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\\dot{L}_t(x),~~~ t>0, ~x\\in\\mathbb{R},\n  \\end{eqnarray*} where $\\dot{L}$ denotes an $\\alpha$-stable white noise on $\\mathbb{R}_+\\times \\mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $\\beta$-H\\\"older continuous for $1<\\alpha<2$ and $0<\\beta<1$.\n  For $G\\equiv0$ and 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