Stochastic differential equations for eigenvalues and eigenvectors of a \(G\)-Wishart process with drift. (English. Russian original) Zbl 1435.62219
Ukr. Math. J. 71, No. 4, 572-588 (2019); translation from Ukr. Mat. Zh. 71, No. 4, 502-515 (2019).
Summary: We propose a system of \(G\)-stochastic differential equations for the eigenvalues and eigenvectors of a \(G\)-Wishart process defined according to a \(G\)-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence between the entries of the \(G\)-Brownian motion matrix, we assume that, in our model, their quadratic covariations are equal to zero. An intermediate result, which states that the eigenvalues never collide, is also obtained. This extends M.-F. Bru’s results obtained for the classical Wishart process in [J. Multivariate Anal. 29, No. 1, 127–136 (1989; Zbl 0687.62048)].
MSC:
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60J65 | Brownian motion |
| 60J60 | Diffusion processes |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Citations:
Zbl 0687.62048References:
| [1] | Bru, MF, Diffusions of perturbed principal component analysis, J. Multivariate Anal., 29, 127-136 (1989) · Zbl 0687.62048 · doi:10.1016/0047-259X(89)90080-8 |
| [2] | Demni, N., The Laguerre process and generalized Hartman-Watson law, Bernoulli, 13, 556-580 (2007) · Zbl 1139.60037 · doi:10.3150/07-BEJ6048 |
| [3] | Denis, L.; Hu, M.; Peng, S., Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, Potential Anal., 34, 139-161 (2011) · Zbl 1225.60057 · doi:10.1007/s11118-010-9185-x |
| [4] | V. L. Girko, Theory of Random Determinants, Kluwer AP, Dordrecht (1990). · doi:10.1007/978-94-009-1858-0 |
| [5] | N. Ikeda and S. Watanabe, Stochastic Differential Equation and Diffusion Processes, North-Holland, Amsterdam (1981). · Zbl 0495.60005 |
| [6] | Katori, M.; Tanemura, H., Complex Brownian motion representation of the Dyson model, Electron. Comm. Probab., 18, 1-16 (2013) · Zbl 1306.60119 |
| [7] | S. N. Majumdar, Handbook of Random Matrix Theory, Oxford University Press (2011). |
| [8] | Mayerhofer, E.; Pfaffel, O.; Stelzer, R., On strong solutions for positive definite jump diffusions, Stoch. Process. Appl., 121, 2072-2086 (2011) · Zbl 1225.60096 · doi:10.1016/j.spa.2011.05.006 |
| [9] | McH. P. Kean, Stochastic Integrals, Academic Press, New York (1969). · Zbl 0191.46603 |
| [10] | L. Pastur and M. Shcherbina, “Eigenvalue distribution of large random matrices,” Math. Surveys Monogr., 171 (2011). · Zbl 1244.15002 |
| [11] | Peng, L.; Falei, W., On the comparison theorem for multi-dimensional G-SDEs, Statist. Probab. Lett., 96, 38-44 (2015) · Zbl 1319.60130 · doi:10.1016/j.spl.2014.09.010 |
| [12] | S. Peng, “Nonlinear expectations and stochastic calculus under uncertainty with robust central limit theorem and G-Brownian motion,” arXiv:1002:4546v1 (2010). |
| [13] | Soner, HM; Touzi, N.; Zhang, J., Martingale representation theorem for the G-expectation, Stoch. Process. Appl., 121, 265-287 (2011) · Zbl 1228.60070 · doi:10.1016/j.spa.2010.10.006 |
| [14] | Z. Sun, X. Zhang, and J. Guo, “A stochastic maximum principle for processes driven by G-Brownian motion and applications to finance,” arxiv:1402.6793v2 [math.OC] (2014). · Zbl 1386.49038 |
| [15] | H. Zhang, “A complex version of G-expectation and its application to conformal martingale,” arxiv: 1502.02787v1 [math.PR] (2015). |
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