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Martingale measures and stochastic calculus
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  • Published: March 1990

Martingale measures and stochastic calculus

  • N. El Karoui1 &
  • S. Méléard2 

Probability Theory and Related Fields volume 84, pages 83–101 (1990)Cite this article

Summary

In this paper, martingale measures, introduced by J.B. Walsh, are investigated. We prove, with techniques of stochastic calculus, that each continuous orthogonal martingale measure is the time-changed image martingale measure of a white noise.

We also exhibit a representation theorem for certain vector martingale measures as stochastic integrals of orthogonal martingale measures. Thus we can study the following martingale problem:

$$f(X_t ) - f(X_0 ) - \int\limits_0^t {\int\limits_E {Lf(s,X_s ,x)q_s (dx)ds} } isaP - martingale,$$

whereL is a second order differential operator andq a predictable random measure-valued process. We prove that this problem is bound to a stochastic differential equation with a term integral with respect to a martingale measure.

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References

  1. Dellacherie, C., Meyer, P.A.: Probabilités et potentiel. Paris: Hermann 1976

    Google Scholar 

  2. El Karoui, N., Lepeltier, J.P.: Représentation des processus ponctuels multivariés à l'aide d'un processus de Poisson. Z. Wahrscheinlichkeitstheor. Verw. Geb.39, 111–133 (1977)

    Google Scholar 

  3. El Karoui, N., Huu Nguyen, D., Jeanblanc-Picqué, M.: Compactification methods in the control of degenerate diffusions: existence of an optimal control. Stochastics20, 169–221 (1987)

    Google Scholar 

  4. El Karoui, N., Jeanblanc-Picqué: Partially observable diffusions with control in the observation process. Prépublication du Laboratoire de Probabilités de l'Université Paris VI.

  5. Funaki, T.: A certain class of diffusion processes associated with non linear parabolic equations. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 331–348 (1984)

    Google Scholar 

  6. Grigelionis, B.: On the representation of integer valued measures by means of stochastic integrals with respect to Poisson measures. Litov. Mat. Sb.11, 93–108 (1971)

    Google Scholar 

  7. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam/Kodansha: North Holland Mathematical Library 1981

    Google Scholar 

  8. Jacod, J., Mémin, J.: Weak and strong solutions of stochastic differential equations: existence and stability. In: Proc. Durham Symp. 1980 (Lect. Notes Math., vol. 851) Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  9. Méléard, S., Roelly-Coppoletta, S.: Systèmes de particules et mesures martingales: un théorème de propagation du chaos. Séminaire de Probabilités XXII (Lect. Notes Math. vol. 1321, pp. 438–448) Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  10. Méléard, S., Roelly-Coppoletta, S.: A generalized equation for a continuous measure branching process. To appear in: Proceedings “Stochastic PDE's and Applications.” Trento 1988 Berlin Heidelberg New York: Springer

    Google Scholar 

  11. Méléard, S., Roelly-Coppoletta, S.: Discontinuous measure-valued branching processes and generalized stochastic equations. Prépublication no. 9, du Laboratoire de Probabilités de l'Université Paris VI.

  12. Neveu, J.: Processus aléatoires gaussiens: Séminaire de Mathématiques supérieures-1968, Presses de l'Université de Montréal

  13. Skorohod, A.V.: Studies in the theory of random processes. Reading, Mass.: Addison Wesley 1965

    Google Scholar 

  14. Walsh, J.B.: An introduction to stochastic partial differential equations. In: Ecole d'été de Probabilités de Saint-Flour XIV-1984. Berlin Heidelberg New York: Springer 1986

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire de Probabilités, Université Paris VI, Tour 56, 4, Place Jussieu, F-75230, Paris Cedex 05, France

    N. El Karoui

  2. Faculté des Sciences, Université du Maine, Route de Laval, F-72017, Le Mans Cedex, France

    S. Méléard

Authors
  1. N. El Karoui
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  2. S. Méléard
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Karoui, N.E., Méléard, S. Martingale measures and stochastic calculus. Probab. Th. Rel. Fields 84, 83–101 (1990). https://doi.org/10.1007/BF01288560

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  • Received: 15 January 1988

  • Revised: 20 January 1989

  • Issue Date: March 1990

  • DOI: https://doi.org/10.1007/BF01288560

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Keywords

  • Differential Equation
  • Stochastic Process
  • White Noise
  • Probability Theory
  • Differential Operator
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