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We give a theoretical framework of stochastic non-canonical Hamiltoniansystems as well as their modified symplectic structure which is namedstochastic K-symplectic structure. The framework can be applied to the study ofthe Lotka--Volterra model perturbed by external noises. In terms of internalproperties of the stochastic Lotka--Volterra model, we propose different kindsof stochastic K-symplectic integrators which could inherit the positivity ofthe solution. The K-symplectic conditions are also obtained to ensure that theproposed schemes admit the same geometric structure as the original system.Besides, the first-order condition of the proposed schemes in $L^1(\Omega)$sense are given based on the uniform boundedness of both the exact solution andthe numerical one. Several numerical examples are illustrated to verify aboveproperties of proposed schemes compared with non-K-symplectic ones.

Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka--Volterra model