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Locally Phi-Integrable Sigma-Martingale Densities for General Semimartingales

Author

Listed:
  • Tahir CHOULLI

    (University of Alberta)

  • Martin SCHWEIZER

    (ETH Zurich and Swiss Finance Institute)

Abstract

A P-sigma-martingale density for a given stochastic process S is a local P-martingale Z>0 starting at 1 such that the product ZS is a P-sigma-martingale. Existence of a P-sigma-martingale density is equivalent to a classic absence-of-arbitrage property of S, and it is invariant if we replace the reference measure P with a locally equivalent measure Q. Now suppose that there exists a P-sigma-martingale density for S. Can we find another P-sigma-martingale density for S having some extra local integrability I_loc(P) under P? We show that the answer is always positive for one part of S that we identify, and we show that the complete answer depends in a precise quantitative way on the local integrability of the drift-to-jump ratio of the remaining "jumpy" part of S. Our proofs provide in addition new ideas and results in infinite-dimensional spaces.

Suggested Citation

  • Tahir CHOULLI & Martin SCHWEIZER, 2015. "Locally Phi-Integrable Sigma-Martingale Densities for General Semimartingales," Swiss Finance Institute Research Paper Series 15-15, Swiss Finance Institute.
    • Handle: RePEc:chf:rpseri:rp1515
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    References listed on IDEAS

    as
    1. Hardy Hulley & Martin Schweizer, 2010. "M6 - On Minimal Market Models and Minimal Martingale Measures," Research Paper Series 280, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Constantinos Kardaras, 2012. "Market viability via absence of arbitrage of the first kind," Finance and Stochastics, Springer, vol. 16(4), pages 651-667, October.
    3. Tahir Choulli & Christophe Stricker, 2006. "More On Minimal Entropy–Hellinger Martingale Measure," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 1-19, January.
    4. Tahir Choulli & Christophe Stricker & Jia Li, 2007. "Minimal Hellinger martingale measures of order q," Finance and Stochastics, Springer, vol. 11(3), pages 399-427, July.
    5. Koichiro Takaoka & Martin Schweizer, 2014. "A note on the condition of no unbounded profit with bounded risk," Finance and Stochastics, Springer, vol. 18(2), pages 393-405, April.
    6. Tahir CHOULLI & Martin SCHWEIZER, 2015. "A Result on Integral Functionals with Infinitely Many Constraints," Swiss Finance Institute Research Paper Series 15-38, Swiss Finance Institute.
    7. S. D. Jacka, 1992. "A Martingale Representation Result and an Application to Incomplete Financial Markets," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 239-250, October.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Sigma-martingale; equivalent martingale measures; Jacod decomposition; mathematical finance;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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