Subreplication Costs and Infima of Risk-Neutral Expectations

Summary

The document poses a question about a proposed equality for a nonnegative contingent claim: whether taking the limit of the infimum of expectations of its truncated versions gives the infimum of expectations of the original claim over equivalent local martingale measures. This equality matters to a proof that the subreplication cost can be characterized by the lowest risk-neutral expected payoff.

The text states the claim is integrable and nonnegative, and identifies the measures over which the infimum is taken. It offers no proof, argument, or counterexample, and gives no market model assumptions beyond that setup. As a result, it serves as a focused mathematical question rather than a resolved pricing method; whether the equality holds may depend on further properties of the measure class and the model.

Key ideas

• The proposed equality compares infima of expectations for truncated and untruncated claims.
• The claim is assumed to be nonnegative and integrable.
• The infimum is taken over equivalent local martingale measures.
• The document leaves the equality unresolved and requests a proof or counterexample.

Tags

# Sub replication of contingent claims


# Sub replication of contingent claims












I am trying to prove subreplication cost is the infimum of risk neutral expectation of the contingent claim. I wonder if the following equality holds, which is the key to the proof. $$ \lim_n \inf_{Q\in \mathcal M}\mathbb E^Q[Z\wedge n] = \inf_{Q\in \mathcal M}\mathbb E^Q[Z] $$ where $Z\in L^1$ is a non-negative contingent claim and $\mathcal M$ is the class of equivalent local martingale measures. I don't know whether this is true or a counterexample can be constructed. Any idea in this regard is much appreciated.

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