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Integral representation of martingales motivated by the problem of endogenous completeness in financial economics

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  • Kramkov, Dmitry
  • Predoiu, Silviu

Abstract

Let Q and P be equivalent probability measures and let ψ be a J-dimensional vector of random variables such that dQdP and ψ are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that every local martingale under Q is a stochastic integral with respect to the J-dimensional martingale St≜EQ[ψ|Ft]. While the drift b=b(t,x) and the volatility σ=σ(t,x) coefficients for X need to have only minimal regularity properties with respect to x, they are assumed to be analytic functions with respect to t. We provide a counter-example showing that this t-analyticity assumption for σ cannot be removed.

Suggested Citation

  • Kramkov, Dmitry & Predoiu, Silviu, 2014. "Integral representation of martingales motivated by the problem of endogenous completeness in financial economics," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 81-100.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:81-100
    DOI: 10.1016/j.spa.2013.06.017
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    References listed on IDEAS

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    1. Harrison, J. Michael & Pliska, Stanley R., 1983. "A stochastic calculus model of continuous trading: Complete markets," Stochastic Processes and their Applications, Elsevier, vol. 15(3), pages 313-316, August.
    2. J. Hugonnier & S. Malamud & E. Trubowitz, 2012. "Endogenous Completeness of Diffusion Driven Equilibrium Markets," Econometrica, Econometric Society, vol. 80(3), pages 1249-1270, May.
    3. Mark Davis & Jan Obloj, 2007. "Market completion using options," Papers 0710.2792, arXiv.org, revised Oct 2008.
    4. Robert M. Anderson & Roberto C. Raimondo, 2008. "Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets," Econometrica, Econometric Society, vol. 76(4), pages 841-907, July.
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    Citations

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    Cited by:

    1. Ehling, Paul & Heyerdahl-Larsen, Christian, 2015. "Complete and incomplete financial markets in multi-good economies," Journal of Economic Theory, Elsevier, vol. 160(C), pages 438-462.
    2. Daniel C. Schwarz, 2015. "Market Completion with Derivative Securities," Papers 1506.00188, arXiv.org.
    3. Kim Weston, 2022. "Existence of an equilibrium with limited participation," Papers 2206.12399, arXiv.org.
    4. Jerome Detemple & Scott Robertson, 2022. "Dynamic Equilibrium with Insider Information and General Uninformed Agent Utility," Papers 2211.15573, arXiv.org, revised Mar 2024.
    5. Dmitry Kramkov, 2015. "Existence of an endogenously complete equilibrium driven by a diffusion," Finance and Stochastics, Springer, vol. 19(1), pages 1-22, January.
    6. Dmitry Kramkov & Sergio Pulido, 2019. "Density of the set of probability measures with the martingale representation property," Post-Print hal-01598651, HAL.
    7. Jerome Detemple & Marcel Rindisbacher & Scott Robertson, 2020. "Dynamic Noisy Rational Expectations Equilibrium With Insider Information," Econometrica, Econometric Society, vol. 88(6), pages 2697-2737, November.
    8. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Working Papers hal-01598651, HAL.
    9. Daniel C. Schwarz, 2017. "Market completion with derivative securities," Finance and Stochastics, Springer, vol. 21(1), pages 263-284, January.
    10. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Papers 1709.07329, arXiv.org, revised Jul 2019.

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