Capacity Limit Theory And Nonlinear Expectation With Applications To Finance

Kolmogorov established the system of axioms about theory of probability by Lebes-gue's theories of measure and integration in1993, which make theory of probability to be the important tool of investigating the random or uncertainty phenomena. However, it has been shown that such additivity assumption of probabilities or linear expectation is not feasible in many areas of applications because the uncertainty and ambiguity phenomena, for example, Allais and Ellsberg paradoxes. The mathematical theory of non-additive measure and integral got its first important contribution with Choquet's Theory of Capacities in1954. Since then, capacities and Choquet integral are studied by many researchers, for example, Huber and Strassen(1973), Walley and Fine(1982), Schmeidler(1989), Denneberg(1994), Maccheroni and Marinacci(2005), Chen(2010), and so on. Peng investigated the theory of nonlinear expectations from a new point of view in2006. This theory not based on probability space, but on nonlinear expectation space. Along with the notion of independence under sublinear expectation, the central limit theorem under sublinear expectation was proved by using a deep interior estimate of fully nonlinear partial differential equation, and G-Brownian motion as well as G-Ito calculus are provided by Peng. From the representation of a sublinear expectation, we know that there is a capacity induced by sublinear expectation. Motivated by the works of Kolmogorov, Choquet, Peng and Chen, we mainly investigate the problems about the limit theories of capacities, G-Brownian motion, and G-Ito calculus as well as their applications in this dissertation. We give a new urn model with ambiguity and obtain strong laws of large numbers and central limit theorem for capacities, a weighted central limit theorem under sublinear expectations. Meanwhile, a Berry-Esseen theorem under linear expectation is proved by borrowing PDE, some properties about sublinear expectation martingale in discrete time and properties of increments for G-Brownian motion are given. At last, under some integral-Lipschitz assumptions, the stability theorems for G-SDE and G-BSDE are proved. The existence and uniqueness of the solution for forward and backward stochastic differential equations driven by G-Brownian motion is also proved. Last but not least, stochastic optimal control problems under G-expectation and optimal portfolio selection model under volatility uncertainty are discussed, the optimal rules and mutual fund theorem are presented. Specifically, this dissertation consists of five chapters, whose main results are summed up as follows:In Chapter1, we mainly consider the central limit theorem for weighted sum of independent random variables under sublinear expectations, Berry-Esseen theorem under sublinear expectation and linear expectation, central limit theorem and weak law of large numbers for capacities. In§1.1, motivated by the work of Peng [79], Li and Shi [63], we investigate a central limit theorem for weighted sum of independent random variables under sublinear expectations and obtain the law of large numbers of independent random variables under sublinear expectations, see Theorem1.1.8and Corollary1.1.12. Using the method of proving weighted central limit theorem, we obtain a Berry-Esseen Theorem under Sublinear Expectation, see Theorem1.1.14. In§1.2, we prove a Berry-Esseen Theorem under linear Expectation by using heat equation and Taylor expansion, see Theorem1.2.2.In§1.3, we investigate the Central limit theorem for capacities induced by sublin-ear expectations as follows:Let{Xi}i=1∞be a sequence of i.i.d. random variables with E[X1]=E[-X1]=0. Then and where y is a point at which V (v) is continuous.In§1.4, we give a new ambiguity urn model and introduce capacities (V, v) as well as so-called maximum-minimum expectations (E, E). Based on the ambiguity urn model, we prove that for random variables{Xi}1≤i≤n in ambiguity urn model and any y∈B, we have and Next we extend the ambiguity urn model to general case see Theorem1.4.12.In Chapter2, we introduce the orthogonal notion under ε and consider some results about<ε L-submartingale as well as some useful inequalities. A typical result is Doob's inequality (see Theorem2.2.10).Peng [77] introduced the G-Brownian motion and the related quadratic variation process in2006. G-Brownian motion has many interesting properties which nontriv-ially generalize the classical case. In Chapter3, some new properties and interesting estimations of mutual variation process for G-Brownian Motion are presented, Kunita-Watanabe inequalities (see Theorem3.1.20) and Tanaka formula (see Theorem3.1.23) for multi-dimensional G-Brownian motion are obtained.In§3.2, following the ideas of Csorgo and Revesz [24], we consider the increas-ing quantitative properties of G-Brownian motion. Some useful corollaries are given. Briefly, suppose that (Bt)t≥0be a1-dimensional G-Brownian motion with E[B12]=σ2,-E[-B12]=σ2. Let aT(T≥0) be a nondecreasing function of T for which (ⅰ)0<aT<T,(ⅱ)aT/T is nonincreasing and (ⅲ) lim/T→∞>0or aT=c(0<c≤T), thenIn Chapter4, we consider the stability theorems of G-stochastic differential equa-tions and G-backward stochastic differential equations under integral-Lipschitz condi-tions, see Theorem4.1.5and Theorem4.1.13. Inspired by the method of Antonelli, under some suitable conditions, we prove the existence and uniqueness of the solution of the following system: where the initial condition x∈R, the terminal data ξ∈LG2(HT;R), and b, h, σ, f, g are given functions satisfying b(·, x, y), h,{·, x, y), σ(·, x, y), f(·, x. y), g(·,x, y)∈MG2([0,T]; R) for any (x, y)∈R2and the Lipschitz condition.In§4.2. we consider the exponential stability for G-stochastic differential equations. Firstly, given an exponentially stable stochastic linear system where the initial condition X0∈LG2(Ht0; Rn), X=(X1,…, Xn)T, A is a constant n x n matrix. Assume that some parameters are excited or perturbed by G-Brownian motion, and the perturbed system has the form where Bt is a d-dimensional G-Brownian motion, and σ: R+×Rn×Ω→Rn×d satisfies the conditions for the existence and uniqueness of the solution, its solution is denoted by X(t, t0, X0), suppose there exist positive constants C and α, such that for all x∈Rn and all sufficiently large t,||σ(t,x)||2≤Ce-2||A||t q.s., and limsup log||eAt||2/t≤-α. Then for all t0≥0and any X0∈LG2(Ht0; Rn). Meanwhile, we also obtain a generalization version of Theorem4.2.1, see Theorem4.2.4.In§4.3, we investigate the stochastic optimal control problems under G-expectation and obtain dynamic programming principle: For any δ∈[0, T—t], we have We prove the value function u(t, x) is a viscosity solution of the following fully nonlinear second-order PDE: where More complicated form can be found in Theorem4.3.14.Merton [72] investigated the optimal portfolio selection problems under the lin-ear expectation and volatility is constant, in Chapter5, an optimal portfolio selection model under volatility uncertainty in the G-expectation space is established, the expres-sions about the optimal investment and consumption rules are presented, see Theorem5.2.2. Meanwhile, we also obtain the mutual fund theorem under volatility uncertainty, see Theorem5.3.1. In order to illustrate the optimal portfolios depend on the maximal and minimal volatility of underlying asset, in§5.4, we only consider two assets (stock and bond) and particular utility function, the explicit optimal portfolio is given.