  We propose indifference pricing to estimate the value of the weak
information. Our framework allows for tractability, quantifying the amount of
additional information, and permits the description of the smallness and the
stability with respect to small perturbations of the weak information. We
provide sharp conditions for the stability with counterexamples. The results
rely on a theorem of independent interest on the stability of the optimal
investment problem with respect to small changes in the physical probability
measure. We also investigate contingent claims that are indifference price
invariant with respect to changes in weak information. We show that, in
incomplete models, the class of information-invariant claims includes the
replicable claims, and it can be strictly bigger. In particular, in complete
models, all contingent claims are information invariant. We augment the results
with examples and counterexamples.
