Filtration reduction and completeness in Brownian motion models

Karen Grigorian; Robert A. Jarrow

doi:10.3934/fmf.2025006

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June 2025, Volume 5, 94-120. Doi: 10.3934/fmf.2025006 open access

This volume Previous Article Risk premium and rough volatility Next Article Supply–demand symmetry

Filtration reduction and completeness in Brownian motion models

Karen Grigorian^1, , and
Robert A. Jarrow^{1, 2,}

1.
Operations Research and Information Engineering, Cornell University, Ithaca, N. Y. 14853, USA
2.
Samuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, N. Y. 14853, USA

^*Corresponding author: Karen Grigorian
^*Corresponding author: Karen Grigorian

Received: September 2024

Revised: January 2025

Published online: April 11, 2025

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Abstract

This paper studies an incomplete Brownian motion market and uses filtration reduction to obtain a fictitious complete market with a unique pricing measure. We uplift this measure to the original market and study valuation and hedging via the uplifted measure. We show how a general market can be decomposed into a market corresponding to the reduced information plus a noise component. This allows us to give a precise meaning to the notion of irrelevant information in the context of a filtration reduction in an incomplete market.

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Mathematics Subject Classification: Primary: 91G15, 91G20, 91G30; Secondary: 60G44.

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