Consistent price systems for subfiltrations
ESAIM: Probability and Statistics, Tome 11 (2007) , p. 35-39
doi : 10.1051/ps:2007004
URL stable : http://www.numdam.org/item?id=PS_2007__11__35_0

Classification:  91B24,  93E11
Asymmetric or partial information in financial markets may be represented by different filtrations. We consider the case of a larger filtration $ℱ$ - the natural filtration of the “model world” - and a subfiltration $\stackrel{^}{ℱ}$ that represents the information available to an agent in the “real world”. Given a price system on the larger filtration that is represented by a martingale measure $Q$ and an associated numeraire $S$, we show that there is a canonical and nontrivial numeraire $\stackrel{^}{S}$ such that the price system generated by $\left(\stackrel{^}{S},Q,\stackrel{^}{ℱ}\right)$ is consistent, in a sense to be made precise, with the price system generated by $\left(S,Q,ℱ\right)$.

### Bibliographie

[1] M. Atlan, H. Geman, D.B. Madan and M. Yor, Correlation and the pricing of risks. Prépublications du Laboratoire de Probabilités et Modèles Aléatoires (2004), No. 877.

[2] T. Björk, Interest rate theory, in Financial Mathematics, W.J. Runggaldier Ed., Lecture Notes in Mathematics, Springer, Berlin (1997) 53-122. Zbl 0904.90007

[3] T. Björk, Arbitrage Theory in Continuous Time. Oxford University Press (1998/2004). Zbl 1140.91038

[4] P. Brémaud and M. Yor, Changes of Filtrations and of Probability Measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete 45 (1978) 269-295. Zbl 0415.60048

[5] F. Delbaen and W. Schachermayer, Non-arbitrage and the fundamental theorem of asset pricing: Summary of main results1997). MR 1737722 | Zbl 0969.91004

[6] H. Geman, N. El Karoui and J.-C. Rochet, Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab. 32 (1995) 443-458. Zbl 0829.90007

[7] A. Gombani, S. Jaschke and W. Runggaldier, A filtered no arbitrage model for term structures from noisy data. Stochastic Processes Appl. 115 (2005) 381-400. Zbl pre02192567