The absolute continuity relationship: a fi» fi =exp fif t (X t a t) X s ds W a j Ft () is well-known (see, e.g. Yor [4], Chapter ). It also holds with

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1 A Clarification about Hitting Times Densities for OrnsteinUhlenbeck Processes Anja Göing-Jaeschke Λ Marc Yor y Let (U t ;t ) be an OrnsteinUhlenbeck process with parameter >, starting from a R, that is the solution of: U t = a + B t U s ds = e a t + e s db s ; t ; (1) where (B t ;t ) denotes a Brownian motion starting from. Below, we give an expression of the density oftb U = infft : U t = bg, forb R, in terms of some integrals involving the BES bridges of lengths t, starting at (b a), for b a, and conditioned to end at at time t. It is seen on this expression that, in the discussion in Leblanc et al. [7], one term has been omitted, which explains why formula in Leblanc et al. [7] is incorrect, and how to correct it, at least in terms of BES bridges. For general discussions of first hitting times of diffusions we refer to Arbib [1], Breiman [], Horowitz [5], Kent [6], Nobile et al. [8], Novikov [9, 1, 11], Pitman and Yor [13, 14], Ricciardi and Sato [15], Rogers [16], Salminen [17], Shepp [18], Siegert [19], Truman and Williams [1] and Yor [3]. More general discussions of inverse local times and occupation times R T 1 (X s»y)ds, whenx is a diffusion and T a particular stopping time, are dealt with in Hawkes and Truman [4], Truman [] and Truman et al. [], with particular emphasis on the Ornstein Uhlenbeck case. Denote by a the law of U, solution of (1), and by W a = Q () a the law of f(a + B t );t g, both laws being defined on the canonical space C(R + ; R), where X t (!) =!(t), and F t = fffx s ;s» tg. Λ Baloise Insurance Group, Aeschengraben 1, CH-4 Basel, anja.goeingjaeschke@basler.ch; formerly: ETH Zurich, Department of Mathematics, CH-89 Zurich. The author gratefully acknowledges support from RiskLab, Switzerland. y Université Pierre et Marie Curie, Laboratoire de Probabilités, 4 Place Jussieu, F-755 Paris Cedex 5. 1

2 The absolute continuity relationship: a fi» fi =exp fif t (X t a t) X s ds W a j Ft () is well-known (see, e.g. Yor [4], Chapter ). It also holds with t being replaced by T,any stopping time being assumed to be finite both under a and W a. Consequently, the following holds:» a (T b dt) = exp» (b a t) W a exp X s ds ; T b dt : In Leblanc et al. [7], the authors use the (obvious, but important) fact that (X s ;s ) under W a is distributed as (b X s ;s ) under W (ba).thus, may be written as: a (T b dt) =» exp» (b a t) W ba exp (b X s ) ds ; T dt : (4) Now, recall that for c>, under W c, the process (X s ;s» T ) conditioned by (T = t), is a BES bridge of length t, starting at c, and ending at, whose law we denote by P c ( jx t = ). Thus, denoting c = jb aj, weobtain: W ba»exp 8 >< = >: h E c E c (b X s ) ds ; T dt R t exp (b X s) ds h R exp t (b + X s) ds i jx t = (5) W c (T dt); if b a; i (6) jx t = W c (T dt); if b» a: h For b 6= ; the conditional expectations E c R i exp t (b X s) ds jx t = differ from that written in Leblanc et al. [7] at the bottom of p.11, and on top of p.111, where we find instead: E c» Z ds t exp X s jx t = ; (7) which isknown to be equal to (see Pitman and Yor [1], or Yor ([4], formula (.5) for ffi = 3): 3» t exp sinh(t) c t (t coth(t) 1)

3 Thus, to summarize, we first obtain:» 3 a (T dt) =jaj exp(a =) p exp ß (t a coth(t)) dt (8) sinh(t) as found in a number of papers, e.g. Yor ([5], Exercise p. 56), Elworthy et al. [3]. Secondly, the knowledge of the density of a (T b dt)=dt, for all a and b is equivalent to the knowledge of the joint Laplace transform of ( R t X sds; R t X s ds) under the BES bridges laws P c ( jx t = ). References [1] Arbib, M.A. Hitting and martingale characterizations of one-dimensional diffusions. Z. Wahrsch. Verw. Gebiete, 4:347, [] Breiman, L. First exit times from a square root boundary. Fifth Berkeley Symposium, ():916, [3] Elworthy, K.D., Li, X.M., and Yor, M. The importance of strictly local martingales: Applications to radial OrnsteinUhlenbeck processes. Probab. Theory Related Fields, 115:35356, [4] Hawkes, J. and Truman, A. Statistics of local time and excursions for the OrnsteinUhlenbeck process. In Stochastic Analysis, volume 167 of L. M. Soc. Lect. Notes Series, 911. Cambridge University Press, [5] Horowitz, J. Measure-valued random processes. Z. Wahrsch. Verw. Gebiete, 7:1336, [6] Kent, J. Time-reversible diffusions. Adv. Appl. Probab., 1:819835, [7] Leblanc, B., Renault, O., and Scaillet, O. A correction note on the first passage time of an OrnsteinUhlenbeck process to a boundary. Finance and Stochastics, 4:19111,. [8] Nobile, A.G., Ricciardi, L.M., and Sacerdote, L. A note on first-passagetime problems. J. Appl. Prob., :346359, [9] Novikov, A.A. A martingale approach to first passage problems and a new condition for Wald's identity. In Lecture Notes in Control and Information Sci., Volume 36 of Stochastic differential systems (Visegrád, 198), Springer-Verlag Berlin New York, [1] Novikov, A.A. A martingale approach inproblems on first crossing time of nonlinear boundaries. Proc. of the Steklov Institute of Mathematics, 4:141163,

4 [11] Novikov, A.A. Limit theorems for the first passage time of autoregression process over a level. Proc. of the Steklov Institute of Mathematics, 4: , [1] Pitman, J. and Yor, M. A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete, 59:45457, 198. [13] Pitman, J. and Yor, M. Laplace transforms related to excursions of a onedimensional diffusion. Bernoulli, 5():4955, [14] Pitman, J. and Yor, M. Hitting, occupation, and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Prépublication 684 du Laboratoire de Probabilités et Mod eles aléatoires, Universités de Paris 6 & Paris 7, 1, [15] Ricciardi, L.M. and Sato, S. First-passage time density and moments of the OrnsteinUhlenbeck process. J.Appl.Prob, 5:4357, [16] Rogers, L.C.G. Characterizing all diffusions with the M X property. Ann. Probab., 9(4):56157, [17] Salminen, P. On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. Adv. in Appl. Probab., ():41146, [18] Shepp, L. First passage problem for the Wiener process. Ann. Math. Statist., 38: , [19] Siegert, A.J.F. On the first passage time probability problem. Phys.Rev., 81(4):61763, [] Truman, A. Excursion measures for one-dimensional time-homogeneous diffusions with inaccessible and accessible boundaries. In Stoch. Anal. and App. in Physics, , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Kluwer Acad. Publ., [1] Truman, A. and Williams, D. A generalised arc-sine law and Nelson's stochastic mechanics of one-dimensional time-homogeneous diffusions. In: Diffusion processes and related problems in analysis. Progr. Probab., :117135, 199. [] Truman, A., Williams, D., and Yu, K.Y. Schrödinger operators and asymptotics for Poisson-Lévy excursion measures for one-dimensional timehomogeneous diffusions. Proc. of Symposia in Pure Maths., 57:145156,

5 [3] Yor, M. On square-root boundaries for Bessel processes and pole-seeking Brownian motion. In Stochastic Analysis and Applications, Volume 195 of Lecture Notes in Mathematics, 117. Springer-Verlag Berlin Heidelberg, [4] Yor, M. Some Aspects of Brownian Motion, Part I: Some special functionals. Lectures in Mathematics, ETH Zürich. Birkhäuser Verlag Basel, 199. [5] Yor, M. Local Times and Excursions for Brownian Motion. Volume of Lecciones en Matemáticas. Universidad Central de Venezuela,