Nonstandard methods in stochastics

The first chapter of this thesis is concerned with measure and probability theory from a Nonstandard Analysis point of view. First we give an overview of general facts in Nonstandard Analysis. The second part of the first chapter is about internal measure spaces. Then we apply these results to the Brownian Motion and show that the Brownian Motion can be obtained as an infinitesimal random walk. In the next part we show for a big class of functions and processes on probability spaces in the nonstandard universe that they in a certain sense correspond to standard entities. In the second chapter we give some applications of nonstandard stochastics in mathematical finance. First we look at European call options in the Cox Ross- Rubinstein model and in the Black-Scholes model. Then we introduce the hyperfinite Cox-Ross- Rubinstein model, where we use random walks with infinitesimal time steps. The hyperfinite Cox-Ross- Rubinstein model extends the ordinary Cox-Ross- Rubinstein model and inherits many of its properties. Then we show that the Black- Scholes model is precisely the standard part of the hyperfinite Cox-Ross-Rubinstein model.