Modelling: Continuous state space and time

In the last lectures we investigated how contingent claims can be valued in two different settings: when the time is discrete and the state space is also discrete, and when the time is discrete but the state space is continuous. We noted that in the first case markets can be complete [equipped with a unique risk neutral probability measure] while in the second case we can only achieve prices which only rule out arbitrage opportunities [that is to say there exists an infinite number of risk neutral probability measures, although all of them price derivatives in a consistent way]. Now we are going to examine models where both time and the sample space are continuous. These models are appealing since they allow us to achieve closed form solutions, using some strong results of stochastic calculus.