Pathwise properties and homeomorphic flows for stochastic differential equations driven by \(G\)-Brownian motion. (English) Zbl 1176.60043
Summary: We study pathwise properties and homeomorphic property with respect to the initial values for stochastic differential equations driven by \(G\)-Brownian motion. We first present a Burkholder-Davis-Gundy inequality and an extension of Itô’s formula for the \(G\)-stochastic integrals. Some moment estimates and Hölder continuity of the \(G\)-stochastic integrals and the solutions of stochastic differential equations with Lipschitzian coefficients driven by \(G\)-Brownian motion are obtained. Homeomorphic property with respect to the initial values is also established.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H05 | Stochastic integrals |
| 60J65 | Brownian motion |
Keywords:
\(G\)-Brownian motion; \(G\)-stochastic differential equation; BDG inequality; Itô’s-formula; moment estimate; Hölder continuity; homeomorphic flowReferences:
| [1] | Denis, L.; Martini, C., A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, The Annals of Applied Probability, 16, 2, 827-852 (2006) · Zbl 1142.91034 |
| [2] | L. Denis, M.S. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: Application to \(G\) arXiv: math.PR/0802.1240v1; L. Denis, M.S. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: Application to \(G\) arXiv: math.PR/0802.1240v1 · Zbl 1225.60057 |
| [3] | Fang, S. Z.; Zhang, T. S., A study of stochastic differential equations with non-Lipschitzian coefficients, Probability Theory and Related Fields, 132, 3, 356-390 (2005) · Zbl 1081.60043 |
| [4] | He, S. W.; Wang, J. G.; Yan, J. A., Semimartingale Theory and Stochastic Calculus (1992), CRC Press: CRC Press Beijing · Zbl 0781.60002 |
| [5] | Huang, Z., Foundation of Stochastic Analysis (2001), Science press of China: Science press of China Beijing |
| [6] | Kunita, H., (Stochastic Differential Equations and Stochastic Flows of Diffeomorphisms. Stochastic Differential Equations and Stochastic Flows of Diffeomorphisms, Lecture Notes in Mathematics, vol. 1097 (1984), Springer: Springer Berlin), 143-303 · Zbl 0554.60066 |
| [7] | Kwapień, S.; Rosiński, J., Sample Hölder continuity of stochastic processes and majorizing measures, (Dalang, R.; Dozzi, M.; Russo, F., Seminar on Stochastic Analysis, Random Fields and Applications IV. Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability, vol. 58 (2004), Birkhäuser: Birkhäuser Boston), 155-163 · Zbl 1061.60038 |
| [8] | Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1989), North-Holland/Kodanska: North-Holland/Kodanska Amsterdam · Zbl 0684.60040 |
| [9] | Ledoux, M.; Talagrand, M., Probability in Banach Space Isoperimetry and Processes (1991), Springer-Verlag · Zbl 0748.60004 |
| [10] | Peng, S., \(G\)-Expectation, \(G\)-Brownian motion and related stochastic calculus of Itô’s type, (Benth; etal., Proceedings of the 2005 Abel Symposium 2 (2006), Springer-Verlag), 541-567 · Zbl 1131.60057 |
| [11] | Peng, S., Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation, Stochastic Processes and their Applications, 118, 2223-2253 (2008) · Zbl 1158.60023 |
| [12] | S. Peng, A new central limit theorem under sublinear expectations, 18 Mar 2008, arXiv:0803.2656v1; S. Peng, A new central limit theorem under sublinear expectations, 18 Mar 2008, arXiv:0803.2656v1 |
| [13] | S. Peng, \(G\) arXiv: math.PR/0711.2834v1; S. Peng, \(G\) arXiv: math.PR/0711.2834v1 |
| [14] | Reveuz, D.; Yor, M., Continuous Martingales and Brownian Motion (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0731.60002 |
| [15] | Xu, J.; Zhang, B., Martingale characterization of \(G\)-Brownian motion, Stochastic Processes and their Applications, 119, 232-248 (2009) · Zbl 1168.60024 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
