Inverse process with independent positive increments

Initial definitions

Let & be set of all continuous non-decreasing functions £ : R+ — R+, such that £(t) —> to (t — to); (Px) (x > 0) be consistent semi-Markov family of probability measures on (&, F), where F is Borel sigma-algebra of subsets of the set generated by topology of homogeneous convergency on all bounded intervals. Let us denote tx(£) the first exit time of the process from the interval [0,x) (x > 0).

A random process £, determined by the family of measures (Px), is said to be an inverse process with independent positive increments (I-process for brevity), if the random function tx(£) as a function of x is a (proper) process with independent positive increments. The natural characteristic of the I-process is its the first exit time distribution which is assumed to be absolutely continuous: Px(ry G dt) = fy(t| x) dt, where x < y and for xo < xi < x2 the following equation holds fxi+X2 (t| xo) = fxi (sl xo) fx2 (t — si xi) ds.

It is well known that for Laplace image of this density Levy formula is true (see [sko64, har01])

Px(exp(—\ry)) = exp(—b(\,x,y)) (y > x > 0), where b(X,x,y) = Xa([x,y)) +

x<xi<y a(dx) is a locally finite measure on the half line R+; n(du x dx) is a locally finite measure on the quadrant (0, to) x [0, to) (it is so called Levy measure); (xi,Ti) is a sequence of pairs, where (xi) (xi > 0) is determinate sequence, describing space points of temporary stops of the process, (t1) is a sequence of independent positive random values, determining intervals of constancy (durations of temporary stops) at the corresponding space points of the trajectory. In the neighborhood of the line {0} x [0, to) Levy measure can be infinite, however it satisfies the conditions

We will assume that the process (tx) (x G R+) is stochastically continuous, thus the third member of exponent power in Levy formula is absent. Besides we consider the case, when measures a and n are absolutely continuous in their domains with respect to Lebesgue measures. In this case for some positive functions a and v the following representations hold

Because of positiveness of increments of the process tx (P0-almost sure) the following property is true: if the function a = 0 then for any x v(u,x) ^ to

A typical /-process is not Markov. According to our terminology (see [har01]) it is monotone continuous semi-Markov process. Violation of the Markov property is connected with intervals of constancy, which either do not have fixed position in the space, or are distributed with respect to an arbitrary law (not necessary exponential). For intervals of constancy one can easier find a reasonable physical interpretation than that for point of jumps. But the main merit of /-process is its analytical form permitting simple evaluation of reliability functionals for degradation problems (see [har04d]).

Moments of the first exit time distributions

Moments of distribution of the random value tx can be find from Levy formula by means of differentiating it with respect to X as X = 0. So we have

un(du x [0,x)) < to, n([1, to) x [0,x)) < to.

as u

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