Retrieving Information from Subordination
We recall some instances of the recovery problem of a signal process hidden in an observation process. Our main focus is then to show that if \((X_{s},\,s\,\geq \,0)\) is a right-continuous process, \(Y _{t} = \int \limits _{0}^{t}X_{s}\mathrm{d}s\) its i
- PDF / 171,779 Bytes
- 10 Pages / 439.36 x 666.15 pts Page_size
- 39 Downloads / 184 Views
Abstract We recall some instances of the recovery problem of a signal process hidden in an observation process. Our R t main focus is then to show that if .Xs ; s 0/ is a right-continuous process, Yt D 0 Xs ds its integral process and D .u ; u 0/ a subordinator, then the time-changed process .Yu ; u 0/ allows to retrieve the information about .Xv ; v 0/ when is stable, but not when is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable. Keywords Recovery problem • Subordination • Bougerol’s identity
Mathematics Subject Classification (2010): 60G35, 60G51
1 Introduction and Motivations Stemming from Hidden Processes Many studies of random phenomena involve several sources of randomness. To be more specific, a random phenomenon is often modeled as the combination C D ˚.X; X 0 / of two processes X and X 0 which can be independent or correlated,
J. Bertoin () Laboratoire de Probabilit´es et Mod`eles Al´eatoires, UPMC, 4 Place Jussieu, 75252 Paris cedex 05, France e-mail: [email protected] M. Yor Institut Universitaire de France and Laboratoire de Probabilit´es et Mod`eles Al´eatoires, UPMC, 4 Place Jussieu, 75252 Paris cedex 05, France e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 5, © Springer-Verlag Berlin Heidelberg 2013
97
98
J. Bertoin and M. Yor
for some functional ˚ acting on pairs of processes. In this framework, it is natural to ask whether one can recover X from C , and if not, what is the information on X that can be recovered from C ? We call this the recovery problem of X given C . Here are two well-known examples of this problem. • Markovian filtering: There C is Rthe observation process defined for every t 0 t by Ct D St C Bt where St D 0 h.Xs /ds is the signal process arising from a Markov process X and B D .Bt ; t 0/ is an independent Brownian motion. Then the recovery problem translates in the characterization of the filtering process, that is the conditional law of Xt given the sigma-field Ct D .Cs ; s t/. We refer to Kunita [6] for a celebrated discussion. In the simplest case when X remains constant as time passes, which yields h.Xt / A where A is a random variable, note that A can be recovered in infinite horizon by A D limt !1 t 1 Ct , but not in finite horizon. More precisely, it is easily shown that for a Borel function f 0, there is the identity R f .a/E a .da/ E.f .A/ j Ct / D R a t Et .da/ where is the law of A and Eta D exp.aCt ta2 =2/; see Chap. 1 in [9]. • Brownian subordination: An important class of L´evy processes may be represented as Ct D Bt ; t 0; where a subordinator and B is again a Brownian motion (or more generally a L´evy process) which is independent of ; see for instance Chap. 6 in [7]. Geman, Madan and Yor [4, 5] solved the recovery problem of hidden in C ; we refer th
Data Loading...
