## Y AAy Y y z y tYZmL Z dByl Az

Since the process Y is left-continuous, the second term in the right-hand side is a martingale. The first term is continuous and therefore predictable. Hence it equals Bh(z).

The second characteristic, Ch, is a compensator of the process (Ah — Bh)2. By the well-known formula for predictable variation of stochastic integrals,

The third characteristic, v, is a compensator of the jump measure p of the process A. Obviously, p(dy, du) = 1 {AA(y)=0}^' (y,AA(y) (dy^ dx) y>0

If U(y) is a continuous adapted process and f (u) is a deterministic continuous function, then z z n "

U(y)f (u)p(dy, du) = V U(Z^)f(Y-1(ZinH))1{z^.<z}

The first term in the right-hand side is continuous and therefore predictable. The second term is a martingale. The first term can be written

The proof is completed.

To formulate the conditions under which the estimator Ai is consistent, we need the following notation:

Direct calculations show that EY(z) also equals nb(z). Indeed, Z(T) = (T — Sj)/Aj for Sj < T < Sj+1, so b(z) = Y^ E

oo oo

EY\Ak 1

Ej2\Ak 1

Theorem 3. Suppose that

(i) infz<zo b(z) > 0, supz<z0 b(z) < <x>,

(ii) suPz<zo \ n 1 Y(z) - b(z) \P ° suPz<zo \ n 1 Y(z) - b(z) \P 0 as n

Then the estimator Ai is uniformly consistent, i.e.

Proof. By Theorem VIII.2.17 of  and Propositionl it suffices to prove the following:

1) supz<20 | Bh(y) - A(z) 1= supz<z0 | ¡J h(Y-i(y))Y(y)dA(y) - A(z) IP

2) supz<zo I C\(y) I= ¡J0 h2(Y-i(z))Y(z)dA(z) P 0;

3) for each bounded continuous non-negative function g, which equals 0 in some neighborhood of 0, sup I ( i g(x)v(dy,dx) I= g{Y-i(z))Y(z)dA(z) P 0.