Local Behaviour of First Passage Probabilities - MIMS - The ...

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in t; of the LHS of (34) is the same as that of the RHS, which isk 7Z x= k 7Z xZ 1y=w + t=0Z 1y=w + t=0e qt Z ts=0e qt U(dt; dy)U(ds; dy)U (dt s; y dw)Z 1s=0e qs U (ds; y dw): (35)Note that if we introduce an independent Exp(q) random variable e q the Wiener-Hopf factorisation allows us to write X eq = S eqe Seq ; where S denotes thesupremum process of X and S ~ denotes an independent copy of the supremumprocess S of X: Let and denote the bivariate Laplace exponents of theladder processes of X and X :Then, using the identity (q; 0) (q; 0) = q=k 7which follows from the Wiener-Hopf factorisation, (see e.g. (3), p166 of [5]),Z 1t=0But (1), p 163 of [5] givessoe qt P (X t 2 dw; t < T x )dtE(e Seq )(q; 0)== q 1 P (X eq 2 dw; e q < T x )= q 1 P (S eq x; S eqe Seq 2 dw)Z x= q 1 P (S eq 2 dy)P (S eq2 y dw)wZ + xP (S eq 2 dy) P (S eq2 y dw)= k 7 :w (q; 0) (q; 0)+Z11(q; ) =P (S eq 2 dy)=(q; 0)0Z 10Z 10e (qt+y) U(dt; dy);e qt U(dt; dy):Using the analogous expression for P (S eq2 y dw); (35) is immediate, and then(34) follows. Specializing this to the stable case then gives (32). Finally, if wewrite n for the characteristic measure of the excursions away from zero of X I;with I denoting the in…mum process of X, then p t (dx) := n(" t 2 dx)=n( > t)is a probabilty measure which coincides with that of the meander of length t attime t in the stable case. (Here denotes the life length of the generic excursion":) In the special case of spectrally negative Lévy processes, we havep t (dx) = Cxt 1 P (X t 2 dx)=n( > t)= Ch x (t)dx=n( > t):The …rst equality here comes from Cor 4 in [2], and the second is the Lévyversion of the ballot theorem (see Corollary 3, p 190 of [5]). Specialising to thestable case and t = 1 gives (33). (I owe this observation to Loic Chaumont.)9
4 Proof of Proposition 12Proof. We will be applying the results in Lemma 10 to formula (17) and wewrite the RHS of (17) as P 1 + P 2 + P 3 ; where with 2 (0; 1=2);andP i = X Xg(r; x z)g (n r; y z); 1 i 3;y^xr2A i z=0A 1 = f0 r ng; A 2 = fn < r (1 )ng; and A 3 = f(1 )n < r ng:(i) We introduce d(); an increasing and continuous function which satis…esd(n) = nc n ; and with bc standing for the integer part function, use Lemma 10to writeXy^xP 1 f(0)=bncXz=0 r=0V (y z)g(r; x z)d(n r)y^xbncf(0) X XV (y z) g(r; x z)d(n(1 ))z=0r=0f(0) Xy^xV (y z)[u(x z)d(n(1 ))z=01Xr=bnc+1g(r; xz)]:We can apply Lemma 10 again to get the estimate, uniform for w=c n ! 0;1Xr=bnc+1Since we know thatg(r; w) 1Xr=bnc+1nu(n)lim infn!1 U(n) > 0;f(0)U(w) Cf(0) U(w):rc r(see e.g. Theorem 8.7.4 in [6]) we see that this is o(u(w)). Also we haveXy^xbncXz=0 r=0V (y z)g(r; x z)d(n r) 1d(n)Xy^xbncXz=0 r=0c ng(r; x z)V (y z);so we see thatd(n)P 1limn; f(0) P y^x= 1; (36)V (y z)u(x z)z=010
Page 1: Local Behaviour of First Passage Pr
Page 5: From this we get immediately a stre
Page 8 and 9: From this we can deduce the followi
Page 12 and 13: where lim n; () = 1 is shorthand fo
Page 14 and 15: (iii) In this case it is P 2 that d
Page 16 and 17: so one part of (45) will follow if
Page 18 and 19: 5.3 Proof when x=c n ! 1Proof. This
Page 20 and 21: 6 The non-lattice caseWe indicate h
Page 22 and 23: An asymptotic lower bound is given
Page 24 and 25: References[1] V. I. Afanasyev, C. B

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