## Bernoulli

• Bernoulli
• Volume 20, Number 1 (2014), 190-206.

### Bridges of Lévy processes conditioned to stay positive

Gerónimo Uribe Bravo

#### Abstract

We consider Kallenberg’s hypothesis on the characteristic function of a Lévy process and show that it allows the construction of weakly continuous bridges of the Lévy process conditioned to stay positive. We therefore provide a notion of normalized excursions Lévy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Lévy process conditioned to stay positive by using the weakly continuous bridges of the Lévy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper semi-continuous versions. Using the bridges of the conditioned Lévy process, the Durrett–Iglehart theorem stating that the Brownian bridge from $0$ to $0$ conditioned to remain above $-\varepsilon$ converges weakly to the Brownian excursion as $\varepsilon \to0$, is extended to Lévy processes. We also extend the Denisov decomposition of Brownian motion to Lévy processes and their bridges, as well as Vervaat’s classical result stating the equivalence in law of the Vervaat transform of a Brownian bridge and the normalized Brownian excursion.

#### Article information

Source
Bernoulli, Volume 20, Number 1 (2014), 190-206.

Dates
First available in Project Euclid: 22 January 2014

https://projecteuclid.org/euclid.bj/1390407285

Digital Object Identifier
doi:10.3150/12-BEJ481

Mathematical Reviews number (MathSciNet)
MR3160578

Zentralblatt MATH identifier
1296.60121

#### Citation

Uribe Bravo, Gerónimo. Bridges of Lévy processes conditioned to stay positive. Bernoulli 20 (2014), no. 1, 190--206. doi:10.3150/12-BEJ481. https://projecteuclid.org/euclid.bj/1390407285

#### References

• [1] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge: Cambridge Univ. Press.
• [2] Biane, P. (1986). Relations entre pont et excursion du mouvement brownien réel. Ann. Inst. Henri Poincaré Probab. Stat. 22 1–7.
• [3] Blumenthal, R.M. (1983). Weak convergence to Brownian excursion. Ann. Probab. 11 798–800.
• [4] Chaumont, L. (1997). Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 377–403.
• [5] Chaumont, L. and Doney, R.A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948–961.
• [6] Chaumont, L. and Doney, R.A. (2008). Corrections to: “On Lévy processes conditioned to stay positive” [Electron J. Probab. 10 (2005) 948–961]. Electron. J. Probab. 13 1–4.
• [7] Chaumont, L. and Doney, R.A. (2010). Invariance principles for local times at the maximum of random walks and Lévy processes. Ann. Probab. 38 1368–1389.
• [8] Chaumont, L., Hobson, D.G. and Yor, M. (2001). Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes. In Séminaire de Probabilités, XXXV. Lecture Notes in Math. 1755 334–347. Berlin: Springer.
• [9] Chaumont, L. and Uribe Bravo, G. (2011). Markovian bridges: Weak continuity and pathwise constructions. Ann. Probab. 39 609–647.
• [10] Chen, Z.Q. and Song, R. (1997). Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. J. Funct. Anal. 150 204–239.
• [11] Denisov, I.V. (1983). Random walk and the Wiener process considered from a maximum point. Teor. Veroyatn. Primen. 28 785–788.
• [12] Doney, R.A. (2007). Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Berlin: Springer.
• [13] Durrett, R.T., Iglehart, D.L. and Miller, D.R. (1977). Weak convergence to Brownian meander and Brownian excursion. Ann. Probab. 5 117–129.
• [14] Fourati, S. (2005). Vervaat et Lévy. Ann. Inst. Henri Poincaré Probab. Stat. 41 461–478.
• [15] Hunt, G.A. (1956). Some theorems concerning Brownian motion. Trans. Amer. Math. Soc. 81 294–319.
• [16] Kallenberg, O. (1981). Splitting at backward times in regenerative sets. Ann. Probab. 9 781–799.
• [17] Knight, F.B. (1996). The uniform law for exchangeable and Lévy process bridges. Astérisque 236 171–188.
• [18] Miermont, G. (2001). Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps. Electron. J. Probab. 6 33 pp. (electronic).
• [19] Millar, P.W. (1977). Zero–one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226 365–391.
• [20] Pitman, J. and Uribe Bravo, G. (2012). The convex minorant of a Lévy process. Ann. Probab. 40 1636–1674.
• [21] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Berlin: Springer.
• [22] Sharpe, M. (1969). Zeroes of infinitely divisible densities. Ann. Math. Statist. 40 1503–1505.
• [23] Vervaat, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 143–149.
• [24] Vondraček, Z. (2002). Basic potential theory of certain nonsymmetric strictly $\alpha$-stable processes. Glas. Mat. Ser. III 37 211–233.
• [25] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research. New York: Springer.