Enlargement of filtration and predictable representation property for semi-martingales

We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a square-integrable semi-martingale X does not transfer from the reference filtration F to a larger filtration G when the information starts growing up to a positive time. Then we study the case when F coincides with the natural filtration of X and G is obtained by adding the natural filtration of a second square-integrable semi-martingale, Y. We establish conditions under which the triplet (X,Y,[X,Y]) enjoys the predictable representation property with respect to G.Comment: to appear on Stochastic

No-arbitrage conditions and absolutely continuous changes of measure

We study the stability of several no-arbitrage conditions with respect to absolutely continuous, but not necessarily equivalent, changes of measure. We first consider models based on continuous semimartingales and show that no-arbitrage conditions weaker than NA and NFLVR are always stable. Then, in the context of general semimartingale models, we show that an absolutely continuous change of measure does never introduce arbitrages of the first kind as long as the change of measure density process can reach zero only continuously.Comment: 14 pages. Arbitrage, Credit and Informational Risks (C. Hillairet, M. Jeanblanc and Y. Jiao, eds.), Peking University Series in Mathematics, Vol. 6, World Scientific, 201

On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.Comment: 28 pages. Added DOI https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final publication, corrected typo (missing gamma) in example 4.1

On the Hedging of Options On Exploding Exchange Rates

We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More precisely, we discuss a change of num\'eraire (change of currency) technique when the underlying is only a local martingale modelling for example an exchange rate. The new pricing operator assigns prices to contingent claims according to the minimal cost for superreplication strategies that succeed with probability one for both currencies as num\'eraire. Within this context, we interpret the lack of the martingale property of an exchange-rate as a reflection of the possibility that the num\'eraire currency may devalue completely against the asset currency (hyperinflation).Comment: Major revision. Accepted by Finance and Stochastics. The original publication is available at http://link.springer.co

On maximal inequalities for purely discontinuous martingales in infinite dimensions

The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces. Such maximal inequalities are important in the study of stochastic partial differential equations with noise of jump type.Comment: 19 pages, no figure