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Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

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  • Peng, Shige
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    Abstract

    We develop a notion of nonlinear expectation-G-expectation-generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a multi-dimensional G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô's type with respect to our G-Brownian motion, and derive the related Itô's formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G-expectation.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 118 (2008)
    Issue (Month): 12 (December)
    Pages: 2223-2253

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    Handle: RePEc:eee:spapps:v:118:y:2008:i:12:p:2223-2253

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    Related research

    Keywords: g-expectation G-expectation G-normal distribution BSDE SDE Nonlinear probability theory Nonlinear expectation Brownian motion Ito's stochastic calculus Ito's integral Ito's formula Gaussian process Quadratic variation process Jensen's inequality G-convexity;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Zengjing Chen & Larry G. Epstein, 2000. "Ambiguity, risk and asset returns in continuous time," RCER Working Papers 474, University of Rochester - Center for Economic Research (RCER).
    2. Chen, Zengjing & Peng, Shige, 2000. "A general downcrossing inequality for g-martingales," Statistics & Probability Letters, Elsevier, vol. 46(2), pages 169-175, January.
    3. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    4. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
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    Cited by:
    1. Yan Dolinsky & Halil Soner, 2013. "Duality and convergence for binomial markets with friction," Finance and Stochastics, Springer, vol. 17(3), pages 447-475, July.
    2. Moreau, Ludovic, 2012. "A contribution in stochastic control applied to finance and insurance," Economics Thesis from University Paris Dauphine, Paris Dauphine University, number 123456789/10711 edited by Bouchard, Bruno.
    3. Ren, Liying, 2013. "On representation theorem of sublinear expectation related to G-Lévy process and paths of G-Lévy process," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1301-1310.
    4. Epstein, Larry G. & Ji, Shaolin, 2014. "Ambiguous volatility, possibility and utility in continuous time," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 269-282.
    5. Nutz, Marcel & van Handel, Ramon, 2013. "Constructing sublinear expectations on path space," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3100-3121.

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