Conditional Law of the Hitting Time for a Lévy Process in Incomplete Observation
Author(s)
Waly Ngom1,2
Affiliation(s)
1 IMT, University of Toulouse, France. 2 F.S.T, University Cheikh Anta Diop, Dakar, Sénégal.
1 IMT, University of Toulouse, France. 2 F.S.T, University Cheikh Anta Diop, Dakar, Sénégal.
ABSTRACT
We study the default risk in incomplete information. That means we model the value of a firm by a Lévy process which is the sum of a Brownian motion with drift and a compound Poisson process. This Lévy process cannot be completely observed, and another process represents the available information on the firm. We obtain a stochastic Volterra equation satisfied by the conditional density of the default time given the available information. The uniqueness of solution of this equation is proved. Numerical examples of (conditional) density are also given.
KEYWORDS
Conditional Density, Default Time, Lévy Processes, Filtering Theory, Stochastic Voltera Equations
Conditional Density, Default Time, Lévy Processes, Filtering Theory, Stochastic Voltera Equations
Cite this paper
Ngom, W. (2015) Conditional Law of the Hitting Time for a Lévy Process in Incomplete Observation. Journal of Mathematical Finance, 5, 505-524. doi: 10.4236/jmf.2015.55041.
Ngom, W. (2015) Conditional Law of the Hitting Time for a Lévy Process in Incomplete Observation. Journal of Mathematical Finance, 5, 505-524. doi: 10.4236/jmf.2015.55041.
References
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[1] Bertoin, J. (1998) Lévy Processes. Vol. 121, Cambridge University Press, Cambridge. [2] Sato, K. (1999) Lévy Processes and Infinitely Divisible Distribution. Vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Translated from the 1990 Japanese Original, Revised by the Author, 1999. [3] Cont, R. and Tankov, P. (2004) Financial Modeling with Jump Processes. Chapman & Hall/CRC, Boca Raton, London, New York. [4] Duffie, D. and Lando, D. (2001) Term Structure of Credit Spreads with Incomplete Accounting Information. Econometrica, 69, 633-664. http://dx.doi.org/10.1111/1468-0262.00208 [5] Kou, S.G. and Wang, H. (2003) First Passage Time of a Jump Diffusion Process. Advances in Applied Probability, 35, 504-531. http://dx.doi.org/10.1239/aap/1051201658 [6] Bernyk, V., Dalang, R.C. and Peskir, G. (2008) The Law of the Supremum of a Stable Lévy Process with No Negative Jumps. The Annals of Probability, 36, 1777-1789. http://dx.doi.org/10.1214/07-AOP376 [7] Guo, X., Jarrow, R. and Zeng, Y. (2009) Credit Model with Incomplete Information (Earlier Version Information Reduction in Credit Risk Models). Mathematics of Operations Research, 34, 320-332. http://dx.doi.org/10.1287/moor.1080.0361 [8] Dorobantu, D. (2007) Modélisation de risque de défaut en enterprise. Thèse de l’Université de Toulouse 3. [9] Roynette, B., Vallois, P. and Volpi, A. (2008) Asymptotic Behavior of the Hitting Time, Overshoot and Undershoot for Some Lévy Processes. ESAIM: PS, 12, 58-93. [10] Gapeev, P.V. and Jeanblanc, M. (2010) Pricing and Filtering in Two-Dimensional Dividend Switching Model. International Journal of Theoretical and Applied Finance, 13, 1001-1017. http://dx.doi.org/10.1142/S021902491000608X [11] Coutin, L. and Dorobantu, D. (2011) First Passage Time Law for Some Lévy Process with Compound Poisson: Existence of a Density. Bernoulli, 17, 1127-1135. http://dx.doi.org/10.3150/10-BEJ323 [12] Zakai, M. (1969) On the Optimal Filtering Diffusion Process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 11, 230-249. http://dx.doi.org/10.1007/BF00536382 [13] Pardoux, E. (1991) Filtrage non linéaire et équations aux dérivées partielles stochastiques associées. Ecole d’Eté de Probabilités de Saint-Flour-1989, Lecture Notes in Mathematics 1464, Springer-Verlag, Heidelberg, New York. [14] Coutin, L. (1996) Filtrage d’un système càd-làg: Application du calcul des variations stochastiques à l'existence d'une densité. Stochastics and Stochastics Reports, 58, 209-243. http://dx.doi.org/10.1080/17442509608834075 [15] Bain, A. and Crisan, D. (2009) Fundamentals of Stochastic Filtering. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-0-387-76896-0 [16] Karatzas, I. and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus. Second Edition, Springer-Verlag, New York. [17] Doney, R.A. and Kiprianou, A. (2005) Overshoot and Undershoot of Lévy Process. The Annals of Applied Probability, 16, 91-106. http://dx.doi.org/10.1214/105051605000000647 [18] Protter, P. (1985) Volterra equations driven by semi martingale. Annals of Probability, 13, 518-530. http://dx.doi.org/10.1214/aop/1176993006 [19] Revuz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion. Third Edition, Springer-Verlag, Berlin, Heldelberg, New York. http://dx.doi.org/10.1007/978-3-662-06400-9 [20] Protter, P. (2003) Stochastic Integration and Differential Equation. Second Edition, Springer, Berlin. [21] Jeanblanc, M. and Rutkowski, M. (2000) Modeling of Default Risk: Mathematical Tools, Mathematical Finance; Theory and Practice. Modern Mathematics Series, High Education Press.