This article addresses the synthesis of static output feedback stabilization via employing event‐based control, for discrete‐time Markov jump systems subject to limited bandwidth and mismatched modes. The event‐triggered communication strategy with an extra dynamic variable is considered in the design of controller to alleviate transmission burden. Then, for the purpose of further improving system performance, a special triggering threshold constructed as a diagonal matrix is introduced. Asynchronous behavior caused by mismatched modes between the controller and controlled systems is depicted by utilizing a hidden Markov model. With the assistance of output feedback theory, an asynchronous event‐based output feedback control law is sufficiently formulated to achieve the stochastic stabilization with desired H∞$$ {H}_{\infty } $$ performance. Ultimately, an illustrative example is implemented to verify the feasibility of theoretical results.

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