In this paper is introduced the concept of $(m,h_{1},h_{2})$-convexity for stochastic processes dominated by other stochastic processes with the same property, some mean square integral Hermite-Hadamard type inequalities for this kind of generalized convexity are established and from the founded results, other mean square integral inequalities for the classical convex, $ s$-convex in the first and second sense, $P$-convex and $ MT$-convex stochastic processes are deduced.