Opuscula Math. 38, no. 2 (2018), 261-283
https://doi.org/10.7494/OpMath.2018.38.2.261
Opuscula Mathematica
Stochastic differential equations for random matrices processes in the nonlinear framework
Sara Stihi
Hacène Boutabia
Selma Meradji
Abstract. In this paper, we investigate the processes of eigenvalues and eigenvectors of a symmetric matrix valued process \(X_{t}\), where \(X_{t}\) is the solution of a general SDE driven by a \(G\)-Brownian motion matrix. Stochastic differential equations of these processes are given. This extends results obtained by P. Graczyk and J. Malecki in [Multidimensional Yamada-Watanabe theorem and its applications to particle systems, J. Math. Phys. 54 (2013), 021503].
Keywords: \(G\)-Brownian motion matrix, \(G\)-stochastic differential equations, random matrices, eigenvalues, eigenvectors.
Mathematics Subject Classification: 60B20, 60H10, 60H05.
- G.W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices, Cambridge University Press, 2009.
- X.P. Bai, Y.Q. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by \(G\)-Brownian motion with integral-Lipschitz coefficients, Acta Math. Appl. Sin. Engl. Ser. 30 (2014), 589-610.
- F. Faizullah, A note on the carathéodory approximation scheme for stochastic differential equations under \(G\)-Brownian motion, Z. Naturforsch. 67a (2012), 699-704.
- F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equation driven by \(G\)-Brownian motion, Stochastic Process. Appl. 119 (2009), 3356-3382.
- P. Graczyk, J. Malecki, Multidimensional Yamada-Watanabe theorem and its applications to particle systems, J. Math. Phys. 54 (2013), 021503.
- M. Katori, H. Tanemura, Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, J. Math. Phys. 45 (2004), 3058-3085.
- X. Li, S. Peng, Stopping times and related Itô's calculus with \(G\)-Brownian motion, Stochastic Process. Appl. 121 (2011), 1492-1508.
- Q. Lin, Local time and Tanaka formula for the \(G\)-Brownian motion, J. Math. Anal. Appl. 398 (2012), 315-334.
- S. Peng, \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications (2007), Abel Symp., 2, Springer, Berlin, 541-567.
- S. Peng, Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation, Stochastic Process. Appl. 118 (2008), 2223-2253.
- S. Peng, A. Bensoussan, J. Sung, Real Options, Ambiguity, Risk and Insurance, IOS Press, Amsterdam-Berlin-Tokyo-Washington DC, 2013.
- P. Wu, Z. Chen, Invariance principles for the law of the iterated logarithm under \(G\)-framework, Sci. China Math. 58 (2011), 1251-1264.
- Sara Stihi
- LaPS Laboratory, Badji-Mokhtar University, PO BOX 12, Annaba 23000, Algeria
- Hacène Boutabia
- LaPS Laboratory, Badji-Mokhtar University, PO BOX 12, Annaba 23000, Algeria
- Selma Meradji
- LaPS Laboratory, Badji-Mokhtar University, PO BOX 12, Annaba 23000, Algeria
- Communicated by Palle E.T. Jorgensen.
- Received: 2017-05-03.
- Revised: 2017-10-08.
- Accepted: 2017-10-10.
- Published online: 2017-12-29.
