Sub-Monotonicity Sublinear Expectations And Martingales

In this paper, we mainly study the inequality of the simple sub-monotone SL- mar-tingale under the linear’s frame. First. a family of mapping defined through sublinear expectation. Under such a family of mapping, we define the martingale, and then dis-cuss the inequality under the simple martingale. In the last part, we set up the general definition of sub-monotone SL-expectation and discuss the related properties.