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Representing filtration consistent nonlinear expectations as $g$-expectations in general probability spaces

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  • Samuel N. Cohen

Abstract

We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations with Lipschitz continuous drivers, where both the martingale and the driver terms are permitted to jump, and the martingale representation is infinite dimensional. To establish this result, we show that this domination condition is sufficient to guarantee that the comparison theorem for BSDEs will hold, and we generalise the nonlinear Doob-Meyer decomposition of Peng to a general context.

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  • Samuel N. Cohen, 2011. "Representing filtration consistent nonlinear expectations as $g$-expectations in general probability spaces," Papers 1102.5287, arXiv.org.
  • Handle: RePEc:arx:papers:1102.5287
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    File URL: http://arxiv.org/pdf/1102.5287
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    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Stefan Thurner & J. Doyne Farmer & John Geanakoplos, 2012. "Leverage causes fat tails and clustered volatility," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 695-707, February.
    3. Barunik, Jozef & Kristoufek, Ladislav, 2010. "On Hurst exponent estimation under heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3844-3855.
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