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This paper presents the mean-square joint state filtering and parameter identification problem for uncertain linear stochastic systems with unknown parameters in both state and observation equations, where the unknown parameters are considered Wiener processes. The original problem is reduced to the filtering problem for an extended state vector that incorporates parameters as additional states. The resulting filtering system is polynomial in state and linear in observations. The obtained mean-square filter for the extended state vector also serves as the mean-square identifier for the unknown parameters. Performance of the designed mean-square state filter and parameter identifier is verified for both, positive and negative, parameter values.