Catálogo de publicaciones - libros

For a price process that has an equivalent risk neutral measure, we investigate if the same property holds when the numéraire is changed. We give necessary and sufficient conditions under which the price process of a particular asset — which should be thought of as a different currency — can be chosen as new numéraire. The result is related to the characterisation of attainable claims that can be hedged. Roughly speaking: the asset representing the new currency is a reasonable investment (in terms of the old currency) if and only if the market does not permit arbitrage opportunities in terms of the new currency as numéraire. This rough but economically meaningful idea is given a precise content in this paper. The main ingredients are a duality relation as well as a result on maximal elements. The paper also generalises results previously obtained by Jacka, Ansel-Stricker and the authors.

We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase Radon-Nikodým theorems for predictable processes.

For a locally bounded local martingale , we investigate the vector space generated by the convex cone of maximal admissible contingent claims. By a maximal contingent claim we mean a random variable (⋅), obtained as a final result of applying the admissible trading strategy to a price process and which is optimal in the sense that it cannot be dominated by another admissible trading strategy. We show that there is a natural, measure-independent, norm on this space and we give applications in Mathematical Finance.

The topic of the present paper is the statement and proof of the subsequent in a 

For ℋ-bounded sequences of martingales, we introduce a technique, related to the Kadeč-Pełczyński decomposition for sequences, that allows us to prove compactness theorems. Roughly speaking, a bounded sequence in ℋ can be split into two sequences, one of which is weakly compact, the other forms the singular part. If the martingales are continuous then the singular part tends to zero in the semi-martingale topology. In the general case the singular parts give rise to a process of bounded variation. The technique allows to give a new proof of the optional decomposition theorem in Mathematical Finance.