The Augmented Jump Chain: a sparse representation of time-dependent Markov jump processes

The paper’s Theorem 2 derives the space–time kernel k(xi, s, xj, t) = q̃ij(t) qi(t) exp(-∫_s^t qi(u) du) for t > s (and 0 otherwise) and identifies the jump operator J and its adjoint J†; this is exactly the construction and formula used by the candidate solution. The paper’s proof combines the instantaneous destination probabilities q̃ij(t) with the non-homogeneous exponential (hazard) density for the next jump time (eqs. (16)–(21) and definitions (17)–(18)) . The candidate solution reproduces the same steps via the hazard ODE for survival, explicitly invokes the strong Markov property at Jn to justify time-homogeneity on X×T, and notes the possible need for a cemetery state if ∫_s^∞ qi(u) du < ∞, which the paper does not discuss explicitly. Minor issues in the paper (e.g., a domain/codomain mismatch in the operator typing and a typographical integrand variable in an exponent) do not affect the main result; the candidate’s version is a slightly more careful presentation. The appendix’s statement lim_{t→∞} F(t)=1 for the non-homogeneous exponential implicitly assumes ∫_0^∞ q(u) du = ∞; the model’s solution correctly allows for the “no further jump” event when this integral is finite . Overall, the two arguments are the same in substance, and both are correct.