Numerical simulations for G-Brownian motion

Abstract

This paper is concerned with numerical simulations for the GBrownian motion (defined by S. Peng in Stochastic Analysis and Applications, 2007, 541–567). By the definition of the G-normal distribution, we first show that the G-Brownian motions can be simulated by solving a certain kind of Hamilton-Jacobi-Bellman (HJB) equations. Then, some finite difference methods are designed for the corresponding HJB equations. Numerical simulation results of the G-normal distribution, the G-Brownian motion, and the corresponding quadratic variation process are provided, which characterize basic properties of the G-Brownian motion. We believe that the algorithms in this work serve as a fundamental tool for future studies, e.g., for solving stochastic differential equations (SDEs)/stochastic partial differential equations (SPDEs) driven by the G-Brownian motions.

Key words： Nonlinear expectation; G-Brownian motion; G-normal distribution; Hamilton-Jacobi-Bellman (HJB) equation

References

1 Artzner P, Delbaen F, Eber J M, Heath D. Coherent measures of risk. Math Finance, 1999, 9: 203–228

2 Avellaneda M, Levy A, Paras A. Pricing and hedging derivative securities in markets with uncertain volatilities. Appl Math Finance, 1995, 2: 73–88

3 Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Anal, 2011, 34: 139–161

4 Einstein A. Investigation on the theory of the Brownian movement (originally in German). Annalen der Physik, 1905, 17: 549–560

5 Fahim A, Touzi N, Warin X. A probabilistic numerical method for fully nonlinear parabolic PDEs. Ann Appl Probab, 2011, 4: 1322–1364

6 Gao F. Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion. Stochastic Process Appl, 2009, 119: 3356–3382

7 Guo W, Zhang J, Zhuo J. A monotone scheme for high dimensional fully nonlinear PDEs. Ann Appl Probab (to appear), arXiv: 1212.0466

8 Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 2001, 43(3): 525–546

9 Holmes M. Introduction to Numerical Methods in Differential Equations. Berlin: Springer-Verlag, 2007

10 Hu M, Ji S, Peng S, Song Y. Backward stochastic differential equations driven by G-Brownian motion. Stochastic Process Appl, 2014, 124: 759–784

11 Hu M, Ji S, Peng S, Song Y. Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stochastic Process Appl, 2014, 124: 1170–1195

12 Hu M, Peng S. On the representation theorem of G-expectation and paths of G-Brownian motion. Acta Math Appl Sin Engl Ser, 2009, 25(3): 539–546

13 Kloeden P, Platen E. Numerical Solution of Stochastic Differential Equations. Berlin: Springer-Verlag, 1999

14 Li X, Peng S. Stopping times and related Itô calculus with G-Brownian motion. Stochastic Process Appl, 2011, 121: 1492–1508

15 Lin Y. Stochastic differential equations driven by G-Brownian motion with reflecting boundary conditions. Electron J Probab, 2013, 18(9): 1–23

16 Peng S. G-expectation, G-Brownian motion and related stochastic calculus of Itô’s type. In: Stochastic Analysis and Applications. Berlin: Springer, 2007, 541–567

17 Peng S. G-Brownian motion and dynamic risk measure under volatility uncertainty. arXiv: 0711.2834v1 [math.PR], 2007

18 Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl, 2008, 118: 2223–2253

19 Peng S. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci China Ser A, 2009, 52(7): 1391–1411

20 Peng S. Nonlinear expectations and stochastic calculus under uncertainty—with robust central limit theorem and G-Brownian motion. arXiv: 1002.4546v1 [math.PR], 2010

21 Rüdiger S. Tools for Computational Finance. 2nd ed. Berlin: Springer-Verlag, 2002

22 Soner M H, Touzi N, Zhang J. Martingale representation theorem under G-expectation. Stochastic Process Appl, 2011, 121: 265–287

23 Song Y. Some properties on G-evaluation and its applications to G-martingale decomposition. Sci China A, 2011, 54(2): 287–300

24 Tan X. Discrete-time probabilistic approximation of path-dependent stochastic control problems. Ann Appl Probab, 2014, 24(5): 1803–1834

25 Xu J, Zhang B. Martingale characterization of G-Brownian motion. Stochastic Process Appl, 2009, 119: 232–248

26 Zhang D, Chen Z. Exponential stability for stochastic differential equation driven by G-Brownian motion. Appl Math Lett, 2012, 25: 1906–1910

27 Zhao W, Fu Y, Zhou T. New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations. SIAM J Sci Comput, 2014, 36(4): A1731–1751