Free lunch and arbitrage possibilities in a financial market model with an insider.
The paper studies filtrations on a Wiener space, enlarged by a random variable \(L\) which depends on the whole Brownian motion trajectory. The prototypical example is the supremum of a Wiener process on an interval. The theory is aimed at solving the portfolio problem of an investor who is informed about \(L\), in addition to the usual knowledge of the price process statistics under the original Wiener filtration. The paper looks for the conditions under which the enlarged filtration preserves the semimartingale property of the price process. These conditions are obtained in terms of the conditional law of \(L\) given the original filtration, by means of a measure-valued Malliavin calculus technique. In particular, the probability change that makes the price process a Wiener one under the enlarged filtration is determined by the Radon-Nikodym derivative of the conditional law of \(L\) with respect to this conditional law. Among other things, the authors give conditions under which the relative entropy of the conditional law of \(L\) with respect to the law of \(L\) is finite and positive. They also prove that their prototypical \(L\) generates an asset market model with arbitrage possibilities.
scientific article; zbMATH DE number 2102378
scientific article