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Malliavin Calculus Approach to Statistical Inference for Lévy Driven SDE’s

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Abstract

By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX t = a θ (X t )dt + dZ t with a Lévy process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.

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References

  • Bally V, Clément E (2011) Integration by parts formula and applications to equations with jumps. Probab Theory Relat Fields 151(3–4):613–657

    Article  MATH  Google Scholar 

  • Bichteler K, Gravereaux J-B, Jacod J (1987) Malliavin calculus for processes with jumps. Gordon and Breach science publishers, NY

    MATH  Google Scholar 

  • Bismut J-M (1981) Calcul des variations stochastique et processus de sauts. Z Warw theor verw Geb 56(4):469–505

    Article  MATH  MathSciNet  Google Scholar 

  • Bouleau N, Denis L (2011) Application of the lent particle method to Poisson-driven SDE’s. Probab Theory Relat Fields 151(3–4):403–433

    Article  MATH  MathSciNet  Google Scholar 

  • Carr P, Geman H, Madan DB, Yor M (2003) Stochastic volatility for Lévy processes. Math Finance 13:345–382

    Article  MATH  MathSciNet  Google Scholar 

  • Chaumont L, Uribe Bravo G (2011) Markovian bridges: weak continuity and pathwise constructions. Ann Probab 39(2):609–647

    Article  MATH  MathSciNet  Google Scholar 

  • Corcuera JM, Kohatsu-Higa A (2011) Statistical inference and Malliavin calculus. Semin Stoch Anal Random Fields Appl VI, Prog probab Springer Basel 63:59–82

    MathSciNet  Google Scholar 

  • Fournié E, Lasry JM, Lebuchoux J, Lions P-L, Touzi N (1999) Some applications of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochast 3:391–412

    Article  MATH  Google Scholar 

  • Gikhman II, Skorokhod AV (1972) Stochastic differential equations. Springer-Verlag, New York

    Book  MATH  Google Scholar 

  • Gobet E (2001a) Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli 7(6):899–912

    Article  MATH  MathSciNet  Google Scholar 

  • Gobet E (2001b) LAN property for ergodic diffusions with discrete observations. Ann I H Poincaré PR 3 5:711–737

    Google Scholar 

  • Ibragimov IA, Hasminskii RZ (1981) Statistical estimation: asymptotic theory. Springer-Verlag, New York

    Book  MATH  Google Scholar 

  • Ivanenko DO (2013) LAN property for solutions of Lévy driven SDE’s. arXiv:1308.3089

  • Novikov EA (1994) Infinitely divisible distributions in turbulence. Phys Rev E 50:R3303–R3305

    Article  Google Scholar 

  • Nualart D (1998) Analysis in Wiener space and anticipating stochastic calculus. Springer Berlin / Heidelberg, Lect Notes Math 1690:123–220

    Article  MathSciNet  Google Scholar 

  • Rosiński J (2007) Tempering stable processes. Stoch Process Appl 117(6):677–707

    Article  MATH  Google Scholar 

  • Simon B, Read M (1972) Methods of Modern Mathematical Physics. Functional Analysis. Academic Press, San Diego

    Google Scholar 

  • Simon T (2000) Support theorem for jump processes. Stoch Process Appl 89:1–30

    Article  MATH  Google Scholar 

  • Skorokhod AV (1957) On differentiability of the measures which correspond to stochastic processes I. Processes with independent increments. Theory Probab Appl 2(4):407–432

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Kyiv National Taras Shevchenko University, Volodymyrska, 64, Kyiv, 01033, Ukraine

    D. O. Ivanenko

  2. Institute of Mathematics, Ukrainian National Academy of Sciences, 01601, Tereshchenkivska, 3, Kyiv, Ukraine

    A. M. Kulik

Authors
  1. D. O. Ivanenko
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  2. A. M. Kulik
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Correspondence to A. M. Kulik.

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Ivanenko, D., Kulik, A. Malliavin Calculus Approach to Statistical Inference for Lévy Driven SDE’s. Methodol Comput Appl Probab 17, 107–123 (2015). https://doi.org/10.1007/s11009-013-9387-y

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  • DOI: https://doi.org/10.1007/s11009-013-9387-y

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