We had two of our postdocs giving the first talks in our Visitor Seminars yesterday.
Hao Xing spoke about stochastic volatility models where (discounted) stock prices are local martingales under the risk neutral measure and showed how derivative prices (solutions of the pricing PDE) can fail to be unique when the stock price is not an actual martingale. One of the motivations for studying this kind of things are models where asset bubbles are identified precisely with such strict local martingales.
Klaas Schulze showed how the idea of an indifferent level of risk aversion can be used to define the actual riskiness of a position. This is inspired by the tongue-in-cheek comment that "risk is what risk-averters hate" and leads to a lot of neat mathematics related to the notion of dual risk measures.
A very promising start for these seminars indeed.