In this chapter we survey risk measures, nonlinear probability models, and their relationships based on recent research by Peng, Chen and coauthors. We also review recent results about limit theorems for nonlinear probability measures. If a market is complete, the fair price of a contingent claim can be represented by a linear expectation of the discounted payoff of the contingent claim. Statistically, such a linear expectation can be calculated by using Monte Carlo simulations. However, in reality, the financial market is incomplete, and thus the question of how one should determine the prices of contingent claims in an incomplete market is an important issue in financial engineering. Risk measures and nonlinear expectations, which are generalizations of linear expectations, can be used to price contingent claims in an incomplete market. For example, coherent risk measures, convex risk measures, Choquet expectations and Peng's g-expectations have been used in option pricing in an incomplete market. Obviously, the different risk measures typically yield different prices. Then, a natural question arises: what are relationships among such nonlinear expectations? Statistically, how should one calculate nonlinear expectations? In this chapter, we provide a review of recently obtained relationships among the four kinds of nonlinear expectations. we also review a recent result on the strong law of large numbers and a law of iterated logarithm under nonlinear probability measures.
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