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Uniqueness in law for pure jump Markov processes
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  • Published: September 1988

Uniqueness in law for pure jump Markov processes

  • R. F. Bass1 

Probability Theory and Related Fields volume 79, pages 271–287 (1988)Cite this article

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Summary

Let A be the operator defined on C 2 functions by

$$Af\left( x \right) = \smallint \left[ {f\left( {x + h} \right) - f\left( x \right) - f'\left( x \right)h 1_{([ - 1, 1])} \left( h \right)} \right]v\left( {x,dh} \right).$$

Sufficient conditions are given for existence and uniqueness for the martingale problem associated with A. In the case of stable-like processes, where v(x, dh) is equal to the Lévy measure for the stable symmetric process of index α(x) for each x, the conditions reduce to α(x) continuous for existence and α(x) Dini continuous for uniqueness.

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References

  1. Aldous, D.: Stopping times and tightness. Ann. Probab. 6, 335–340 (1978)

    MATH  MathSciNet  Google Scholar 

  2. Bass, R.: Local times for a class of purely discontinuous martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 433–459 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bass, R.: Occupation time densities for stable-like processes and other pure jump Markov processes. Stochastic Proc. Appl. (in press)

  4. Folland, G.: Introduction to partial differential equations. Princeton: Princeton University Press 1976

    Google Scholar 

  5. Komatsu, T.: Markov processes associated with certain integro-differential operators. Osaka J. Math. 10, 271–303 (1973)

    MATH  MathSciNet  Google Scholar 

  6. Komatsu, T.: Markov processes associated with pseudo-differential operators. In: Ito, K., Prokhorov, I.V. (eds.) Probability theory and mathematical statistics. Proceedings of the fourth USSR-Japan Symposium, Tbilisi 1982. (Lect. Notes Math., vol. 1021, pp. 289–298) New York Berlin Heidelberg: Springer 1983

    Google Scholar 

  7. Komatsu, T.: On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21, 113–132 (1984)

    MATH  MathSciNet  Google Scholar 

  8. Skorokhod, A.V.: Studies in the theory of random processes. Reading, Mass.: Addison-Wesley 1965

    Google Scholar 

  9. Stroock, D.W.: Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 209–244 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  10. Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. New York Berlin Heidelberg: Springer 1979

    Google Scholar 

  11. Tanaka, H., Tsuchiya, M., Watanabe, S.: Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14, 70–92 (1974)

    MathSciNet  Google Scholar 

  12. Tsuchiya, M.: On a small drift of a Cauchy process. J. Math. Kyoto Univ. 10, 473–492 (1970)

    MathSciNet  Google Scholar 

  13. Tsuchiya, M.: On some perturbations of stable processes. In: Marvyama, G., Prokhorov, Yu.V. (eds.) Proceedings of the Second Japan-USSR Symposium on Probability Theory. Lect. Notes Math., vol. 330, pp. 490–497) New York: Springer 1973

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    R. F. Bass

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  1. R. F. Bass
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Partially supported by NSF grant DMS 85-00581

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Cite this article

Bass, R.F. Uniqueness in law for pure jump Markov processes. Probab. Th. Rel. Fields 79, 271–287 (1988). https://doi.org/10.1007/BF00320922

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  • Received: 09 October 1987

  • Revised: 12 February 1988

  • Issue Date: September 1988

  • DOI: https://doi.org/10.1007/BF00320922

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Keywords

  • Stochastic Process
  • Probability Theory
  • Markov Process
  • Mathematical Biology
  • Martingale Problem
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