A representation formula for the probability density in stochastic dynamical systems with memory

Fang Yang, Xu Sun

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the representation formula for the density of a Marcus SDDE by converting it to a non-delayed k-stage cascade (method of steps), then expressing PA(x,t) via the transition densities Qk of the cascade, under assumptions (H2)–(H3) (existence/uniqueness and existence/positivity of densities). The main identities (27)–(29) are stated in Theorem 2 and derived using conditional densities and the cascade structure, with equations (30)–(33) worked out in the appendix (including (31)–(32)) . The candidate solution uses the same core idea: segment the path on τ-windows, invoke independent Lévy increments and the Markov property of the non-delayed cascade, factor the joint law, and integrate out intermediate variables to obtain (ii) and the boundary case (iii). This matches the paper’s approach in substance, differing mainly in presentation (sigma-field factorization vs. conditional density ratios).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript establishes a clean representation formula for the density of solutions to Marcus SDDEs with discrete delay by identifying them with a non-delayed cascade and expressing the SDDE density via transition kernels of the cascade. The assumptions are clearly stated (existence/uniqueness and existence/positivity of transition densities), and the main result is correct. The contribution is primarily methodological and clarifies how to reduce non-Markovian delayed dynamics to a Markovian, higher-dimensional non-delayed system for density computations. Some steps (e.g., conditional density manipulations and normalization) could be made more explicit, and a brief discussion of when (H3) holds would strengthen the presentation.