Abstracts

We consider a general jump-diffusion model (with possibly infinite activity jumps) and study the viability of the financial market, defined as the ability to solve a portfolio optimization problem. Our main goal consists in characterizing the notion of market viability in terms of martingale measures, in a sense to be made precise in the following, studying under which conditions the (normalized) marginal utility of terminal wealth gives rise to a martingale measure. We refrain from a-priori imposing no-arbitrage restrictions on the model and, moreover, we suppose that market participants have only access to a partial information flow.
In order to solve portfolio optimization problems under partial information, we employ necessary and sufficient maximum principles for stochastic control problems under partial information. This approach allows us to characterize the optimal solution via an associated BSDE, which in turn requires a good control on the integrability properties of the processes involved. Since such integrability conditions are not satisfied in general, a localization procedure is needed.
Our main contributions can be outlined as follows:
1) we show that the financial market under partial information is locally viable, in the sense that a portfolio optimization problem admits a solution up to a stopping time, if and only if there exists a partial information equivalent martingale measure (PIEMM) up to a stopping time. Moreover, the density of such PIEMM is given by the (normalized) marginal utility of the optimal terminal wealth, thus recovering the classical result of financial economics;
2) we prove that, if the financial market under partial information is globally viable, in the sense that it is locally viable for a sequence of stopping times, then there exists a partial information local martingale deflator (PILMD). Furthermore, such a PILMD can be constructed by aggregating the densities of all local PIEMMs if and only if the locally optimal portfolios satisfy a consistency condition. In the special case of bounded coefficients, this PILMD is actually the density process of a PIEMM on the global time horizon;
3) by means of a simple example we show that, even for regular utility functions and continuous processes with good integrability properties but unbounded coefficients, a PIEMM may fail to exist globally but, nevertheless, a PILMD exists.