Chapter

Stochastic Methods in Finance

Volume 1856 of the series Lecture Notes in Mathematics pp 165-253

Date:

Nonlinear Expectations, Nonlinear Evaluations and Risk Measures

  • Shige PengAffiliated withInstitute of Mathematics, Shandong University Email author 

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Contents.

  • 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. \(\mathcal{F}_t\)-Consistent Nonlinear Expectations

    • 2.3. \(\mathcal{F}_t\)-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 \(\mathcal {F}\)-Expectation a g-Expectation?
    • 4.1. \(\mathcal {E}^\mu \)-Dominated \(\mathcal {F}\)-Expectations

    • 4.2. \(\mathcal{F}_t\)-Consistent Martingales

    • 4.3. BSDE under \(\mathcal{F}_t\)-Consistent Nonlinear Expectations

    • 4.4. Decomposition Theorem for \(\mathcal {E}\)-Supermartingales

    • 4.5. Representation Theorem of an \(\mathcal {F}\)-Expectation by a g-Expectation

    • 4.6. How to Test and Find g?

    • 4.7. A General Situation: \(\mathcal{F}_t\)-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 \(\mathcal {E}^g\)-Supermartingale

  • References

  • References on BSDE and Nonlinear Expectations

Mathematics Subject Classification (2000):

60G99 60-06 91-06 91B06 91B16 91B24 91B28 91B30 91B70 93-06 93E11 93E20