| NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS |
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| 作者姓名： | PENG Shige |
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| 作者单位： | PENG Shige School of Mathematics and System Science，Shandong University，Jinan 250100，China. |
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| 基金项目： | Project supported by the National Natural Science Foundation of China(No.10131040). |
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| 摘 要： | §1.IntroductionLet(?,F)be a measurable space and let Lb(F)be the space of F-measurable andbounded real functions.A nonlinear expectation is a continuous functionalE·]:Lb(F)?→Rthat is order preserving(i.e.,EX1]≥EX2],if X1≥X2)and constant preserving
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| 关 键 词： | Backward stochastic di?erential equations Nonlinear expectation Non- linear expected utilities Measure of risk g-expectation Nonlinear Mar- kov chain g-martingale Nonlinear martingale Nonlinear Kolmogorov’s consistent theorem Doob-Meyer decomposition |
| 收稿时间： | 9/4/2008 12:00:00 AM |
NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS |
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| PENG Shige.NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS[J].Chinese Annals of Mathematics,Series B,2005,26(2):159-184. |
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| Authors: | PENG Shige |
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| Institution: | School of Mathematics and System Science,Shandong University,Jinan 250100,China |
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| Abstract: | This paper deals with nonlinear expectations. The author obtains a nonlinear generalization of the well-known Kolmogorov's consistent theorem and then use it to construct filtration-consistent nonlinear expectations via nonlinear Markov chains. Compared to the author's previous results, i.e., the theory of g-expectations introduced via BSDE on a probability space, the present framework is not based on a given probability measure. Many fully nonlinear and singular situations are covered. The induced topology is a natural generalization of Lp-norms and L∞-norm in linear situations.The author also obtains the existence and uniqueness result of BSDE under this new framework and develops a nonlinear type of von Neumann-Morgenstern representation theorem to utilities and present dynamic risk measures. |
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| Keywords: | Backward stochastic differential equations Nonlinear expectation Nonlinear expected utilities Measure of risk g-expectation Nonlinear Markov chain g-martingale Nonlinear martingale Nonlinear Kolmogorov's consistent theorem Doob-Meyer decompositio |
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