Abstract
We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale parts in the multiplicative Doob–Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.
Highlights
Optional projections of martingales onto smaller filtrations retain the martingale property; for the class of local martingales, this preservation may fail
The projection of a nonnegative local martingale can only be guaranteed to be a supermartingale in the smaller filtration, but might fail to be a local martingale; see Stricker [41] and Föllmer and Protter [14]
The existence of a strictly positive G-local martingale Y such that Y S is a G-local martingale is equivalent to the so-called absence of arbitrage of the first kind
Summary
Optional projections of martingales onto smaller filtrations retain the martingale property; for the class of local martingales, this preservation may fail. 3 contains the proof of Theorem 3.1 and related results The previous motivates another natural question: when does the projection of Y lose the local martingale property, that is, under which circumstances is it the case that K∞ > 0? Let FW and FB denote the smallest rightcontinuous filtrations making W and B adapted, respectively Both W and B are standard Brownian motions with the predictable representation property in FW. The last result provides an interesting corollary: the existence of two nested filtrations F ⊆ G and a one-dimensional continuous stock price process S, adapted to F, such that the market is complete under G and under F, but not under some “intermediate information” model.
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