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Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Anita Diana Behme
Anita Diana Behme*
Affiliation:
Technische Universität Braunschweig
*
∗ Postal address: Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstrasse 14, D-38106 Braunschweig, Germany. Email address: a.behme@tu-bs.de
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Abstract

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For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

Keywords

Stochastic differential equationgeneralized Ornstein-Uhlenbeck processstationaritymoment conditionstail behavior

MSC classification

Primary: 60G10: Stationary processes 60G51: Processes with independent increments; L'evy processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 688 - 711
Copyright
Copyright © Applied Probability Trust 2011 

References

Alsmeyer, G., Iksanov, A. and Rösler, U. (2009). On distributional properties of perpetuities. J. Theoret. Prob. 22, 666–682.Google Scholar
Behme, A., Lindner, A. and Maller, R. (2011). Stationary solutions of the stochastic differential equation dV t =V t -dU t + dL t with Lévy noise. Stoch. Process. Appl. 121, 91–108.CrossRefGoogle Scholar
Bertoin, J., Lindner, A. and Maller, R. (2008). On continuity properties of the law of integrals of Lévy processes. In Séminaire de Probabilités XLI (Lecture Notes Math. 1934), eds Donati-Martin, C. et al., Springer, Berlin, pp. 137–159.CrossRefGoogle Scholar
Bichteler, K. (2002). Stochastic Integration with Jumps. Cambridge University Press.CrossRefGoogle Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.CrossRefGoogle Scholar
Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480.CrossRefGoogle Scholar
Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob. 28, 1195–1218.CrossRefGoogle Scholar
Grincevičius, A. K. (1974). On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines. Theory Prob. Appl. 19, 163–168.CrossRefGoogle Scholar
Grincevičius, A. K. (1980). Products of random affine transformations. Lithuanian Math. J. 20, 279–282.CrossRefGoogle Scholar
Jaschke, S. (2003). A note on the inhomogeneous linear stochastic differential equation. Insurance Math. Econom. 32, 461–464.CrossRefGoogle Scholar
Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta. Math. 131, 207–248.Google Scholar
Lindner, A. and Maller, R. (2005). Lévy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes. Stoch. Process. Appl. 115, 1701–1722.Google Scholar
Maller, R. A., Müller, G. and Szimayer, A. (2009). Ornstein–Uhlenbeck processes and extensions. In Handbook of Financial Time Series, eds Andersen, T. et al., Springer, Berlin, pp. 421–437.Google Scholar
Meyer, P.-A. (1978). Inégalités de normes pour les intégrales stochastiques. In Séminaire de Probabilités XII (Lecture Notes Math. 649), eds Dold, A. and Eckmann, B., Springer, Berlin, pp. 757–762.Google Scholar
Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Protter, P. E. and Shimbo, K. (2008). No arbitrage and general semimartingales. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Inst. Math. Statist. Collect. 4), Institute of Mathematical Statistics, Beachwood, OH, pp. 267–283.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar