# On infinitely divisible semimartingales

- 356 Downloads
- 3 Citations

## Abstract

Stricker’s theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker’s theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosiński (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of càdlàg infinitely divisible processes given in Basse-O’Connor and Rosiński (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.

## Keywords

Semimartingales Infinitely divisible processes Stationary processes Fractional processes## Mathematics Subject Classification

60G48 60H05 60G51 60G17## Notes

### Acknowledgments

Jan Rosiński’s research was partially supported by a Grant #281440 from the Simons Foundation.

## References

- 1.Barndorff-Nielsen, O.E.: Superposition of Ornstein–Uhlenbeck type processes. Teor. Veroyatnost. i Primenen.
**45**(2), 289–311 (2000)MathSciNetCrossRefGoogle Scholar - 2.Basse, A.: Gaussian moving averages and semimartingales. Electron. J. Probab.
**13**(39), 1140–1165 (2008)MathSciNetzbMATHGoogle Scholar - 3.Basse, A.: Spectral representation of Gaussian semimartingales. J. Theor. Probab.
**22**(4), 811–826 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Basse, A., Pedersen, J.: Lévy driven moving averages and semimartingales. Stoch. Process. Appl.
**119**(9), 2970–2991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Basse-O’Connor, A.: Representation of Gaussian semimartingales with application to the covariance function. Stochastics
**82**(4), 381–401 (2010)MathSciNetzbMATHGoogle Scholar - 6.Basse-O’Connor, A., Graversen, S.-E.: Path and semimartingale properties of chaos processes. Stoch. Process. Appl.
**120**(4), 522–540 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Basse-O’Connor, A., Rosiński, J.: Characterization of the finite variation property for a class of stationary increment infinitely divisible processes. Stoch. Process. Appl.
**123**(6), 1871–1890 (2013)Google Scholar - 8.Basse-O’Connor, A., Rosiński, J.: On the uniform convergence of random series in Skorohod space and representations of càdlàg infinitely divisible processes. Ann. Probab.
**41**(6), 4317–4341 (2013)Google Scholar - 9.Beiglböck, M., Schachermayer, W., Veliyev, B.: A direct proof of the Bichteler–Dellacherie theorem and connections to arbitrage. Ann. Probab.
**39**(6), 2424–2440 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Bender, C., Lindner, A., Schicks, M.: Finite variation of fractional Lévy processes. J. Theor. Probab.
**25**(2), 594–612 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Bichteler, K.: Stochastic integration and \(L^p\)-theory of semimartingales. Ann. Probab.
**9**(1), 49–89 (1981)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989)Google Scholar
- 13.Cheridito, P.: Gaussian moving averages, semimartingales and option pricing. Stoch. Process. Appl.
**109**(1), 47–68 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Cherny, A. : When is a moving average a semimartingale? MaPhySto Research Report 2001-28 (2001)Google Scholar
- 15.Çinlar, E., Jacod, J., Protter, P., Sharpe, M.J.: Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete
**54**(2), 161–219 (1980)Google Scholar - 16.Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete
**49**(3), 335–347 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Emery, M.: Covariance des semimartingales gaussiennes. C. R. Acad. Sci. Paris Sér. I Math.
**295**(12), 703–705 (1982)MathSciNetzbMATHGoogle Scholar - 18.Fasen, V., Klüppelberg, C.: Extremes of supOU processes. In: Stochastic Analysis and Applications, Abel Symp., vol. 2, pp. 339–359. Springer, Berlin (2007)Google Scholar
- 19.Gal’chuk, L.I.: Gaussian semimartingales. In: Statistics and Control of Stochastic Processes (Moscow, 1984), Transl. Ser. Math. Eng., pp. 102–121. Optimization Software, New York (1985)Google Scholar
- 20.Hida, T., Hitsuda, M.: Gaussian processes. Translations of Mathematical Monographs, vol. 120. American Mathematical Society, Providence (1993). (Translated from the 1976 Japanese original by the authors)Google Scholar
- 21.Hudson, W.N., Tucker, H.G.: Asymptotic independence in the multivariate central limit theorem. Ann. Probab.
**7**(4), 662–671 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 2nd edn, vol. 288. Springer, Berlin (2003)Google Scholar
- 23.Jain, N.C., Monrad, D.: Gaussian quasimartingales. Z. Wahrsch. Verw. Gebiete
**59**(2), 139–159 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Jeulin, T., Yor, M.: Moyennes mobiles et semimartingales. In: Séminaire de Probabilités, XXVII. Lecture Notes in Math., vol. 1557, pp. 53–77. Springer, Berlin (1993)Google Scholar
- 25.Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York). Springer, New York (1997)Google Scholar
- 26.Kardaras, C., Platen, E.: On the semimartingale property of discounted asset-price processes. Stoch. Process. Appl.
**121**(11), 2678–2691 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Knight, F.B.: Foundations of the Prediction Process, Oxford Studies in Probability, vol. 1. The Clarendon Press Oxford University Press, Oxford Science Publications, New York (1992)Google Scholar
- 28.Ledoux, M., Talagrand, M.: Probability in Banach spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23. Springer, Berlin (1991). (Isoperimetry and processes)Google Scholar
- 29.Liptser, R.S., Shiryayev, A.: Theory of Martingales, Mathematics and its Applications (Soviet Series), vol. 49. Kluwer Academic Publishers Group, Dordrecht (1989)Google Scholar
- 30.Meyer, P.-A.: Un résultat d’approximation. In: Séminaire de Probabilités, XVIII. Lecture Notes in Math., vol. 1059, pp. 268–270. Springer, Berlin (1984)Google Scholar
- 31.Mijatović, A., Urusov, M.:On the loss of the semimartingale property at the hitting time of a level. J. Theor. Probab. (2014). doi: 10.1007/s10959-013-0527-7
- 32.Protter, P.E.: Stochastic integration and differential equations. Stochastic Modelling and Applied Probability, 2nd edn, vol. 21 of Applications of Mathematics (New York). Springer, Berlin (2004)Google Scholar
- 33.Rajput, B.S., Rosiński, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields
**82**(3), 451–487 (1989)CrossRefzbMATHGoogle Scholar - 34.Rosiński, J.: On path properties of certain infinitely divisible processes. Stoch. Process. Appl.
**33**(1), 73–87 (1989)CrossRefzbMATHGoogle Scholar - 35.Rosiński, J.: On series representations of infinitely divisible random vectors. Ann. Probab.
**18**(1), 405–430 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 36.Rosiński, J.: Series representations of Lévy processes from the perspective of point processes. In: Lévy Processes, pp. 401–415. Birkhäuser, Boston (2001)Google Scholar
- 37.Sato, K.: Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999). (Translated from the 1990 Japanese original, revised by the author)Google Scholar
- 38.Schnurr, A.: On the semimartingale nature of Feller processes with killing. Stoch. Process. Appl.
**122**(7), 2758–2780 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 39.Stricker, C.: Semimartingales gaussiennes—application au problème de l’innovation. Z. Wahrsch. Verw. Gebiete
**64**(3), 303–312 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 40.Surgailis, D., Rosiński, J., Mandrekar, V., Cambanis, S.: Stable mixed moving averages. Probab. Theory Relat. Fields
**97**(4), 543–558 (1993)CrossRefzbMATHGoogle Scholar