A Two-Part Controller Synthesis Approach for Nonlinear Stochastic Systems Perturbed by Lévy Noise Using Renewal Theory and HJB-Based Impulse Control

The paper cleanly states the two formulas (Theorem 4) for the expected time between pattern occurrences in a discrete-state Markov chain—no-overlap and exactly one border case—and these align with standard results and their iid specialization. However, the paper defers all proof details to an external preprint, so the argument in this PDF is not shown. The candidate model’s derivation, while aiming to reproduce the same formulas, contains critical errors: (i) it misapplies Kac’s formula to a set B_k starting from a single point a; (ii) it ignores entrance distributions when decomposing returns through B_k; and (iii) it produces internally inconsistent expressions for E_{x_1}[T_end], equating quantities that are generally unequal. Therefore, the result statements in the paper appear correct, but the model’s proof is unsound, despite arriving at the same final formulas.