Dynamically consistent nonlinear evaluations with their generating functions in \(L^p\)
The paper is concerned with an important problem of stochastic finance: the evaluation of the discounted value of a derivative asset. This problem is treated as a dynamically consistent nonlinear evaluation (expectation) and the first results in this direction was given by \textit{S. Peng} [C. R., Math., Acad. Sci. Paris 339, No. 8, 585--589 (2004; Zbl 1065.60087); Acta Math. Appl. Sin., Engl. Ser. 20, No. 2, 191--214 (2004; Zbl 1061.60063)] in the form the well-known \(g\)-evaluation. The author follows a result of \textit{F. Hu} and \textit{Z. Chen} [Stat. Probab. Lett. 80, No. 3--4, 191--195 (2010; Zbl 1181.60084)] and study dynamically consistent nonlinear evaluations in \({\mathcal L}\); they obtain that, under a domination condition, an \({\mathcal F}_t\)-consistent evaluation is an \({\mathcal E}^g\)-evaluation in \({\mathcal L}\), which generalizes the famous results given by S. Peng [loc. cit.] in 2004--2005. Furthermore, without the assumption that the generating function \(g(t,\omega,y,z)\) is continuous with respect to \(t\), the author gives some useful characterizations of \({\mathcal E}^g\)-evaluations and extends some similar results given by \textit{E. Rosazza Gianin} [Insur. Math. Econ. 39, No. 1, 19--34 (2006; Zbl 1147.91346)] and \textit{L. Jiang} [Ann. Appl. Probab. 18, No. 1, 245--258 (2008; Zbl 1145.60032)]. In the second section, the author shows that, for each well-defined function \(g\), the solution of a corresponding backward stochastic differential equation satisfies some adequate conditions of dynamically consistent nonlinear evaluations. An important result is given in Theorem 2.3 which extends similar results given by F. Hu and Z. Chen [loc. cit.]. In the next section, the author proves (Theorem 3.1) the first main result which we mentioned above. In Section 4, some important results (Propositions 4.1--4.6) are proved which show that the behavior of a \(g\)-evaluation is perfectly reflected by its generating function \(g\). Also, two interesting examples are given. In the last section, the author gives the proofs of the propositions from the previous section. The basic idea of the proofs comes from a result by \textit{P. Shige} from 2006 [``Modelling derivatives pricing mechanisms with their generating functions'', preprint, \url{arXiv:math/0605599}].
backward stochastic differential equations
dynamically consistent nonlinear evaluation
scientific article; zbMATH DE number 6149098
scientific article