . §4. CONSTRUCTION OF A SOLUTION 23 Indeed, since RS (T2) = RS (TI) R T, (T2) Qsx-almost surely, we find that for A E % (XA is the indicator function of the event A) Rs(T2)QSX(dw) = ESXXARS(T2) PSX(A) = A = ESX{XARs(7'I )ESX{RT, ESXXARS(TI) = f R(T1)Q(dw) = P X'(A), which is what was required.

44) is established, and the theorem is proved. §7. Construction of a solution in the general case Assume that a function b(t, x), t > 0, x E Rm, whose values are symmetric m x m matrices satisfies the following conditions which will be called Conditions (C) below. CONDITIONS (C). 1) The function b(t, x) is jointly continuous. , (b(t, x)8 , 0) for all 0 E Rm with 6 # 0 ; 3) Forall t>0 and 0,xERm, (b(t, x)0, 0) < c21012, where c2 is a positive constant. As proved in [53], these conditions also ensure the existence and unique- ness for each s > 0 and x c Rm of a probability measure QSX on (S2, A,/S) such that QSX{x(s) = x} = 1 and the process (x(t) - x(s), s , QSX) , t c [s, oo), is a square-integrable martingale with characteris- tic For n = 1, 2, ...

B) The process x(t) - x(s) t - a(u, x(u)) du, t E [s, oo), S is a square-integrable martingale with respect to (,9ts , PsX), with characteristic t b(u, x(u)) du. c) (x (t) , Os , S PsX) is a Markov process. §7. 55) can be taken as an example of such a measure. 4. 4, then for all t > s , x E Rm , and r = 1, 2, .. 49). CHAPTER II The Analytic Method In this chapter we construct a solution of a stochastic differential equation under the assumption that the diffusion matrix is bounded, uniformly nonsingular, and Holder, while the drift coefficient has the same integrability property as in Chapter I.

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