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Explicit Construction of a Dynamic Bessel Bridge of Dimension 3

Le : 18/04/2011 11h00
Par : Albina Danilova (London School of Economics)
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Lien web :
Résumé : Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies V (t) > t for all t\geq 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V(s), where s:= inf {t > 0 : Z_t = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V (s). We call this a dynamic Bessel bridge since V (s) is not known in advance. Our study is motivated by insider trading models with default risk.(This is a joint work with Luciano Campi and Umut Cetin)