Time consistency of dynamic risk measures in markets with transaction costs

Set-valued dynamic risk measures are defined on L^p_d(F_T ) with 0 ≤ p ≤ ∞ and with an image space in the power set of L^p_d(F_t). Primal and dual representations of dynamic risk measures are deduced. Definitions of different time consistency properties in the set-valued framework are given. It is shown that the recursive form for multivariate risk measures as well as an additive property for the acceptance sets is equivalent to a stronger time consistency property called multi-portfolio time consistency.