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Market viability and martingale measures under partial information

Claudio Fontana 1 Bernt Øksendal 2 Agnès Sulem 1
1 MATHRISK - Mathematical Risk handling
Inria Paris-Rocquencourt, UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech
Abstract : We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow $(\varepsilon_t)_{t\geq0}\subseteq(\mathcal{F}_t)_{t\geq0}$. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the problem of maximizing the expected utility of terminal wealth has a solution up to a stopping time, if and only if the marginal utility of the terminal wealth is the density of a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control under partial information. We then show that the financial market is globally viable if and only if there exists a partial information local martingale deflator (PILMD), which can be explicitly constructed. In the case of bounded coefficients, the latter turns out to be the density process of a global PIEMM. We illustrate our results by means of an explicit example.
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Claudio Fontana, Bernt Øksendal, Agnès Sulem. Market viability and martingale measures under partial information. [Research Report] RR-8243, INRIA. 2015. ⟨hal-00789517⟩

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