Financial Investment Decisions And Risk Measures Under Knightian Uncertainty

Investors are inevitably facing with financial decision-making confusion in financial un-certainty environments. After financial crisis, more and more people have realized the un-certainty of financial risk, which makes the investors use more prudent risk measurement methods and more robust financial investment strategy. Knightian uncertainty engages a set of probability distributions to represent an agent’s uncertainty. Besides, some famous para-dox, such as Allais paradox and Ellsberg paradox, also motivate people to employ nonlinear expectations framework considering economic and financial problems. The dissertation aims at using the nonlinear expectation theory to study the financial investment decisions and risk measurements under Knightian uncertainty financial market.In Chapter1, some research backgrounds and main contributions of the dissertation are briefly introduced. Starting from Chapter2, we use the nonlinear expectation theory to con-duct deep explorations of several problems in risk measurement theory and financial decisions. Some significant progresses are obtained.Chapter2examines the comparative statics of "risk averse" and "ambiguity averse" to-gether in a continuous-time framework. We demonstrate that, the more the Arrow-Pratt co-efficient of absolute risk aversion under ambiguity the less optimal demand on risky asset (Theorem2.1). Risk aversion and ambiguity aversion jointly affect the demand on the risky asset in a monotonic theory. The comparative statics allow for distinct estimated model param-eters across agents and multi-economy (Theorem2.2-Theorem2.3). These results accurately describe that the risk-preference, the ambiguity aversion, the volatility, the risk-free interest rate, and Sharpe ratio all these factors contribute to this monotonic theory. As a special case, we consider one agent who invests two different economies:domestic economy and foreign economy. Some lights are shed on the home bias investment puzzle. The results of Chapter2enrich the Arrow-Pratt’s static results, and deepen the results of Borell [2007] and Xia [2011] in a continuous-time setting.Chapter3considers the uncertainty orders theory in a sublinear expectation space, we describe the uncertainty orders from two different viewpoints. One is from the view of the sublinear operator, we give the characterizations of uncertainty orders for the maximal dis-tributions, G-normal distributions and G-distributions, which are the most important random vectors in the sublinear expectation space theory (Theorem3.1-Theorem3.3). On the other hand, from the viewpoint of a family of probability measures, we discuss some characteriza-tions for the bounded variables in sublinear expectation space by capacity orders (Theorem3.4-Theorem3.5). Chapter3’s results extend the classical results for the stochastic orders in a probability space.Chapter4establishes some representation theorems for the data-based convex and qua-siconvex risk statistics with scenario analysis. Motivated by risk measures, we obtain the representation theorems for the comonotonic convex risk statistics, law-invariance convex risk statistics, comonotonic quasiconvex risk statistics and law-invariance quasiconvex risk statistics respectively by using the dual convex analysis theory (Theorem4.1-Theorem4.4). Some connections with the risk measures are also discussed. It provides some basis for the study of the comonotonic quasiconvex risk measures and risk management. Chapter4’s re-sults generalize the results for the natural risk statistics in Ahmed-Filipovic-Svindland [2008] and Kou-Peng-Heyde [2013].Chapter5discusses the existence and uniqueness of Lp(p≥1) solutions for a class of backward Stochastic differential equations(BSDEs). It is very helpful for extending an important nonlinear expectation, g-expectation, in Lp space. An existence result of Lp(p＞1) solution for BSDE with a kind of discontinuity generator is obtained (Theorem5.1). And an existence and uniqueness result of L1solution for BSDE with quasi-Holder continuity generator is proved (Theorem5.2). The results of Chapter5improve the results of Jia [2006], Jia [2008b], Fan-Liu [2010] and Fan-Jiang (2012b, in Chinese).