Resolvent Operators for One-Dimensional Lévy Processes: A Unified Treatment

We give a unified treatment of the resolvent operator $\mathscr{R}_q$ of a one-dimensional Lévy process, organised around its Wiener–Hopf factorisation $\mathscr{R}_q = q\,\mathscr{R}_q^+ \mathscr{R}_q^-$ into the resolvents of the running supremum and the running infimum. For three rational families — Kou double-exponential, hyperexponential and meromorphic — the factors act on exponentials through the Wiener–Hopf factors $\Phi_q^\pm$, finite or convergent infinite products over the roots of $\psi(r) = q$ and the jump rates, and the supremum and infimum at an exponential time are mixtures of exponential laws. We prove a comparison principle for the optimal-stopping variational inequality, identify its solution with the stopping value, and show that the perpetual put is governed by the infimum: the optimal threshold is $b^{\ast} = K + \mathbb{E}[\underline X_{\mathbf{e}(q)}]$ for the payoff $(K – x)^+$ and $e^{b^{\ast}} = K\,\mathbb{E}[e^{\underline X_{\mathbf{e}(q)}}]$ for $(K – e^x)^+$. Under Kou jumps both close in the two negative roots of $\psi(r) = q$; at the benchmark $b^{\ast} = -0.0897$. Every closed form is checked by an independent method — exact Monte Carlo of the process killed at an exponential time, and a finite-difference solve of the integro-differential equations — and agrees within the Monte Carlo error and at the grid’s second-order rate.