  In this paper, we consider the discrete-time setting, and the market model
described by (S,F,T)$. Herein F is the ``public" flow of information which is
available to all agents overtime, S is the discounted price process of
d-tradable assets, and T is an arbitrary random time whose occurrence might not
be observable via F. Thus, we consider the larger flow G which incorporates F
and makes T an observable random time. This framework covers the credit risk
theory setting, the life insurance setting and the setting of employee stock
option valuation. For the stopped model (S^T,G) and for various vulnerable
claims, based on this model, we address the super-hedging pricing valuation
problem and its intrinsic Immediate-Profit arbitrage (IP hereafter for short).
Our first main contribution lies in singling out the impact of change of prior
and/or information on conditional essential supremum, which is a vital tool in
super-hedging pricing. The second main contribution consists of describing as
explicit as possible how the set of super-hedging prices expands under the
stochasticity of T and its risks, and we address the IP arbitrage for (S^T,G)
as well. The third main contribution resides in elaborating as explicit as
possible pricing formulas for vulnerable claims, and singling out the various
informational risks in the prices' dynamics.
