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Fractional Wishart Processes and $varepsilon$-Fractional Wishart Processes with Applications

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 Added by Nan-Jing Huang
 Publication date 2016
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and research's language is English




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In this paper, we introduce two new matrix stochastic processes: fractional Wishart processes and $varepsilon$-fractional Wishart processes with integer indices which are based on the fractional Brownian motions and then extend $varepsilon$-fractional Wishart processes to the case with non-integer indices. Both of two kinds of processes include classic Wishart processes when the Hurst index $H$ equals $frac{1}{2}$ and present serial correlation of stochastic processes. Applying $varepsilon$-fractional Wishart processes to financial volatility theory, the financial models account for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets returns but also between their volatilities and for stochastic serial correlation of the relevant assets.



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Let $X^{(delta)}$ be a Wishart process of dimension $delta$, with values in the set of positive matrices of size $m$. We are interested in the large deviations for a family of matrix-valued processes ${delta^{-1} X_t^{(delta)}, t leq 1 }$ as $delta$ tends to infinity. The process $X^{(delta)}$ is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.
120 - Jason Leung 2020
This article is concerned with the joint law of an integrated Wishart bridge process and the trace of an integrated inverse Wishart bridge process over the interval $ left[0,tright] $. Its Laplace transform is obtained by studying the Wishart bridge processes and the absolute continuity property of Wishart laws.
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We consider eigenvalues of generalized Wishart processes as well as particle systems, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, central limit theorems for the Dysons Brownian motion and the eigenvalues of the Wishart process are recovered under slightly more general initial conditions, and a central limit theorem for the eigenvalues of a symmetric Ornstein-Uhlenbeck matrix process is obtained.
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