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.