Some aspects of modeling and statistical inference for financial models

Modeling the stock price development as a geometric Brownian motion or, more generally, as a stochastic exponential of a diffusion, requires the use of specific statistical methods. For instance, the observations seldom reach us in the form of a continuous record and we are led to infer about diffusion coefficients from discrete time data. Next, often the classical assumption that the volatility is constant has to be dropped. Instead, a range of various stochastic volatility models is formed by the limiting transition from known volatility models in discrete time towards their continuous time counterparts. These are the main topics of the present survey. It is closed by a quick look beyond the usual Gaussian world in continuous time modeling by allowing a Levy process to be the driving process.