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The transformation theorem for two-parameter pure jump martingales
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  • Published: September 1991

The transformation theorem for two-parameter pure jump martingales

  • Peter Imkeller1 

Probability Theory and Related Fields volume 89, pages 261–283 (1991)Cite this article

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Summary

LetM be a martingale of pure jump type, i.e. the compensation of the process describing the total of the point jumps ofM in the plane.M can be uniformly approximated by martingales of bounded variation jumping only on finitely many axial parallel lines. Using this fact we prove a change of variables formula in which forC 4-functions f the processf(M) is described by integrals off (k) (M),k=1, 2, with respect to stochastic integrators of the types expected: a martingale, two processes behaving as martingales in one direction and as processes of bounded variation in the other, and one process of bounded variation. Hereby we are led to investigate two types of random measures not considered so far in this context. By combination with the integrators already known, they might complete the set needed for a general transformation formula.

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References

  1. Al Hussaini, A., Elliott, R.J.: Stochastic calculus for a two-parameter jump process. In: Korezlioglu, H., Mazziotto, G., Szpirglas, J. (eds.) Processus aléatoires à deux indices. LNM, Vol. 863, pp. 233–244, Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  2. Bakry, D.: Limites quadrantales des martingales. In: Korezlioglu, H., Mazziotto, G., Szpirglas, J. (eds.) Processus aléatoires à deux indices. LNM Vol. 863, pp. 40–49 Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  3. Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Math.134, 111–183 (1975)

    Google Scholar 

  4. Chevalier, L.: Martingales continues à deux paramètres. Bull. Sci. Math. (2)106, 19–62 (1982)

    Google Scholar 

  5. Imkeller, P.: Stochastic integrals of point processes and the decomposition of two-parameter martingales. J. Multivariate Anal.30, 98–123 (1989)

    Google Scholar 

  6. Imkeller, P.: A class of two-parameter stochastic integrators. Stochastics and Stochastics Rep.27, 167–188 (1989)

    Google Scholar 

  7. Imkeller, P.: Two-parameter martingales and their quadratic variation. LNM vol 1308. Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  8. Imkeller, P.: Regularity and integrator properties of variation processes of two-parameter martingales with jumps. Séminaire de Probabilités XXIII, LNM, vol. 1372, pp. 536–565. Berlin Heidelberg New York: Springer 1989

    Google Scholar 

  9. Jacod, J.: Calcul stochastique et problèmes de martingales (Lect. Notes Math., vol. 714), Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  10. Mazziotto, G., Szpirglas, J.: Equations du filtrage pour un processus de Poisson mélangé à deux indices. Stochastics4, 89–119 (1980)

    Google Scholar 

  11. Métivier, M.: Semimartingales. A course on stochastic processes. Berlin New York: W. de Gruyter 1982

    Google Scholar 

  12. Millet, A., Sucheston, L.: On regularity of multiparameter amarts and martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 21–45 (1981)

    Google Scholar 

  13. Mishura, Yu. S.: On some properties of discontinuous two-parameter martingales. Theory Probab. Math. Stat.29, 87–100 (1984)

    Google Scholar 

  14. Mishura, Yu. S.: A generalized Ito formula for two-parameter martingales. I. Theory Probab. Math. Stat.30, 127–142 (1985)

    Google Scholar 

  15. Mishura, Yu. S.: A generalized Ito formula for two-parameter martingales. II. Theory Probab. Math. Stat.32, 77–94 (1986)

    Google Scholar 

  16. Nualart, D.: On the quadratic variation of two-parameter continuous martingales. Ann. Probab.12, 445–457 (1984)

    Google Scholar 

  17. Nualart, D.: Une formule d'Ito pour les martingales à deux indices et quelques applications. Ann. Inst. Henri Poinearé20, 251–275 (1984)

    Google Scholar 

  18. Sanz, M.: r-variations for two-parameter continuous martingales and Ito's formula. Preprint, University Barcelona (1988)

  19. Wong, E., Zakai, M.: Weak martingales and stochastic integrals in the plane. Ann. Probab.4, 570–586 (1976)

    Google Scholar 

  20. Wong, E., Zakai, M.: Differentiation formulas for stochastic integrals in the plane. Stochastic Processes Appl.6, 339–349 (1978)

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, der Ludwig-Maximilians-Universität, Theresienstrasse 39, W-8000, München 2, Federal Republic of Germany

    Peter Imkeller

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  1. Peter Imkeller
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Imkeller, P. The transformation theorem for two-parameter pure jump martingales. Probab. Th. Rel. Fields 89, 261–283 (1991). https://doi.org/10.1007/BF01198787

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  • Received: 03 November 1988

  • Revised: 16 September 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01198787

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Parallel Line
  • Bounded Variation
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