nach oben

Finance and Stochastics

Erschienen in:

04.12.2019

Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process

verfasst von: Yuri Kabanov, Serguei Pergamenshchikov

Erschienen in: Finance and Stochastics | Ausgabe 1/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We study the asymptotics of the ruin probability for a process which is the solution of a linear SDE defined by a pair of independent Lévy processes. Our main interest is a model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let \(\beta >0\) be the root of the cumulant-generating function \(H\) of the increment \(V_{1}\) of the log-price process. We show that the ruin probability admits the exact asymptotic \(Cu^{-\beta }\) as the initial capital \(u\to \infty \), assuming only that the law of \(V_{T}\) is non-arithmetic without any further assumptions on the price process.

Vorheriger Artikel A Black–Scholes inequality: applications and generalisations

Nächster Artikel Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

That is, the distribution is not concentrated on a set \({{\mathbb{Z}}d}=\{0,\pm d, \pm 2d,\dots \}\) for some \(d\).

Other truncation functions are also used in the literature; see e.g. [32].

Albrecher, H., Badescu, A., Landriault, D.: On the dual risk model with taxation. Insur. Math. Econ. 42, 1086–1094 (2008) CrossRef

Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific, Singapore (2010) CrossRef

Avanzi, B., Gerber, H.U., Shiu, E.S.W.: Optimal dividends in the dual model. Insur. Math. Econ. 41, 111–123 (2007) MathSciNetCrossRef

Bankovsky, D., Klüppelberg, C., Maller, R.: On the ruin probability of the generalised Ornstein–Uhlenbeck process in the Cramér case. J. Appl. Probab. 48A, 15–28 (2011) MathSciNetCrossRef

Bayraktar, E., Egami, M.: Optimizing venture capital investments in a jump diffusion model. Math. Methods Oper. Res. 67, 21–42 (2008) MathSciNetCrossRef

Behme, A., Lindner, A., Maejima, M.: On the range of exponential functionals of Lévy processes. In: Donati-Martin, C., et al. (eds.) Séminaire de Probabilités XLVIII. Lecture Notes in Mathematics, vol. 2168, pp. 267–303. Springer, Berlin (2016) CrossRef

Belkina, T.A., Konyukhova, N.B., Kurochkin, S.V.: Dynamical insurance models with investment: constrained singular problems for integrodifferential equations. Comput. Math. Math. Phys. 56, 43–92 (2016) MathSciNetCrossRef

Bichteler, K., Jacod, J.: Calcul de Malliavin pour les diffusions avec sauts: existence d’une densité dans le cas unidimensionnel. In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités XVII. Lecture Notes in Mathematics, vol. 986, pp. 132–157. Springer, Berlin (1983)

Buraczewski, D., Damek, E.: A simple proof of heavy tail estimates for affine type Lipschitz recursions. Stoch. Process. Appl. 127, 657–668 (2017) MathSciNetCrossRef

10.

Cont, R., Tankov, P.: Financial Modeling with Jump Processes. Financial Mathematics Series. Chapman & Hall/CRC Press, London/Boca Raton (2004) MATH

11.

Dufresne, D.: The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuar. J. 1990, 39–79 (1990) MathSciNetCrossRef

12.

Elton, J.H.: A multiplicative ergodic theorem for Lipschitz maps. Stoch. Process. Appl. 34, 39–47 (1990) MathSciNetCrossRef

13.

Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn. Wiley, New York (1971) MATH

14.

Frolova, A., Kabanov, Yu.: Pergamenshchikov S.: in the insurance business risky investments are dangerous. Finance Stoch. 6, 227–235 (2002) MathSciNetCrossRef

15.

Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166 (1991) MathSciNetCrossRef

16.

Grandell, I.: Aspects of Risk Theory. Springer, Berlin (1990) MATH

17.

Grincevic̆ius, A.K.: One Limit Theorem for a Random Walk on the Line. Institute of Physics and Mathematics, Academy of Sciences of the Lithuanian SSR, vol. 15, pp. 79–91 (1975)

18.

Guivarc’h, Y., Le Page, E.: On the homogeneity at infinity of the stationary probability for an affine random walk. In: Bhattacharya, S., et al. (eds.) Recent Trends in Ergodic Theory and Dynamical Systems. Contemporary Mathematics, pp. 119–130. AMS, Providence (2015)

19.

Iksanov, A., Polotskiy, S.: Tail behavior of suprema of perturbed random walks. Theory Stoch. Process. 21(37)(1), 12–16 (2016) MathSciNetMATH

20.

Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2002) MATH

21.

Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky investments are dangerous: the case of negative risk sums. Finance Stoch. 20, 355–379 (2016) MathSciNetCrossRef

22.

Kalashnikov, V., Norberg, R.: Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98, 211–228 (2002) MathSciNetCrossRef

23.

Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973) MathSciNetCrossRef

24.

Klüppelberg, C., Kyprianou, A.E., Maller, R.A.: Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14, 1766–1801 (2004) MathSciNetCrossRef

25.

Lamberton, D., Lapeyre, B.: Introduction to Stochastic Calculus Applied to Finance, 1st edn. Chapman & Hall, London (1996) MATH

26.

Marinelli, C., Röckner, M.: On maximal inequalities for purely discontinuous martingales in infinite dimensions. In: Donati-Martin, C., et al. (eds.) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics, vol. 2123, pp. 293–315. Springer, Berlin (2014) CrossRef

27.

Novikov, A.A.: On discontinuous martingales. Theory Probab. Appl. 20, 11–26 (1975) MathSciNetCrossRef

28.

Nyrhinen, H.: On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83, 319–330 (1999) MathSciNetCrossRef

29.

Nyrhinen, H.: Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, 265–285 (2001) MathSciNetCrossRef

30.

Paulsen, J.: Risk theory in a stochastic economic environment. Stoch. Process. Appl. 46, 327–361 (1993) MathSciNetCrossRef

31.

Paulsen, J.: Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Process. Appl. 75, 135–148 (1998) MathSciNetCrossRef

32.

Paulsen, J.: On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Probab. 12, 1247–1260 (2002) MathSciNetCrossRef

33.

Paulsen, J., Gjessing, H.K.: Ruin theory with stochastic return on investments. Adv. Appl. Probab. 29, 965–985 (1997) MathSciNetCrossRef

34.

Pergamenshchikov, S., Zeitouni, O.: Ruin probability in the presence of risky investments. Stoch. Process. Appl. 116, 267–278 (2006). Erratum to: Ruin probability in the presence of risky investments 119, 305–306 (2009) MathSciNetCrossRef

35.

Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) MATH

36.

Saxén, T.: On the probability of ruin in the collective risk theory for insurance enterprises with only negative risk sums. Scand. Actuar. J. 1948, 199–228 (1948) MathSciNetCrossRef

Titel: Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process
verfasst von: Yuri Kabanov
Serguei Pergamenshchikov
Publikationsdatum: 04.12.2019
Verlag: Springer Berlin Heidelberg
Erschienen in: Finance and Stochastics / Ausgabe 1/2020
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-019-00413-3

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 1/2020

Pathwise superhedging on prediction sets

The value of a liability cash flow in discrete time subject to capital requirements

Linear credit risk models

Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

A Black–Scholes inequality: applications and generalisations

On the quasi-sure superhedging duality with frictions