Explicit construction of a dynamic Bessel bridge of dimension $3$

Luciano Campi (University Paris 13)
Umut Cetin (London School of Economics)
Albina Danilova (London School of Economics)


Given a deterministically time-changed Brownian motion $Z$ startingfrom $1$, whose time-change $V(t)$ satisfies $V(t) > t$ for all $t > 0$, 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 $V(\tau)$, where $\tau := \inf\{t>0: Z_t =0\}$. 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 $V(\tau)$.  We call this a dynamic  Bessel bridge since $V(\tau)$ is not known in advance.  Our study is motivated by insider trading models with default risk, where the insider observes the firm's value continuously on time. The financial application, which uses results proved in the present paper, has been developed in a companion paper.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-25

Publication Date: February 27, 2013

DOI: 10.1214/EJP.v18-1907


  • Back, K., and H. Pedersen (1998): Long-lived information and intraday patterns. phJournal of Financial Market, 1, 385-402.
  • Bertoin, J.; Doney, R. A. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994), no. 4, 2152--2167. MR1331218
  • Bertoin, Jean; Pitman, Jim; Ruiz de Chavez, Juan. Constructions of a Brownian path with a given minimum. Electron. Comm. Probab. 4 (1999), 31--37 (electronic). MR1703609
  • Bielecki, Tomasz R.; Rutkowski, Marek. Credit risk: modelling, valuation and hedging. Springer Finance. Springer-Verlag, Berlin, 2002. xviii+500 pp. ISBN: 3-540-67593-0 MR1869476
  • Campi, Luciano; Çetin, Umut. Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling. Finance Stoch. 11 (2007), no. 4, 591--602. MR2335835
  • Campi, L., Cetin, U., and Danilova, A. (2012): Equilibrium model with default and insider's dynamic information. Published online in Finance and Stochastics, http://dx.doi.org/10.1007/s00780-012-0196-x
  • Carr, Peter; Linetsky, Vadim. A jump to default extended CEV model: an application of Bessel processes. Finance Stoch. 10 (2006), no. 3, 303--330. MR2244347
  • Chaumont, L. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996), no. 1, 39--54. MR1419491
  • Chaumont, L.; Doney, R. A. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005), no. 28, 948--961. MR2164035
  • Doob, J. L. Classical potential theory and its probabilistic counterpart. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 262. Springer-Verlag, New York, 1984. xxiv+846 pp. ISBN: 0-387-90881-1 MR0731258
  • Föllmer, Hans; Wu, Ching-Tang; Yor, Marc. Canonical decomposition of linear transformations of two independent Brownian motions motivated by models of insider trading. Stochastic Process. Appl. 84 (1999), no. 1, 137--164. MR1720102
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2 MR1876169
  • Kallianpur, Gopinath. Stochastic filtering theory. Applications of Mathematics, 13. Springer-Verlag, New York-Berlin, 1980. xvi+316 pp. ISBN: 0-387-90445-X MR0583435
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 MR1121940
  • Kurtz, T. G.; Ocone, D. L. Unique characterization of conditional distributions in nonlinear filtering. Ann. Probab. 16 (1988), no. 1, 80--107. MR0920257
  • Mansuy, Roger; Yor, Marc. Random times and enlargements of filtrations in a Brownian setting. Lecture Notes in Mathematics, 1873. Springer-Verlag, Berlin, 2006. xiv+158 pp. ISBN: 978-3-540-29407-8; 3-540-29407-4 MR2200733
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Version 2.1. Corrected third printing. Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. xiv+419 pp. ISBN: 3-540-00313-4 MR2273672
  • Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4 MR0532498
  • Yor, Marc. Some aspects of Brownian motion. Part II. Some recent martingale problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1997. xii+144 pp. ISBN: 3-7643-5717-7 MR1442263

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.