Resume

How an investor measures his daily risky choices in a financial market? This a fundamentally important problem. According to the well–known expected utility theory under von Neumann and Morgenstern’s axioms an agent prefers a random choice X to another one Y can be described by E[u(X)] ≥ E[u(Y)]. where u is a fixed increasing function, called utility function and E[·] is the ex-pectation in a probability space (Ω,F, P). But statistical tests and theoretical analysis show that this expectation has to be nonlinear in many important situ-ations. This phenomenon becomes even more obvious in the situation where the statistic data is incomplete: it is often the case for investors in a financial market. In this lectures, we will study this theory nonlinear expectation within the model of continuous time. Let (Ω,F) be a measurable space and let Lb(F) be the space of F–measurable and bounded real functions. A nonlinear expectation is a continuous functional E [·] : Lb(F) − → R that is order preserving (i.e., E [X1] ≥ E [X2], if X1 ≥ X2) and constant pre-serving (i.e., E [c] = c). If furthermore E [·] is a linear functional, then it is a classical expectation under the (additive) probability measure P on (Ω,F) induced b