Numerical schemes for G-Expectations

Yan Dolinsky (ETH Zurich and Hebrew University of Jerusalem)

Abstract


We consider a discrete time analog of $G$-expectations and we prove that in the case where the time step goes to zero the corresponding values converge to the original $G$-expectation. Furthermore we provide error estimates for the convergence rate. This paper is continuation of Dolinsky,  Nutz, and Soner (2012). Our main tool is a strong approximation theorem which we derive for general discrete time martingales.

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Pages: 1-15

Publication Date: November 6, 2012

DOI: 10.1214/EJP.v17-2284

References

  • Denis, Laurent; Hu, Mingshang; Peng, Shige. Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion paths. Potential Anal. 34 (2011), no. 2, 139--161. MR2754968
  • Denis, Laurent; Martini, Claude. A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 (2006), no. 2, 827--852. MR2244434
  • Deparis, Simone; Martini, Claude. Superheading strategies and balayage in discrete time. Seminar on Stochastic Analysis, Random Fields and Applications IV, 205--219, Progr. Probab., 58, Birkhäuser, Basel, 2004. MR2096290
  • Dolinsky, Yan; Nutz, Marcel; Soner, H. Mete. Weak approximation of $G$-expectations. Stochastic Process. Appl. 122 (2012), no. 2, 664--675. MR2868935
  • Y.Hu and S.Peng, Some Estimates for Martingale Representation under G--Expectation, preprint.
  • M.Nutz and H.M.Soner, Superhedging and Dynamic Risk Measures under Volatility Uncertainty, to appear in SIAM Journal on Control and Optimization.
  • Peng, Shige. $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type. Stochastic analysis and applications, 541--567, Abel Symp., 2, Springer, Berlin, 2007. MR2397805
  • Peng, Shige. Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 (2008), no. 12, 2223--2253. MR2474349
  • S.Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint.
  • S.Peng Tightness, weak compactness of nonlinear expectations and application to CLT, preprint.
  • A.I Sakhanenko, A New Way to Obtain Estimates in the Invariance Principle, High Dimensional Probability II, (2000), 221--243.
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 (2011), no. 2, 265--287. MR2746175
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