Jian Smith

Set the matter: the equivalence between sensitivities at fixed points and KKT must be proven conditionally, not decisively. Highly volatile capital markets, as seen with the sudden devaluation of certain Asian currencies, introduce extreme nonlinearities that break these assumptions. These situations require ongoing empirical validation, not just theoretical proof.

It is time to set the standard: a proof does not create an equivalence; it demonstrates it fully.
The equivalence between sensitivities at fixed points and KKT is an intrinsic property, not a fabrication.
For our actuarial commitments, we must rely on existing properties, not on a creation.
For example, the law of gravity was not created by Newton, but explained by his theory; similarly, this proof clarifies a pre-existing relationship.

Here's the problem: believing that a mathematical proof 'creates' an equivalence is a somewhat simplistic story; it demonstrates this only under idealized conditions. In our markets, the major turning point occurs when real policies disrupt the scene. A concrete example is when the China Securities Regulatory Commission (CSRC) introduces price limits: this changes the entire theoretical equivalence. These interventions are not secondary characters but the main actors who transform the planned course. Proof is a fixed act, while real life is the spectacle.

Who has the incentive to believe that a proof 'creates' the equivalence rather than demonstrating it? The reality is that a proof like that of the mirror descent establishes a conditional validity, it does not shape a new mathematical truth.
If everyone adopted this logic, trust in formalization would erode, somewhat like claiming that regulation of the CVM in Brazil creates market risk.
These regulations impact the payoffs and structure, but the existence of the equivalence remains independent.
An explicit proof simply establishes a preexisting relationship, making it usable under certain conditions, such as the sensitivity of gold flows to central bank decisions.

Who has what incentive to consider formal proof as an intrinsic creation rather than a useful clarification?
The balance here is that the proof of equivalence for the mirror gradient reveals a correlation; it does not generate it ex nihilo; preexisting conditions are crucial.
If everyone acted as if the proof alone was sufficient, we would ignore operational limits, creating a risk of strategic error.
For example, the equivalence between physical gold and futures contracts on the COMEX is proven, but if the logistics of Brazilian delivery fail at a bottleneck point, the theoretical equivalence will have no practical meaning on the spot prices.
The gain lies in understanding the conditions of application, not in deifying the proof.

The issue of regulatory compliance is paramount here. The approval pathway for such equivalence often depends on guidelines from the Financial Markets Authority or similar organizations, which may require additional robustness testing. More specifically, the European jurisdiction, for example, emphasizes demonstrating model stability in the event of unforeseen exogenous shocks, beyond traditional stopping conditions.

Who has an interest in considering the equivalence of sensitivities as an unconditional creation rather than a necessary condition?
For example, even if an equivalence proof between sensitivities based on fixed points and those based on KKT conditions is established, its application in a volatile market like Brazil does not de facto create a new market reality.
The incentives diverge strongly between academic theory and trading practice, where commodity shocks or changes in monetary policy can render these equivalences invalid.
The true payment comes from understanding when these equivalences hold in the face of market frictions and not just by assuming their existence.
If actors ignore local macroeconomic signals in favor of a mathematical abstraction, the risk of an underperforming equilibrium is high.