JOJts

/ fy+h(u\y) fy(s\ x) dsdu + / fy+h(u| y) fy(s\ x) ds du

/ v(u,y) fy(s\x) dsdu + / v(u,y) fy(s\x) dsdu (h ^ 0) Jo Jt-u Jt Jo because both the functions ft_u fy(s\ x) ds/u in the first integral, and

Jo fy(s\x) ds/u in the second integral are continuous and bounded. Consequently, if the conditions of lemma 1 are fulfilled, then

JO Jt-s

Theorem 1. If condition (6) is fulfilled, and the functions fy (i\ x), v(t,x) are continuous in their arguments, besides fy(t\x) is differentiable with respect to t, and fg u v(u, x) du < to, then iyfy (t\x) = — dt where

dyy fy(t\x) = J (fy(t—'s\x)—fy(t\x)) v(s,y) ds—fy(t\x) J v(s,y) ds. (10)

Proof. Let us note that for any point (t, x) (t, x > 0) we have

This identity is valid for any non-decreasing process, beginning from the point (0,0). Moreover, for 0 < ti < t and 0 < xi < x we have

0 = Po(rx < t) + Po(i(t) <x) — (Po(rx < ti) + Po(£(ti) < x)) —

— (Po(rXl < t) + Po(C(t) <xi)) + 1 = = (Po(Tx < t) — Po(Tx < ti)) + (Po(i(t) <x) — Po(£(t) < xi)) — —Po(£(ti) < x) — Po(txi < t) + 1 =

= / fx(s) ds ^ / gt(y) dy — gH (y) dy — fxt (s) ds + 1, Jt 1 Jx1 Jo Jo where for brevity we write fx(t\0) = fx(t) and gt(x\0) = gt(x). If (1/h)(fx(s\0) — fx-h(s\0)) tends uniformly with respect to s G (ti, t] to corresponding partial derivative as h ^ 0, we have

On statistics of inverse gamma process as a model of wear 195 dxJ

rt d

Consequently, if dfx(s)/dx is continuous at the point t we obtain

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