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An Ideal Class to Construct Solutions for Skew Brownian Motion Equations

Abstract

This paper contributes to the study of stochastic processes of the class \((\Sigma )\). First, we extend the notion of the above-mentioned class to càdlàg semi-martingales, whose finite variation part is considered càdlàg instead of continuous. Thus, we present some properties and propose a method to characterize such stochastic processes. Second, we investigate continuous processes of the class \((\Sigma )\). More precisely, we derive a series of new characterization results. In addition, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the class \((\Sigma )\).

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References

  1. Azéma, J., Yor, M.: Sur les zéros des martingales continues. Séminaire de probabilités (Strasbourg) 26, 248–306 (1992)

    Article  Google Scholar 

  2. Bouhadou, S., Ouknine, Y.: On the time inhomogeneous skew Brownian motion. Bulletin des Sciences Mathématiques 137(7), 835–850 (2013)

    MathSciNet  Article  Google Scholar 

  3. Cheridito, P., Nikeghbali, A., Platen, E.: Processes of class sigma, last passage times, and drawdowns. preprint arXiv:0910.5493v1

  4. Dellacherie, C., Meyer, P.A. : Probabilités et Potentiel. Chapitres V à VIII. Théorie des Martingales. Revised Edition, Hermann, Paris, (1980)

  5. Etoré, P., Martinez, M.: on the existence of a time inhomogeneous skew Brownian motion and some related laws. Electron. J. Probab. 17(19), 1–27 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Eyi-Obiang, F., Ouknine, Y., Moutsinga, O., Trutnau, G.: Some contributions to the study of stochastic processes of the classes \(\Sigma (H)\) and \((\Sigma )\). Stochastics 89(8), 1253–1269 (2017)

    MathSciNet  Article  Google Scholar 

  7. Harrison, J.M., Shepp, L.A.: On skew Brownian motion. Ann. Probab. 9(2), 309–313 (1981)

    MathSciNet  MATH  Google Scholar 

  8. Itô, K., McKean, H.P.: Diffusion and Their Sample Paths, 2nd edn. Springer, Berlin (1974)

    MATH  Google Scholar 

  9. Meyer, P.A., Stricker, C., Yor, M.: Sur une formule de la théorie du balayage. Séminaire de probabilités (Strasbourg) 13, 478–487 (1979)

    Article  Google Scholar 

  10. Najnudel, J., Nikeghbali, A.: A new construction of the \(\Sigma \)- finite measures associated with submartingales of class \((\Sigma )\). C. R. Math. Acad. Sci. Paris 348, 311–316 (2010)

    MathSciNet  Article  Google Scholar 

  11. Najnudel, J., Nikeghbali, A.: A remarkable sigma-finite measure associated with last passage times and penalisation results. In: Contemporary Quantitative Finance, Essays in Honour of Eckhard Platen, pp. 77–98. Springer (2010)

  12. Najnudel, J., Najnudel, A.: On some properties of a universal sigma finite measure associated with a remarkable class of submartingales. Publ. Res. Inst. Math. Sci. (Kyoto Univ.) 47(4), 911–936 (2011)

    MathSciNet  Article  Google Scholar 

  13. Najnudel, J., Nikeghbali, A.: On some universal sigma-finite measures and some extensions of Doob’s optional stopping theorem. Accepted in Stochastic processes and their applications

  14. Nikeghbali, A.: A class of remarkable submartingales. Stoch. Process. Appl. 116, 917–938 (2006)

    MathSciNet  Article  Google Scholar 

  15. Nikeghbali, A.: Multiplicative decompositions and frequency of vanishing of nonnegative submartingales. J. Theor. Probab. 19(4), 931–949 (2006)

    MathSciNet  Article  Google Scholar 

  16. Ouknine, Y.: “Skew-Brownian motion” and derived processes. Theory Probab. Appl. 35, 163–169 (1990)

    MathSciNet  Article  Google Scholar 

  17. Walsh, J.B.: A diffusion with a discontinuous local time. In: Temps locaux, Astérisques, pp. 37–45. Société Mathématique de France (1978)

  18. Weinryb, S.: Etude d’une équation différentielle stochastique avec temps local. C. R. Acad. Paris Sér. I Math. 296(6), 319–321 (1983)

    MathSciNet  MATH  Google Scholar 

  19. Yor, M.: Les inégalités de sous-martingales, comme conséquences de la relation de domination. Stochastics 3(1), 1–15 (1979)

    MathSciNet  MATH  Google Scholar 

  20. Yor, M.: Sur le balayage des semi-martingales continues. Séminaire de probabilités (Strasbourg) 13, 453–471 (1979)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

We thank the referee for the careful reading of the paper and for valuable remarks.

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Correspondence to Fulgence Eyi Obiang.

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Eyi Obiang, F., Moutsinga, O. & Ouknine, Y. An Ideal Class to Construct Solutions for Skew Brownian Motion Equations. J Theor Probab 35, 894–916 (2022). https://doi.org/10.1007/s10959-021-01078-5

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  • DOI: https://doi.org/10.1007/s10959-021-01078-5

Keywords

  • Class \((\Sigma )\)
  • Skew Brownian motion
  • Balayage formula
  • Honest time
  • Relative martingales

Mathematics Subject Classification

  • 60G07
  • 60G20
  • 60G46
  • 60G48