On Choquet Integral And Set-valued Choquet Integral With Their Applications In Finance

In the research of economic and financial problems, we often meet some phe-nomena that can not be explained incompletely by the theory of classical probabil-ity measures and linear mathematical expectations. For this reason, nonadditivecapacities and nonlinear Choquet integrals have been got more and more atten-tion. The main goal of this thesis is to improve some results of Choquet integraland set-valued Choquet integral, and to discuss its applications in finance.This thesis is mainly concerned with four parts. The first part is about sev-eral theoretical results of Choquet integral. We firstly study the properties ofthe submodular capacity and also introduce several special submodular capac-ities. Under submodular capacity, we then introduce the concept of conditionalChoquet expectation and also we prove its some properties, which include positivehomogeneity, translation invariance, asymmetry, subadditivity and monotonicity.Moreover, we state that Crinequality, Jensen’s inequality, Ho¨lder’s inequality andMinkowski’s inequality still hold for this kind of conditional Choquet expectations.Furthermore, by using backward stochastic diferential equalities we show severalinequalities of upper and lower probability, which are upper and lower bound of aset of risk neutral probability measures respectively. Finally we study the Choquetintegral with respect to bi-capacity, and this kind of integral can be used to modelthe new paradox that Grabisch put forward when he studied the multi-criteriadecision making problem in2006.The second part is about two applications of the Choquet integral in finance. Firstly, we propose a new measure approach of ambiguous risk aversion undersome capacity μ. By using the Choquet expectation value and Choquet variance,we introduce the concept of ambiguous risk premium ρuXof a risk asset X for a riskaversive individual whose utility function is u, and we investigate some propertiesof the ambiguous risk premium by using the properties of Choquet integral andJensen’s inequality. In particular, we introduce the simple case and an exampleof ambiguous risk aversion under distorted probability. Secondly we discuss theapplication of the Wang transform, which is a kind of special distorted probability,in European option pricing.The third part is about set-valued Choquet integral with its application.Firstly we investigate some properties of set-valued Choquet integral. Then wediscuss the convergence and uniform integrability of the sequence of set-valuedrandom variables in capacity space. We extend several convergence theorems ofrandom variables in capacity space, such as Egorof’s theorem, Lebesgue’s theoremand Riesz’s theorem, to the cases of set-valued random variables. We also inves-tigate the space of the Choquet integrably bounded set-valued random variables,and in this space, we introduce the concepts of convergence in mean and uni-form integrability. Furthermore, we prove Fatou’s Lemmas, Lebesgue dominatedconvergence theorem and monotone convergence theorem of set-valued Choquetintegrals under the weaker conditions than those in previous works. Finally wepropose a new approach of risk measurement based on the integral of Choquetintegrably bounded set-valued random variable with respect to capacity. In orderto show this method of risk measurement to be applicable, we consider a realisticcase study using stock data in Shanghai stock exchanges. The fourth part is about the set-valued capacity and the Choquet integralwith respect to set-valued capacity. The set-valued capacity is not only the gener-alization of capacity, but also the extension of set-valued measure in one dimension.We firstly discuss some properties of set-valued capacity and define the concept ofthe selector set of set-valued capacity; then we introduce the Choquet integral withrespect to a set-valued capacity; finally we study some properties of the Choquetintegral with respect to a set-valued capacity.