An integral representation theorem of g-expectations

There are two classes of nonlinear expectations, one is the Choquet expectation given by Choquet (1955), the other is the Peng’s g-expectation given by Peng (1997) 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 gexpectation (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.