Geodesic
On exponential functionals of processes with independent increments
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 2, pp. 330-357 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we study the exponential functionals of the processes $X$ with independent increments, namely, $I_t= \int_0^t\exp\{-X_s\}\,ds,\ t\geq 0$, and also $I_{\infty}= \int_0^{\infty}\exp\{-X_s\}\,ds$. When $X$ is a semimartingale with absolutely continuous characteristics, we derive necessary and sufficient conditions for the existence of the Laplace exponent of $I_t$, and also the sufficient conditions of finiteness of the Mellin transform $\mathbf{E}(I_t^{\alpha})$ with $\alpha\in \mathbf{R}$. We give recurrent integral equations for this Mellin transform. Then we apply these recurrent formulas to calculate the moments. We also present the corresponding results for the exponential functionals of Lévy processes, which hold under less restrictive conditions than in [J. Bertoin and M. Yor, Probab. Surv., 2 (2005), pp. 191–212]. In particular, we obtain an explicit formula for the moments of $I_t$ and $I_{\infty}$, and we give the precise number of finite moments of $I_{\infty}$.
Keywords: exponential functional, process with independent increments, Mellin transform
Mots-clés : Lévy process, moments. 
@article{TVP_2018_63_2_a5,
     author = {P. Salminen and L. Vostrikova},
     title = {On exponential functionals of processes with independent increments},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {330--357},
     year = {2018},
     volume = {63},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_2_a5/}
}
TY  - JOUR
AU  - P. Salminen
AU  - L. Vostrikova
TI  - On exponential functionals of processes with independent increments
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2018
SP  - 330
EP  - 357
VL  - 63
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2018_63_2_a5/
LA  - en
ID  - TVP_2018_63_2_a5
ER  - 
%0 Journal Article
%A P. Salminen
%A L. Vostrikova
%T On exponential functionals of processes with independent increments
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2018
%P 330-357
%V 63
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2018_63_2_a5/
%G en
%F TVP_2018_63_2_a5
P. Salminen; L. Vostrikova. On exponential functionals of processes with independent increments. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 2, pp. 330-357. http://geodesic.mathdoc.fr/item/TVP_2018_63_2_a5/

[1] S. Asmussen, Ruin probabilities, Adv. Ser. Stat. Sci. Appl. Probab., 2, World Sci. Publ., River Edge, NJ, 2000, xii+385 pp. | DOI | MR | Zbl

[2] O. E. Barndorff-Nielsen, A. Shiryaev, Change of time and change of measure, Adv. Ser. Stat. Sci. Appl. Probab., 13, World Sci. Publ., Hackensack, NJ, 2010, xvi+305 pp. | DOI | MR | Zbl

[3] A. Behme, “Exponential functionals of Lévy processes with jumps”, ALEA Lat. Am. J. Probab. Math. Stat., 12:1 (2015), 375–397 | MR | Zbl

[4] A. Behme, A. Lindner, “On exponential functionals of Lévy processes”, J. Theoret. Probab., 28:2 (2015), 681–720 | DOI | MR | Zbl

[5] J. Bertoin, Lévy processes, Cambridge Tracts in Math., 121, Cambridge Univ. Press, Cambridge, 1996, x+265 pp. | MR | Zbl

[6] J. Bertoin, A. Lindner, R. Maller, “On continuity properties of the law of integrals of Lévy processes”, Séminaire de probabilités XLI, Lecture Notes in Math., 1934, Springer, Berlin, 2008, 137–159 | DOI | MR | Zbl

[7] J. Bertoin, M. Yor, “Exponential functionals of Lévy processes”, Probab. Surv., 2 (2005), 191–212 | DOI | MR | Zbl

[8] A. N. Borodin, P. Salminen, Handbook of Brownian motion — facts and formulae, Probab. Appl., 2nd ed., Birkhäuser Verlag, Basel, 2002, xvi+672 pp. ; 2nd ed. corrected, 2015, xvi+677 pp.; A. N. Borodin, P. Salminen, Spravochnik po brounovskomu dvizheniyu. Fakty i formuly, Lan, SPb., 2000, 639 pp. | DOI | MR | Zbl

[9] P. Carmona, F. Petit, M. Yor, “On the distribution and asymptotic results for exponential functionals of Lévy processes”, Exponential functionals and principal values related to Brownian motion, Bibl. Rev. Mat. Iberoamericana, Univ. Autónoma de Madrid, Madrid, 1997, 73–130 | MR | Zbl

[10] P. Carr, L. Wu, “Time-changed Lévy processes and option pricing”, Journal of Financial Economics, 71:1 (2004), 113–141 | DOI

[11] D. Dufresne, “The distribution of a perpetuity, with applications to risk theory and pension funding”, Scand. Actuar. J., 1990:1-2 (1990), 39–79 | DOI | MR | Zbl

[12] I. Epifani, A. Lijoi, I. Prünster, “Exponential functionals and mean of neutral-to-the-right priors”, Biometrika, 90:4 (2003), 791–808 | DOI | MR | Zbl

[13] K. B. Erickson, R. A. Maller, “Generalised Ornstein–Uhlenbeck processes and the convergence of Lévy integrals”, Séminaire de probabilités XXXVIII, Lecture Notes in Math., 1857, Springer, Berlin, 2005, 70–94 | DOI | MR | Zbl

[14] H. K. Gjessing, J. Paulsen, “Present value distributions with applications to ruin theory and stochastic equations”, Stochastic Process. Appl., 71:1 (1997), 123–144 | DOI | MR | Zbl

[15] J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, Grundlehren Math. Wiss., 288, Springer-Verlag, Berlin, 1987, xviii+601 pp. | DOI | MR | MR | MR | Zbl | Zbl

[16] M. Jeanblanc, M. Yor, M. Chesnay, Mathematical methods for financial markets, Springer Finance, Springer-Verlag London, Ltd., London, 2009, xxvi+732 pp. | DOI | MR | Zbl

[17] Yu. Kabanov, S. Pergamenshchikov, “In the insurance business risky investments are dangerous: the case of negative risk sums”, Finance Stoch., 20:2 (2016), 355–379 | DOI | MR | Zbl

[18] C. Kardaras, S. Robertson, “Continuous-time perpetuities and time reversal of diffusions”, Finance Stoch., 21:1 (2017), 65–110 ; arXiv: 1411.7551v1 | DOI | MR | Zbl

[19] A. Kuznetsov, J. C. Pardo, M. Savov, “Distributional properties of exponential functionals of Lévy processes”, Electron. J. Probab., 17 (2012), paper No 8, 35 pp. | DOI | MR | Zbl

[20] A. E. Kyprianou, Fluctuations of Lévy processes with applications. Introductory lectures, Universitext, 2nd ed., Springer, Heildelberg, 2014, xviii+455 pp. | DOI | MR | Zbl

[21] J. Lamperti, “Semi-stable Markov processes. I”, Z. Wahrscheinlichkeitstheor. verw. Geb., 22:3 (1972), 205–225 | DOI | MR | Zbl

[22] A. Novikov, N. Kordzakhia, “Pitman estimators: an asymptotic variance revisited”, Theory Probab. Appl., 57:3 (2013), 521–529 | DOI | DOI | MR | Zbl

[23] J. C. Pardo, V. Rivero, K. van Schaik, “On the density of exponential functionals of Lévy processes”, Bernoulli, 19:5A (2013), 1938–1964 | DOI | MR | Zbl

[24] P. Patie, M. Savov, Bernstein-gamma functions and exponential functionals of Lévy processes, 2016, arXiv: 1604.05960v2

[25] J. Paulsen, “Ruin models with investment income”, Probab. Surv., 5 (2008), 416–434 | DOI | MR | Zbl

[26] P. Salminen, L. Vostrikova, On moments of integral exponential functionals of additive processes, 2018, arXiv: 1803.04859

[27] P. Salminen, O. Wallin, “Perpetual integral functionals of diffusions and their numerical computations”, Stochastic analysis and applications, Abel Symp., 2, Springer, Berlin, 2007, 569–594 | DOI | MR | Zbl

[28] P. Salminen, M. Yor, “Perpetual integral functionals as hitting and occupation times”, Electron. J. Probab., 10 (2005), paper No 11, 371–419 | DOI | MR | Zbl

[29] K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Stud. Adv. Math., 68, 2nd rev. ed., Cambridge Univ. Press, Cambridge, 2013, xiv+521 pp. ; т. 2, Теория, МЦНМО, М., 2016, 458 с. | MR | Zbl

[30] A. N. Shiryaev, Essentials of stochastic finance. Facts, models, theory, Adv. Ser. Stat. Sci. Appl. Probab., 3, World Sci. Publ., River Edge, NJ, 1999, xvi+834 pp. | DOI | MR | Zbl

[31] A. N. Shiryaev, A. S. Cherny, “Vector stochastic integrals and the fundamental theorems of asset pricing”, Proc. Steklov Inst. Math., 237 (2002), 6–49 | MR | Zbl