Research On Theories And Applications Of VaR Based On Nonlinear Expectations

The most classical risk measure is VaR?Value-at-Risk?.But VaR has some shortcomings,for example,it does not satisfy subadditivity,that is,the diversification of investment will increase the risk value of the portfolio.Recognizing the shortcoming of VaR,scholars proposed a series of risk measure models,such as,coherent risk measure,convex risk measure,WVaR.Expected utility theory is one of the classical theories in economics,but it was challenged by some empirical experiments such as Allais's paradox and Ellsberg's paradox,due to the linearity of mathematical expectation.In order to overcome the shortcoming of linear expected utility theory,many scholars turned to study nonlinear expectations and nonadditive probability measures.At present,the main nonlinear expectations are Choquet expectation,g-expectation and G-expectation.In this paper,we mainly study the representation theorems for WVaR with respect to a capacity and the properties of VaR with respect to a g-probability.In Chapter 1,we briefly introduce the research background and status of nonlinear expectations and related risk measure,and also introduce the main results of our paper.In Chapter 2,we mainly study the representation theorems for WVaR with respect to a capacity.In Theorem 2.1,we prove that WVaR with respect to a capacity can be represented as Choquet integral with respect to a corresponding distorted capacity.In Theorem 2.2,we prove that for a concave capacity with continuity from above,WVaR with respect to a capacity can be represented as the maximum value of a family of linear expectations.Moreover,we study a special kind of WVaR with respect to a capacity.In Chapter 3,we discuss the properties of VaR with respect to ag-probability and we prove that VaR with respect to ag-probability satisfies monotonicity,translation invariance and positive homogeneity.In Theorem 3.1 and Theorem 3.2,under some generator assumptions and terminal conditions,we give the value of VaR with respect to a g-probability of Brownian motion a^1/2B_T/a and -B_T,and tell the difference between classical VaR and VaR with respect to a g-probability.In Chapter 4,we summarizes the contents of our paper.