In mainstream economic theory agents are supposed to maximise the expected
utility of the return on their investments under the objective probability measure.
Furthermore, they are assumed to be risk-averse (which results in concave optimisation problems).
Real investors are often found to be risk-seeking and they distort the objective probability, creating non-concave optimisation problems involving non-linear expectations which lead to more exciting mathematics as well as to pitfalls. We will present some recent results and open problems.