4,301 research outputs found

    On the almost sure running maxima of solutions of affine stochastic functional differential equations

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    This paper studies the large fluctuations of solutions of scalar and finite-dimensional affine stochastic functional differential equations with finite memory as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the running maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process

    The size of the largest fluctuations in a market model with Markovian switching

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    This paper considers the size of the large fluctuations of a stochastic differential equation with Markovian switching. We concentrate on processes which obey the Law of the Iterated Logarithm, or obey upper and lower iterated logarithm growth bounds on their almost sure partial maxima. The results are applied to financial market models which are subject to random regime shifts. We prove that the security exhibits the same long-run growth properties and deviations from the trend rate of growth as conventional geometric Brownian motion, and also that the returns, which are non-Gaussian, still exhibit the same growth rate in their almost sure large deviations as stationary continuous-time Gaussian processes

    Symmetry of Anomalous Dimension Matrices for Colour Evolution of Hard Scattering Processes

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    In a recent paper, Dokshitzer and Marchesini rederived the anomalous dimension matrix for colour evolution of ggā†’gggg \to gg scattering, first derived by Kidonakis, Oderda and Sterman. They noted a weird symmetry that it possesses under interchange of internal (colour group) and external (scattering angle) degrees of freedom and speculated that this may be related to an embedding into a context that correlates internal and external variables such as string theory. In this short note, I point out another symmetry possessed by all the colour evolution anomalous dimension matrices calculated to date. It is more prosaic, but equally unexpected, and may also point to the fact that colour evolution might be understood in some deeper theoretical framework. To my knowledge it has not been pointed out elsewhere, or anticipated by any of the authors calculating these matrices. It is simply that, in a suitably chosen colour basis, they are complex symmetric matrices.Comment: 3 page

    Logistics imbalance: Authority vs systems

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    Spectra of phase point operators in odd prime dimensions and the extended Clifford group

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    We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ESL(2,FN){\rm ESL}(2, \mathbb{F}_N).Comment: Latex, 19page
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