We study terminal perturbation bounds for jump-diffusion processes in the Wasserstein distance $W_1$. In the finite-variance setting, let $\nu_0$ and $\nu_1$ denote the jump kernels and define$$\theta_\nu:=\int_0^Td_{\mathrm{FM}}\left(y^2\nu_0(t,\mathrm{d}y),y^2\nu_1(t,\mathrm{d}y)\right)\mathrm{d}t.$$Here $d_{\mathrm{FM}}$ denotes the Fortet--Mourier distance. We prove that the rate $W_1=O(\theta_\nu^{1/3})$ is optimal in the degenerate class. If $r_\nu(t)$ denotes the instantaneous jump discrepancy and $r_\nu\in L^q(0,T)$, with $1\leq q<\infty$, one-sided smoothing improves the exponent to $q/(q+2)$. For additive processes, a two-sided Duhamel argument removes the terminal singularity and yields the optimal linear rate $W_1=O(\theta_\nu)$ under uniform Gaussian smoothing. We also extend the method to multidimensional additive processes, finite-rank state-dependent kernel perturbations, and infinite-variance models. In the latter case, for stable smoothing of order $\beta\in(1,2)$, the one-sided exponent is $1/(3-\beta+\beta/q)$, while two-sided smoothing again gives a linear estimate.
Alexandre Autran· Zenodo (CERN European Organi...· 0 citations
We study terminal perturbation bounds for jump-diffusion processes in the Wasserstein distance $W_1$. In the finite-variance setting, let $\nu_0$ and $\nu_1$ denote the jump kernels and define$$\theta_\nu:=\int_0^Td_{\mathrm{FM}}\left(y^2\nu_0(t,\mathrm{d}y),y^2\nu_1(t,\mathrm{d}y)\right)\mathrm{d}t.$$Here $d_{\mathrm{FM}}$ denotes the Fortet--Mourier distance. We prove that the rate $W_1=O(\theta_\nu^{1/3})$ is optimal in the degenerate class. If $r_\nu(t)$ denotes the instantaneous jump discrepancy and $r_\nu\in L^q(0,T)$, with $1\leq q<\infty$, one-sided smoothing improves the exponent to $q/(q+2)$. For additive processes, a two-sided Duhamel argument removes the terminal singularity and yields the optimal linear rate $W_1=O(\theta_\nu)$ under uniform Gaussian smoothing. We also extend the method to multidimensional additive processes, finite-rank state-dependent kernel perturbations, and infinite-variance models. In the latter case, for stable smoothing of order $\beta\in(1,2)$, the one-sided exponent is $1/(3-\beta+\beta/q)$, while two-sided smoothing again gives a linear estimate.
Alexandre Autran· Zenodo (CERN European Organi...· 0 citations