Top-of-market deviations in the monostable regime

Characterize the behavior of top-of-market deviations for the rank-based entry-and-exit reaction-diffusion equation when the reaction term is monostable, including the failure of uniform attraction to a single traveling front.

Background

The paper analyzes a monostable regime motivated by the 2000s data, in which the normalized reaction term is negative on the interior and the upper endpoint is unstable. Unlike the bistable case, the associated traveling-wave equation admits a half-line of admissible speeds rather than a unique speed, and the selected speed and tail may depend on the initial data.

Because the linearized essential spectrum enters the right half-plane at the unstable upper state, the authors state that no single front attracts all cumulative-distribution initial data uniformly. The unresolved issue concerns the resulting deviations at the top of the market, where concentration and tail behavior are not determined by a universal attracting wave.

References

Stability degenerates as well: with \tilde f'(1) > 0 the essential spectrum of the linearization enters the right half-plane (the equality at the edge in eq:ess-spec does not use the signs of the slopes), so no single front attracts all CDF data uniformly, and deviations at the top of the market are unresolved for a monostable reaction term.

eq:ess-spec:

σess(L)  ⊆  {Re⁡λ≤max⁡(f~′(0),f~′(1))},\sigma_{\mathrm{ess}}(\mathcal{L}) \;\subseteq\; \big\{\operatorname{Re}\lambda \leq \max(\tilde f'(0), \tilde f'(1))\big\},

— Traveling Waves in Equity Markets with Rank-Based Entry and Exit  (2608.27156 - Baker et al., 27 Aug 2026) in Section 3.4, subsection “Comparisons with Known Cases,” paragraph “The Monostable Case”