Jx Jo

Lemma 1. If condition (6) is fulfilled, and functions fy (i| x) and v(t,x) are continuous in their arguments, and ^ u v(u, x) du < tt, then for any t > 0

fy (t| x) ^ v (t,x) (y I x), y-x besides for any continuous bounded function ip y-x p(t) t fy (t| x) dx ^ / p(t) tv(t,x) dx (y j x)

Proof. We have

Px(e-Xry )=exp(— / (1 — e-Xu) v(u,s) duds). Jx Jo

From here it follows that for any Ao > 0 uniformly for all A < Ao

1 fTO 1

Lc(A,x) = - (1 — e-Xu) fx+c(u| x) du = - (1 — Ex(e-XTx+c)) = c J o c

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