g-Expectations with application to risk measures

Abstract

Peng introduced a typical ltration consistent nonlinear expectation, called a g-expectation in [40]. It satis es all properties of the classical mathematical expectation besides the linearity. Peng's conditional g-expectation is a solution to a backward stochastic di erential equation (BSDE) within the classical framework of It^o's calculus, with terminal condition given at some xed time T. In addition, this g-expectation is uniquely speci ed by a real function g satisfying certain properties. Many properties of the g-expectation, which will be presented, follow from the speci cation of this function. Martingales, super- and submartingales have been de ned in the nonlinear setting of g-expectations. Consequently, a nonlinear Doob-Meyer decomposition theorem was proved. Applications of g-expectations in the mathematical nancial world have also been of great interest. g-Expectations have been applied to the pricing of contingent claims in the nancial market, as well as to risk measures. Risk measures were introduced to quantify the riskiness of any nancial position. They also give an indication as to which positions carry an acceptable amount of risk and which positions do not. Coherent risk measures and convex risk measures will be examined. These risk measures were extended into a nonlinear setting using the g-expectation. In many cases due to intermediate cash ows, we want to work with a multi-period, dynamic risk measure. Conditional g-expectations were then used to extend dynamic risk measures into the nonlinear setting. The Choquet expectation, introduced by Gustave Choquet, is another nonlinear expectation. An interesting question which is examined, is whether there are incidences when the g-expectation and the Choquet expectation coincide.