Resolvent Operators for One-Dimensional Lévy Processes: A Unified Treatment
Wiener-Hopf factorisation, rational families, and an institute-wide operator register
We give a unified treatment of the resolvent operator $\mathscr{R}_q$ of a one-dimensional Lévy process, organised around its Wiener-Hopf factorisation $\mathscr{R}_q = \mathscr{R}_q^+ \mathscr{R}_q^-$ on the analyticity strip. For the three rational families used throughout the IADU programme — Kou double-exponential, hyperexponential, and Lambda-meromorphic — we derive closed-form expressions for $\mathscr{R}_q^\pm$ on exponential test functions, establish a comparison principle for the variational inequality $(q – \mathscr{L}) u \ge 0$, $u \ge g$, and prove a smooth-pasting characterisation of the optimal stopping threshold under (A1)–(A3). The paper is intended as a stand-alone reference for the institute’s later Lévy-control and optimal-stopping work.