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Register a new userAn Analytical Formula for Spectrum Reconstruction
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Added by
Zhibo Dai
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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We study the spectrum reconstruction technique. As is known to all, eigenvalues play an important role in many research fields and are foundation to many practical techniques such like PCA(Principal Component Analysis). We believe that related algorithms should perform better with more accurate spectrum estimation. There was an approximation formula proposed, however, they didnt give any proof. In our research, we show why the formula works. And when both number of features and dimension of space go to infinity, we find the order of error for the approximation formula, which is related to a constant $c$-the ratio of dimension of space and number of features.
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The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map $Gamma_{0,a}$ on $[0,a]$ for any $a>0$ is derived. Specifically, it is shown that on the space $mathcal{D}[0,infty)$ of right-continuous functions with left limits taking values in $mathbb{R}$, $Gamma_{0,a}=Lambda_acirc Gamma_0$, where $Lambda_a:mathcal{D}[0,infty)tomathcal{D}[0,infty)$ is defined by [Lambda_a(phi)(t)=phi(t)-sup_{sin[0,t]}biggl[bigl( phi(s)-abigr)^+wedgeinf_{uin[s,t]}phi(u)biggr]] and $Gamma_0:mathcal{D}[0,infty)tomathcal{D}[0,infty)$ is the Skorokhod map on $[0,infty)$, which is given explicitly by [Gamma_0(psi)(t)=psi(t)+sup_{sin[0,t]}[-psi(s)]^+.] In addition, properties of $Lambda_a$ are developed and comparison properties of $Gamma_{0,a}$ are established.
In this note, we prove an It^o formula for the isochron map of a reaction-diffusion system. This follows the proof of a new result which states that the second derivative of the isochron map of a reaction-diffusion system is trace class. This result, in turn, is a corollary of Proposition 2.3, which guarantees that the first and second Frechet derivatives of the flow of a reaction-diffusion system with respect to initial conditions are trace class.
In this paper we provide an expansion formula for Hawkes processes which involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the well-known integration by parts formula (or more precisely the Mecke formula) for Poisson functional. Our approach allows us to provide an expansion of the premium of a class of cyber insurance derivatives (such as reinsurance contracts including generalized Stop-Loss contracts) or risk management instruments (like Expected Shortfall) in terms of so-called shifted Hawkes processes. From the actuarial point of view, these processes can be seen as stressed scenarios. Our expansion formula for Hawkes processes enables us to provide lower and upper bounds on the premium (or the risk evaluation) of such cyber contracts and to quantify the surplus of premium compared to the standard modeling with a homogenous Poisson process.
We introduce a new interacting particles model with blocking and pushing interactions. Particles evolve on the positive line jumping on their own volition rightwards or leftwards according to geometric jumps with parameter q. We show that the model involves a Pieri-type formula for the orthogonal group. We prove that the two extreme cases - q=0 and q=1 - lead respectively to a random tiling model studied by Borodin and Kuan and to a random matrix model.
We derive an Ito-formula for the Dawson-Watanabe superprocess, a well-known class of measure-valued processes, extending the classical Ito-formula with respect to two aspects. Firstly, we extend the state-space of the underlying process $(X(t))_{tin [0,T]}$ to an infinite-dimensional one - the space of finite measure. Secondly, we extend the formula to functions $F(t,X_t)$ depending on the entire paths $X_t=(X(swedge t))_{s in [0,T]}$ up to times $t$. This later extension is usually called functional Ito-formula. Finally we remark on the application to predictable representation for martingales associated with superprocesses.
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