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

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  • Peng, Shige

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.

Suggested Citation

  • Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:12:p:2223-2253
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    References listed on IDEAS

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    1. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    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, July.
    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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