Math. Program., Ser. B (2018) 168:347–367
FULL LENGTH PAPER
Shadow price of information in discrete time stochastic
· Ari-Pekka Perkkiö
Received: 19 January 2016 / Accepted: 10 May 2017 / Published online: 30 May 2017
© The Author(s) 2017. This article is an open access publication
Abstract The shadow price of information has played a central role in stochastic
optimization ever since its introduction by Rockafellar and Wets in the mid-seventies.
This article studies the concept in an extended formulation of the problem and gives
relaxed sufﬁcient conditions for its existence. We allow for general adapted decision
strategies, which enables one to establish the existence of solutions and the absence of
a duality gap e.g. in various problems of ﬁnancial mathematics where the usual bound-
edness assumptions fail. As applications, we calculate conjugates and subdifferentials
of integral functionals and conditional expectations of normal integrands. We also give
a dual form of the general dynamic programming recursion that characterizes shadow
prices of information.
Keywords Stochastic optimization · Shadow price of information · Subdifferential ·
Mathematics Subject Classiﬁcation 46A20 · 52A41 · 90C15 · 90C46
Dedicated to R. T. Rockafellar on his 80th Birthday.
The second author is grateful to the Einstein Foundation for the ﬁnancial support.
Department of Mathematics, King’s College London, London, UK
Department of Mathematics, Technische Universität Berlin, Berlin, Germany