Page 108 - Textos de Matemática Vol. 47

98 A. C. ROSA AND M. E. NOGUEIRA
From the result obtained for FT (·/x), we deduce the consistency of the following estimator of q↵(x):
q↵,T(x)=inf{z2R:FT(z/x) ↵}, ↵2]0,1[. Let us begin by introducing the following -fields:
• Gt = {Xs :0s<t},t>0,andG0 = (X0);
• Ft = {(Xs,Ys):0s<t},t>0,andF0 = (X0,Y0);
• St, = {(Xs,Ys),Xr :0s<t, trt+ }, t > 0, > 0, and S0, = {(X0,Y0),Xr :0r }, >0.
In the sequel, E will denote the real interval [a, b], a < b.
For easy reference, the assumptions needed to derive the announced results are gathered thereafter.
(H1) Forevery t>0, >0,
(i) the conditional density of Xt with respect to Gt , fGt , exists
Xt and is lipschitzian on Rd (if t < 0, take Gt = G0);
(ii) the conditional density of Xt with respect to Ft , fFt , exists
Xt
(iii) the conditional density of (Xt,Yt) with respect to Ft , fFt ,
(ift <0,takeFt =F0); exists; Z T
(Xt ,Yt )
(ift <0,takeSt , =S0, ).
(H3) Forallx2Rd,thereexistrealconstantsC>0, >0and >0such
that, for all (y1, y2) 2 E2 and (x1, x2) 2 V(x) ⇥ V(x), we have
|F(y1/x1) F(y2/x2)|C |y1 y2| +kx1 x2k ,
where V(x) denotes a neigbourhood of x and k · k the euclidean norm
of Rd.
(H4) The kernel K verifies:
(i) K is lipschitzian and has compact support, k⇤ = sup |K(x)|;
(iv) lim 1 fGt (x)dt=g(x)a.s., x2Rd. T!+1T0 Xt
(H2) 8t 0, 8 >0, 8y2E,
E✓G✓y Yt ◆.St , ◆=E✓G✓y Yt ◆.Xt◆ a.s. hT hT
Z
(ii)
kxk K(x) dx < +1.
x2Rd
Rd