D d 0 dxfxt m9tx

We continue our proof for non-uniform case. Let fx(t) satisfy Lipschitz condition in some neighborhood of the point ti. We have for ti < t2

i't2 rti gt2 (x) —gt1 (x) = / fx(s) v (u,x) duds— fx (s) v (u,x) duds

J t2—ti Jti—u Jo Jti—u the first member in this sum has an order

The sum of the second and fifth members can be represented as follows ti t2 / v(u,x) (fx(s) — fx(s — u)) dsdu = t2—ti ti

= (t2 - tl) (fx(tl) - fx(h - u)) v(u,x) du + o(t2 - tl) ■Jo because the ratio (fx(s) -fx(s-u))/u is uniformly bounded on the integration domain. The sum of the third and sixth members has the form pt2-tl pt2

/ v(u,x) (fx(s) - fx(s - t2 + tl)) dsdu = J0 Jt2-u rt2-ti rt,2

(t2-ti) v(u,x) (fx(s)-fx(s-t2+ti))/(t2-ti) dsdu = o(t2-ti),

Jo Jt2-u

>o Jt2—u due to the Lipschitz condition and integrability of the function u v(u, x) in the neighborhood of zero. The fourth member is estimated as t pt2 — t1 pt2 s

Jo Jti rt2-ti

< max(v(u,x) : u G (ti,t2)) t — ti) fx(s) ds = o(t2 — ti)

o due to integrability of the density fx(s). So we obtain the formula d f- f dtgt(x) = J (fx(t) — fx(t — s)) v(s,x) ds + fx(t) J v(s,x) ds■ (11)

Evidently, this formula is true for any other initial point xo (0 < xo < x). Comparing this formula with (11), we obtain proof of the theorem.

Multi-dimensional distribution

Let us show that under conditions of lemma 1 there exists a multi-dimensional distribution density of /-process. Let to <ti < ■ ■■,tn, xo < xi < . ..,xn and 0 < hi < xi+i — xi (xn+i = to). Then we have gt I , ■■■, tn

= , „1-TPxo ((Txi <ti , Txi+hi > t1),■■■, (Txn < tn, Txn + hn ^ t n)), hi^o,...,hn^o hi ■■■ hn and also

Pxo ((txi <t\,TXl+hl > ti),...,(TXn <tn,TXn+hn > tn)) =

■J0 Jt,- s xPxi+hi ((Tx2 <t'2,Tx2+h2 > ^^■■A'rxn <t'n,Txn+hn > t'n))

where ti = ti — s — u. From here the formula follows gti,..,tn(xi, ■ ■ ■ ,xnI xo) =

= fxi (s| xo) v(u,xi) gt'2,..,t'n (x2, ■ ■ ■ ,xnI xi) duds■

So we obtain

/n n Axfc(tk - Uk-i,nk — uk-i\ xk-i) duk, (12) k=1

where An = (t\,t2) X ■ ■■ X (tn-l ,tn) X (tn, to), uo = 0,

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