Abstract
In this paper, we consider a special nonlinear expectation problem on the special parameter space and give a necessary and sufficient condition for the existence of the solution. Meanwhile, we generalize the necessary and sufficient condition to the two-dimensional moment problem. Moreover, we use the maximum entropy method to carry out a kind of concrete solution and analyze the convergence for the maximum entropy solution. Numerical experiments are presented to compute the maximum entropy density functions.
Highlights
The sublinear expectation Ê introduced by Peng [1,2] can be regarded as the supremum of a family of linear expectations { Eθ : θ ∈ Θ}, that is, Ê[ φ( X )] = sup Eθ [ φ( X )], (1)θ∈Θ where φ( x ) is a local Lipschitz continuous function and Θ is the parameter space.It is evident that the sublinear expectation defined by (1) depends on the choice of parameter space Θ.Different spaces will result in different nonlinear expectations
When N = 1, Peng [3] gave the definition of the independent and identically distributed random variable and proved the weak law of large numbers (LLN) under the sublinear expectation and Condition (2)
The new LLN and central limit theorem (CLT) are the theoretical foundations in the framework of sublinear expectation
Summary
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