Assumptions, Theorems & Proof Architecture (mathfin-identification-strategy)

Note on framing

Mathematical Finance is a theory-first journal: papers are evaluated on methodological novelty and rigor, not empirical causal identification. The "identification" that matters here is mathematical identification — pinning down the right assumptions, the precise theorem, and a complete proof. This skill therefore covers assumptions, results, proof exposition, and generality. (Empirical causal design is out of scope for this venue.)

When to trigger

• A "model" is proposed but its formal properties (existence, uniqueness, no-arbitrage) are unproved
• The assumptions are vague (which filtration? which integrability? which regularity?)
• A proof has a gap, an unstated measurability/integrability condition, or a circular step
• You are unsure your generality is the right level for the contribution

The rigor bar (the journal requires self-contained full proofs)

1. State assumptions precisely. Probability space, filtration and its conditions (usual conditions?), integrability ($L^p$, square-integrability), regularity, market structure (complete/incomplete), admissibility of strategies. Number them (A1, A2, ...) and reuse them.
2. State the theorem cleanly. Hypotheses → conclusion, with the object's existence, uniqueness, and characterization separated. Avoid burying conditions in prose.
3. Make the proof self-contained. Full proofs of all formal results are required; cite external theorems with exact hypotheses and check they apply (e.g., that a martingale is genuinely a martingale, not just a local one).
4. Get the generality right. Too narrow → looks like a special case (see mathfin-literature-positioning); too broad → the proof breaks. Justify each assumption: is it essential, or a convenience that could be relaxed?
5. Guard the standard pitfalls. Local vs. true martingale, integrability of stochastic integrals, applicability of Itô / Girsanov / Feynman–Kac, well-posedness of SDEs/BSDEs, verification of HJB solutions, smooth-fit at free boundaries, NFLVR/FTAP conditions.

Branch paths

• Pricing / no-arbitrage: establish the (equivalent) martingale measure; verify NFLVR / FTAP hypotheses; confirm the discounted price is a true martingale.
• Stochastic control / portfolio: state the HJB / verification theorem; check admissibility and the transversality/integrability conditions; prove the candidate is optimal, not just stationary.
• BSDE / duality: existence–uniqueness under stated drivers; comparison theorem if used; rigorous duality gap = 0 argument.
• Optimal stopping / free boundary: Snell envelope or variational inequality; smooth-pasting justified, not assumed.

Assumption-block and statement templates

A house-style assumption block fixes the stochastic basis once and lets every result refer to it by label:

\begin{assumption}\label{ass:basis}
$(\Omega,\mathcal F,(\mathcal F_t)_{t\in[0,T]},\mathbb P)$ is a filtered probability space
satisfying the usual conditions and supporting a $d$-dimensional Brownian motion $W$.
\end{assumption}

\begin{assumption}\label{ass:coeff}
$b,\sigma$ are progressively measurable; $\sigma\sigma^{\top}$ is uniformly elliptic and
$\mathbb E\!\int_0^T \big(|b_t|^2 + |\sigma_t|^4\big)\,dt < \infty$.
\end{assumption}

\begin{theorem}\label{thm:main}
Under Assumptions \ref{ass:basis}--\ref{ass:coeff}, the value function ... Moreover, the
optimal strategy $\pi^{\star}$ is admissible and unique up to indistinguishability.
\end{theorem}

Separating the basis assumption from the coefficient assumption lets you weaken one without touching results that need only the other — referees notice and reward this modularity.

Where each lemma lives

• Main text: the lemma carrying the new idea (a novel estimate, a new compactness or selection argument) — referees should meet it before the main proof, with a sentence saying why existing estimates fail.
• Appendix: routine verifications (moment bounds, measurability of value functions, standard localization steps) — each still proved in full, never waved at.
• Inline remark: one-line consequences of cited results, with the citation pinned to the exact theorem number and a sentence confirming its hypotheses hold here.
• Never split one proof across main text and appendix mid-argument: give a sketch in the text and defer the complete proof as a single unit.

Anti-patterns

• "It is well known that..." standing in for a required step.
• Assuming an integrability/measurability condition only where convenient.
• Treating a local martingale as a martingale without a uniform-integrability argument.
• Stating maximal generality the proof cannot support.
• Relegating a load-bearing lemma to "the reader can verify."

Output format

【Main theorem】hypotheses → conclusion (one line)
【Assumptions】[A1, A2, ...] with role of each
【Proof architecture】lemmas → main steps → where external theorems enter
【Generality check】each assumption: essential / relaxable
【Pitfalls cleared】[martingale, integrability, well-posedness, smooth-fit, ...]
【Gaps remaining】[...]
【Next step】mathfin-contribution-framing