Download e-book for iPad: Statistics of Random Processes II: Applications by R. S.; Shiryayev, A. N. Liptser

By R. S.; Shiryayev, A. N. Liptser

ISBN-10: 1475742932

ISBN-13: 9781475742930

ISBN-10: 1475742959

ISBN-13: 9781475742954

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Extra info for Statistics of Random Processes II: Applications

Example text

Then, for t < r /\ the values (Yt = Yt- 1 are defined which satisfy- the equation (yo = Yo 1, of 1. O. ). 46), ~)dS}{bo + f~ex{2 {al(u, ~)du ] Af(s, ~) _ b 2b 2(s ( B2(S,~) s I , S exp 2 Therefore, P{ T 00 ~)lds ~»)dS} }[bo + iT B~(S: A 2(S~) ] ~) ds < 0 00. T} = O. ). , ff~o, ~o. WI. w2-measurable at any t) of this system of equations. It is easy to bring out the conditions under which this system has a unique continuous strong solution. 4. Let g(t, x) denote any of the nonanticipative functionals aj(t, x), Aj(t, x), bit, x), B(t, x), i = 0, l,j = 1,2,0 s t s T, x E CT' Assume that: (1) for any x, y E C T , Ig(t, x) - g(t, yW s Ll (2) g2(t, x) ::; Ll f~(xs - 1(1 + Ysf dK(s) x;)dK(s) + L 2(x t - Yt)2; + L 2(1 + x~), where K(s) is some nondecreasing right continuous function, 0 and L l • L2 are constants; s K(s) s 1, (3) (4) M(8'f,n + ~~n) < 00 for some integer n ;::: 1.

The vector m is a vector of the mean values, m = M~, and the matrix R is a matrix of covariances R == cov(~, ~) = M(~ - m)(~ - m)*. Let us note a number of simple properties of Gaussian vectors. 17) and Ml1 = a (2) 2 Let(O,~) = [(0 1 , m~ = M~, M(~ - m~)(~ ••• , + Am, cov('1, '1) = Ok),(~I"'" A cov(~, ~)A*. 18) ~,)]beaGaussianvectorwithm8 = MO, D88 = cov(O,O) = M«(} - m8)«(} - m8)*' D~~ = cov(~, ~) = - m,)* and D8~ = cov«(}, ~) = M«(} - m8)(~ - m~). In algebraic operations, vectors a are regarded as columns, and vectors a* are regarded as rows.

Then I +I IY1(t) - Yz(t) I :::;; 2 t ( 0 t 0 24 lal(s, 2 (s, e) e)1 + 1bB(s, e) A 1(s, e) I) IY1(S) - Y2(s)lds Ai(s, e) B2(S, e) [Yl(S) + Yz(s)] IY1(S) - Y2(s)lds. 2 Uniqueness of solutions of filtering equations Denote ( r1(s,~) = 2 la 1(s, ~)I b2(S, ~) + 1B(s, ~) A1(S,~) I) + AB2(S, i(s, ~) ~) [Y1(S) + Yz(s)J. 44) can be rewritten as follows: f~r1(S' ~)IY1(S) - IY1(t) - Yz(t)1 ::; Y2(s)lds. 43). 42). 45) where r2(s,~) = la1(s, ~)I + 1 b2(S, ~) 1 y(s)A i(s, ~) B(s, ~) A1(S,~) + B2(S, ~) . 45), we find that X1(t) for any t, 0::; t ::; T.

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Statistics of Random Processes II: Applications by R. S.; Shiryayev, A. N. Liptser


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