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Observability estimate and state observation problems for stochastic hyperbolic equations

This paper is devoted to a study of the boundary and internal state observation problems for stochastic hyperbolic equations. For this, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by the global Carleman estimate for stochastic hyperbolic equations. By these inequalities, we get stability estimates for the state observation problems. As a consequence, we also establish a unique continuation property for stochastic hyperbolic equations.