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By Gouet R., Lopez F.J., Sanz G.

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In general however, the influence of the external noise depends on the state of the system. 13). 13), then X^ is one order less irregular than the white noise. It would have been obtained by the integration of the white noise input and hence would have been smoothed. 13) proceeds via the equivalent integral equation ^^: ^ . 14) 0 Formulated in this manner, the question is now how to arrive at a consistent definition of the stochastic integral \g(X^ 4 ds. For the sake of concreteness, let us for the time being consider only the case of continuously varying external parameters, which will in any case occupy a central role in this monograph.

86) and Xf(co) = , 1 t=T{(0) 0 otherwise In words, while the process Xf is zero for all instants of time, the process Xf is obtained from it by modifying it at one particular random point in time, namely by setting the sample path, labeled co, equal to one for t = T{aj). Since the random time variable r is uniformly distributed over the interval [0,1], the probability that it takes the particular value t e [0,1] is of course equal to zero, P[T = t] = 0. , the two processes are equivalent. However, whereas Xf has continuous sample paths, this is not true for Xf.

Oo, z^^) = F(Xi, /'j; . . , the lower members of the hierarchy can be obtained from the higher ones. ;x^,t^)= ldx[... 83) — OO this corresponds to integrating out the variables {x^+i, . . , x j PKP^I 5 ^1 > • • • J ^m> ^m) ~ ] ^-^m + l • • " J CtX^p[Xi, R Tj, . . , X ^ , f^, R . . , X^, f^j . 84) The fundamental theorem of Kolmogorov establishes that the inverse is also true. 82), there exists a probability triple (Q, d, P) and a stochastic process X^ defined on it that possesses the given distribution functions.

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A Characteristic Martingale Relatedtothe Counting Process of Records by Gouet R., Lopez F.J., Sanz G.

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