Bernoulli

  • Bernoulli
  • Volume 6, Number 3 (2000), 401-434.

Stochastic integral equations without probability

Thomas Mikosch and Rimas Norvaiša

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Abstract

A pathwise approach to stochastic integral equations is advocated. Linear extended Riemann-Stieltjes integral equations driven by certain stochastic processes are solved. Boundedness of the p-variation for some 0<p<2 is the only condition on the driving stochastic process. Typical examples of such processes are infinite-variance stable Lévy motion, hyperbolic Lévy motion, normal inverse Gaussian processes, and fractional Brownian motion. The approach used in the paper is based on a chain rule for the composition of a smooth function and a function of bounded p-variation with 0<p<2.

Article information

Source
Bernoulli, Volume 6, Number 3 (2000), 401-434.

Dates
First available in Project Euclid: 10 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1081616698

Mathematical Reviews number (MathSciNet)
MR2001h:60100

Zentralblatt MATH identifier
0963.60060

Keywords
chain rule extended Riemann-Stieltjes integral fractional Brownian motion Lévy process p-variation stable process stochastic integral equation

Citation

Mikosch, Thomas; Norvaiša, Rimas. Stochastic integral equations without probability. Bernoulli 6 (2000), no. 3, 401--434. https://projecteuclid.org/euclid.bj/1081616698


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References

  • [1] Barndorff-Nielsen, O.E. (1978) Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist., 5, 151-157.
  • [2] Barndorff-Nielsen, O.E. (1986) Sand, wind and statistics. Acta Mech., 64, 1-18.
  • [3] Bardorff-Nielsen, O.E. (1995) Normal/inverse Gaussian processes and the modelling of stock returns. Research Report No. 300, Department of Theoretical Statistics, Aarhus University.
  • [4] Barndorff-Nielsen, O.E. (1997) Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist., 24, 1-13.
  • [5] Bertoin, J. (1996) Lévy Processes. Cambridge: Cambridge University Press.
  • [6] Bretagnolle, J. (1972) p-variation de fonctions aléatoires, 2ème partie: processus à accroissements indépendants. In Séminaire de Probabilités VI, Lecture Notes in Math. 258, pp. 64-71. Berlin: Springer-Verlag.
  • [7] Cutland, N.J., Kopp, P.E. and Willinger, W. (1995) Stock price returns and the Joseph effect: a fractional version of the Black-Scholes model. In E. Bolthausen, M. Dozzi and F. Russo (eds), Seminar on Stochastic Analysis, Random Fields and Applications, Progress in Probability 36, pp. 327-351. Basel: Birkhäuser.
  • [8] Dai, W. and Heyde, C.C. (1996) Itô's formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stochastic Anal., 9, 439-448.
  • [9] Doléans-Dade, C. (1970) Quelques applications de la formule de changement de variables pour les semimartingales. Z. Wahrscheinlichkeitstheorie Verw. Geb., 16, 181-194.
  • [10] Doob, J.L. (1937) Stochastic processes depending on a continuous parameter. Trans. Amer. Math. Soc., 42, 107-140.
  • [11] Doob, J.L. (1942) The Brownian movement and stochastic equations. Ann. Math., 43, 351-369.
  • [12] Dudley, R.M. (1992) Fréchet differentiability, p-variation and uniform Donsker classes. Ann. Probab., 20, 1968-1982.
  • [13] Dudley, R.M. and Norvaisa, R. (1999a) Product integrals, Young integrals and p-variation. In R.M. Dudley and R. Norvaisa, Differentiability of Six Operators on Nonsmooth Functions and p- Variation, Lecture Notes in Math. 1703. New York: Springer-Verlag.
  • [14] Dudley, R.M. and Norvaisa, R. (1999b) A survey on differentiability of six operators in relation to probability and statistics. In R.M. Dudley and R. Norvaisa, Differentiability of Six Operators on Nonsmooth Functions and p-Variation, Lecture Notes in Math. 1703. New York: Springer-Verlag.
  • [15] Dudley, R.M., Norvaisa, R. and Jinghau Qian (1999) Bibliographies on p-variations and ö-variation. In R.M. Dudley and R. Norvaisa, Differentiability of Six Operators on Nonsmooth Functions and p-Variation, Lecture Notes in Math. 1703. New York: Springer-Verlag.
  • [16] Eberlein, E. and Keller, U. (1995) Hyperbolic distributions in finance. Bernoulli, 1, 281-299.
  • [17] Fernique, X. (1964) Continuité des processus gaussiens. C. R. Acad. Sci. Paris, 258, 6058-6060.
  • [18] Föllmer, H. (1981) Calcul d'Itô sans probabilités. In J. Azéma and M. Yor (eds), Séminaire de Probabilités XV, Lecture Notes in Math. 850, pp. 144-150. Berlin: Springer-Verlag.
  • [19] Freedman, M.A. (1983) Operators of p-variation and the evolution representation problem. Trans. Amer. Math. Soc., 279, 95-112.
  • [20] Fristedt, B. and Taylor, S.J. (1973) Strong variation for the sample functions of a stable process. Duke Math. J., 40, 259-278.
  • [21] Harrison, J.M. (1977) Ruin problems with compounding assets. Stochastic Process. Appl., 5, 67-79.
  • [22] Hildebrandt, T.H. (1938) Definitions of Stieltjes integrals of the Riemann type. Amer. Math. Monthly, 45, 265-278.
  • [23] Hildebrandt, T.H. (1959) On systems of linear differentio-Stieltjes-integral equations. Illinois J. Math., 3, 352-373.
  • [24] Hildebrandt, T.H. (1963) Introduction to the Theory of Integration. New York: Academic Press.
  • [25] Itô, K. (1969) Stochastic Processes, Aarhus University Lecture Notes No. 16. Aarhus: Matematisk Institut, Aarhus Universitet.
  • [26] Jacod, J. (1979) Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Math. 714. Berlin: Springer-Verlag.
  • [27] Jain, N.C. and Monrad, D. (1983) Gaussian measures in Bp. Ann. Probab., 11, 46-57.
  • [28] Janicki, A. and Weron, A. (1993) Simulation and Chaotic Behaviour of a-Stable Stochastic Processes. New York: Marcel Dekker.
  • [29] Karatzas, I. and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus, 2nd edition. New York: Springer-Verlag.
  • [30] Kawada, T. and KoAno, N. (1973) On the variation of Gaussian processes. In G. Maruyama and Y.V. Prokhorov (eds), Proceedings of the Second Japan-USSR Symposium on Probability Theory, Lecture Notes in Math. 330, pp. 176-192. Berlin: Springer-Verlag.
  • [31] Klingenhofer, F. and Zähle, M. (1999) Ordinary differential equations with fractal noise. Proc. Amer. Math. Soc., 127, 1021-1028.
  • [32] Küchler, U., Neumann, K., Sørensen, M. and Streller, A. (1994) Stock returns and hyperbolic distributions. Discussion paper 23, Humboldt Universitàít Berlin.
  • [33] Lebesgue, H. (1973) Leçons sur l'Intégration et la Recherche des Fonctions Primitives, 3rd edition. New York: Chelsea.
  • [34] Lépingle, D. (1976) La variation d'ordre p des semi-martingales. Z. Wahrscheinlichkeitstheorie Verw. Geb., 36, 295-316.
  • [35] Lin, S.J. (1995) Stochastic analysis of fractional Brownian motions. Stochastics Stochastics Rep., 55, 121-140.
  • [36] Liptser, R.S. and Shiryaev, A.N. (1986) Theory of Martingales. Moscow: Nauka (In Russian). English translation (1989), Dordrecht: Kluwer.
  • [37] Lyons, T. (1994) Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young. Math. Res. Lett., 1, 451-464.
  • [38] Norvaisa, R. (1998) p-variation and integration of sample functions of stochastic processes. Preprint.
  • [39] Samorodnitsky, G. and Taqqu, M.S. (1994) Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. London: Chapman & Hall.
  • [40] Taylor, S.J. (1972) Exact asymptotic estimates of Brownian path variation. Duke Math. J., 39, 219- 241.
  • [41] Wiener, N. and Paley, R.E.A.C. (1934) Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ. 19. Providence, RI: American Mathematical Society.
  • [42] Young, L.C. (1936) An inequality of the Hölder type, connected with Stieltjes integration. Acta Math., 67, 251-282.