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An integral representation theorem of g-expectations

An integral representation theorem of g-expectations There are two classes of nonlinear expectations, one is the Choquet expectation given by Choquet (Ann. Inst. Fourier (Grenoble) 5 (1955), 131–295), the other is the Peng's g-expectation given by Peng (BSDE and Related g-Expectation, Longman, Harlow, 1997, pp. 141–159) via backward differential equations (BSDE). Recently, Peng raised the following question: can a g-expectation be represented by a Choquet expectation? In this paper, we provide a necessary and sufficient condition on g-expectations under which Peng's g-expectation can be represented by a Choquet expectation for some random variables (Markov processes). It is well known that Choquet expectation and g-expectation (also BSDE) have been used extensively in the pricing of options in finance and insurance. Our result also addresses the following open question: given a BSDE (g-expectation), is there a Choquet expectation operator such that both BSDE pricing and Choquet pricing coincide for all European options? Furthermore, the famous Feynman–Kac formula shows that the solutions of a class of (linear) partial differential equations (PDE) can be represented by (linear) mathematical expectations. As an application of our result, we obtain a necessary and sufficient condition under which the solutions of a class of nonlinear PDE can be represented by nonlinear Choquet expectations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Risk and Decision Analysis IOS Press

An integral representation theorem of g-expectations

Risk and Decision Analysis , Volume 2 (4) – Jan 1, 2011

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References (9)

Publisher
IOS Press
Copyright
Copyright © 2011 by IOS Press, Inc
ISSN
1569-7371
eISSN
1875-9173
DOI
10.3233/RDA-2011-0047
Publisher site
See Article on Publisher Site

Abstract

There are two classes of nonlinear expectations, one is the Choquet expectation given by Choquet (Ann. Inst. Fourier (Grenoble) 5 (1955), 131–295), the other is the Peng's g-expectation given by Peng (BSDE and Related g-Expectation, Longman, Harlow, 1997, pp. 141–159) via backward differential equations (BSDE). Recently, Peng raised the following question: can a g-expectation be represented by a Choquet expectation? In this paper, we provide a necessary and sufficient condition on g-expectations under which Peng's g-expectation can be represented by a Choquet expectation for some random variables (Markov processes). It is well known that Choquet expectation and g-expectation (also BSDE) have been used extensively in the pricing of options in finance and insurance. Our result also addresses the following open question: given a BSDE (g-expectation), is there a Choquet expectation operator such that both BSDE pricing and Choquet pricing coincide for all European options? Furthermore, the famous Feynman–Kac formula shows that the solutions of a class of (linear) partial differential equations (PDE) can be represented by (linear) mathematical expectations. As an application of our result, we obtain a necessary and sufficient condition under which the solutions of a class of nonlinear PDE can be represented by nonlinear Choquet expectations.

Journal

Risk and Decision AnalysisIOS Press

Published: Jan 1, 2011

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