Stochastic differential equations for orthogonal eigenvectors of (G,ε)-Wishart process related to multivariate G-fractional Brownian motion
DOI:
https://doi.org/10.5269/bspm.51618Abstract
In the present paper, we introduce a new process called multivariate G-fractional Brownian motion (B_{t}^{H}) where the Hurst parameter H is a diagonal matrix. Moreover, we give an approximation (R_{t}^{ε}) of Riemann-Liouville process of (B_{t}^{H}) by G-Itô's processes. Then we give stochastic differential equations for orthogonal eigenvectors of (G,ε)-Wishart fractional process defined by R_{t}^{ε}(R_{t}^{ε})^{∗}, which has 0 and |R_{t}^{ε}|² as eigenvalues. An intermediate asymptotic comparison result concerning the eigenvalue |R_{t}^{ε}|² is also obtained
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[3] Achard S, Gannaz I. Multivariate wavelet Whittle estimation in long-range dependence. Journal of Time Series Analysis, Wiley-Blackwell, 37 (4) 476 − 512 (2016). https://doi.org/10.1111/jtsa.12170
[4] Boutabia H, Grabsia I. Chaotic expansion in the G−expectation space. Opuscula Mathematica 33(4)647 − 666 (2013). https://doi.org/10.7494/OpMath.2013.33.4.647
[5] Chen W. Fractional G−White Noise Theory, Wavelet Decomposition for Frac- tional G−Brownian Motion and Bid-Ask Pricing for European Contingent Claim Under Uncertainty, Preprint arXiv:1306.4070v1 [math:PR] (2013).
[6] Bru MF.Diffusions of perturbed principal component analysis. Journal of Multivariate Analysis 29127 − 136 (1998).
[7] Dunga NT, Thao TH. An approximate approach to fractional stochastic integration and its applications. Brazilian Statistical Association 24 (1) 57 − 67. (2010) https://doi.org/10.1214/08-BJPS013
[8] Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation application to G−Brownian motion paths. Potential Analysis. 34 (1) 139 − 161. https://doi.org/10.1007/s11118-010-9185-x
[9] Didier G, Pipiras V. Integral representations of operator fractional Brownian motion. Bernoulli 17 (1) : 1 − 33. (2010) https://doi.org/10.3150/10-BEJ259
[10] Gao F. Pathwise properties and homomorphic flows for stochastic differential equations driven by G−Brownian motion. Stochastic Processes and their Applications . 119 (10) 3356 − 3382 (2009) . https://doi.org/10.1016/j.spa.2009.05.010
[11] Hu Y, Øksendal B. Fractional white noise calculus and applications to finance. Infinite Dimensional Analysis. Quantum Probability and Related Topics 6 (1) 1 − 32 (2003). https://doi.org/10.1142/S0219025703001110
[12] Boutabia H, Meradj Si, Stihi S. Stochastic differential equations for eigenvalues and eigenvectors of a G−Wishart process with drift. Ukrainian Mathematical Journal 71 (4) 502 − 515 (2019). https://doi.org/10.1007/s11253-019-01664-1
[13] Majumdar SN. Handbook of Random Matrix Theory. Oxford University Press: (2011).
[14] Mandelbrot B, Van Ness J. Fractional Brownian motions, fractional noises and applications. Society for Industrial and Applied Mathematics Review. 10 (4) 422 − 437 (1968). https://doi.org/10.1137/1010093
[15] Peng S. Multi-dimensional G−Brownian motion and related stochastic calculus under G−expectation, Stochastic Processes and their Applications 118 (12) 2223 − 2253 (2008). https://doi.org/10.1016/j.spa.2007.10.015
[16] Peng S. G−Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty; arXiv 0711.2834v1 [math.PR] (2007).
[17] Peng S. G−expectation, G−Brownian motion and related stochastic calculus of Itˆo type. Stochastic Analysis and Application, Abel Symposia 2541 − 567 (2007). https://doi.org/10.1007/978-3-540-70847-6
[18] Stihi S, Boutabia H, Meradji S. Stochastic diferential equations for Random matrices processes in the nonlinear framework, Opuscula Mathematica 38 (2) 261 − 283 (2018). https://doi.org/10.7494/OpMath.2018.38.2.261
[19] Yue J, Huang NJ. Fractional Wishart Processes and ε−Fractional Wishart Processes with Applications. Computers and Mathematics with Applications 75 (8) 2955 − 2977 (2018). https://doi.org/10.1016/j.camwa.2018.01.024
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2022-12-21
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