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- W63572788 abstract "The asymptotic behavior of the linear stochastic differential equation in Rd $$dx = Ax dt + mathop Sigma limits_{i = 1}^m B_i x o dW_i (t), x(O) = x_O ne O,$$ is studied. It is known (see [2]) in these Proceedings) that the projection of the solution x(t;xo) onto the unit sphere has a unique invariant probability, while $$lambda = mathop {lim }limits_{t to infty } tfrac{1}{t} log |x(t;x_O )|$$ exists a.s. and is essentially independent of chance and of xo. Here we prove that $$g(p) = mathop {lim }limits_{t to infty } tfrac{1}{t} log E|x(t;x_O )|^P , p in R,$$ exists and is independent of xo. Further, g: R → R is convex and analytic with g(p)/p increasing (to γ, say) with g(O)=O and g' (O)=λ. The cases γ<∞ and γ=∞ are characterized. This answers the question of when sample stability (λ<O) implies moment stability (g(p)<O) for all p>O. In case trace A = trace Bi=O for all i we have g(−d)=O, this enabling us to characterize the cases λ=O and λ>O by a simple criterion." @default.
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- W63572788 date "1986-01-01" @default.
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- W63572788 title "Almost sure and moment stability for linear ito equations" @default.
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- W63572788 doi "https://doi.org/10.1007/bfb0076837" @default.
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