In this paper, we present some properties of m-convex stochastic processes. The most important results are: a generalization of the sandwich theorem and a result on Hyers-Ulam stability, given for m-convex functions. The first result allows us to bound an m-convex stochastic process by two convex stochastic processes, and the second allows us to approximate controlled perturbations of an m-convex stochastic process by an m-convex function. As a consequence of these two results, we obtain a Hermite-Hadamard type inequality for m-convex stochastic processes.
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