Narrow Results

The development of the theory of the expansion of filtrations took place mostly three decades ago, in the 1980s. Researchers developed two types of expansions: Initial expansions, where one adds information to the σ algebra ℱ₀, and progressive expansions, where information is added dynamically to turn a positive random variable (such as a last exit time) into a stopping time. The goal was to preserve the semimartingale property in the enlarged filtration. In this paper, we propose a new type of expansion, that of expansion with a stochastic process. This has antecedents in the work of Jeulin, Kohatsu-Higa, and a few others, but the theoretical and systematic approach given here is new. We begin by showing one can enlarge a filtration with a point process rather easily, and that semimartingales remain semimartingales under the expansions, if done correctly. Since one can approximate most stochastic processes with marked point processes, we then prove convergence theorems of the sequence of point processes together with their corresponding enlarged filtrations. To do this, we rely on a theory of the convergence of filtrations. We next need to give conditions such that semimartingales remain semimartingales in the limit. This is delicate, and we obtain partial results, but they are sufficient for our needs; this involves a kind of continual initial expansion procedure. One drawback of this procedure is that we do not always obtain the semimartingale decompositions for the enlarged filtrations of the limiting case, except in a class of examples that are in a Brownian motion paradigm. Recently, there have been attempts to model mathematically insider trading via a theory of enlarged filtrations. There are many technical issues involved, and creating a model with the presence of arbitrage is a constant problem, a situation one wishes to avoid. We show how one can use this procedure to create models of insider trading that do not have arbitrage opportunities, but under which the risk neutral measure for the insider is different than is the risk neutral measure for the rest of the market.

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration and a change of measure. We study and implement a particular type of enlargement, initial expansion of filtration, for stochastic volatility models with and without jumps and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale. We provide examples of initial enlargement that effect this change.