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  A sufficient condition for a general nonlinear stochastic system to have an L_2-L_∞ gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton Jacobi inequality(HJI) .By a slight modification,an equivalent form of this condition is also proposed.Based on this modified criterion,the existence condition of an L_2-L_∞ filter is given by a second-order nonlinear HJI,and the filter matrices can be obtained by solving such an HJI.……