Nonlinear Expectations, Nonlinear Evaluations and Risk Measures

Abstract

• 1. Introduction

  • 1.1. Searching the Mechanism of Evaluations of Risky Assets

  • 1.2. Axiomatic Assumptions for Evaluations of Derivatives

  • 1.3. Organization of the Lecture

• 2. Brownian Filtration Consistent Evaluations and Expectations

  • 2.1. Main Notations and Definitions

  • 2.2. -Consistent Nonlinear Expectations

  • 2.3. -Consistent Nonlinear Evaluations

• 3. Backward Stochastic Differential Equations: g-Evaluations and g-Expectations

  • 3.1. BSDE: Existence, Uniqueness and Basic Estimates

  • 3.2. 1-Dimensional BSDE

  • 3.3. A Monotonic Limit Theorem of BSDE

  • 3.4. g-Martingales and (Nonlinear) g-Supermartingale Decomposition Theorem

• 4. Finding the Mechanism: Is an -Expectation a g-Expectation?

  • 4.1. -Dominated -Expectations

  • 4.2. -Consistent Martingales

  • 4.3. BSDE under -Consistent Nonlinear Expectations

  • 4.4. Decomposition Theorem for -Supermartingales

  • 4.5. Representation Theorem of an -Expectation by a g-Expectation

  • 4.6. How to Test and Find g?

  • 4.7. A General Situation: -Evaluation Representation Theorem

• 5. Dynamic Risk Measures

• 6. Numerical Solution of BSDEs: Euler’s Approximation

• 7. Appendix

  • 7.1. Martingale Representation Theorem

  • 7.2. A Monotonic Limit Theorem of Itô’s Processes

  • 7.3. Optional Stopping Theorem for -Supermartingale

• References

• References on BSDE and Nonlinear Expectations

Shige Peng: The author would like to acknowledge the partial support from the Natural ScienceFoundation of China, grant No. 10131040. He would like to give his specialthanks to the organizers as well as the audience of CIME-EMS school, in thebeautiful town of Bressanone, for their warm hospitality and enthusiasm. Thismemorable Italian trip and lecture experience could never been realized withoutthe persistence and the efforts of the organizers to overcome the author’s‘Shengen-Italy-visa-paradox’. He would like to thank Li Juan as well as XuMingyu for their careful examinations and suggestions to the manuscript.