Fredholm Methods for Optimal Stopping under Lévy Noise
A Wiener-Hopf Cascade for Real Options under Lévy Noise
Abstract
We develop a Fredholm-operator framework for optimal stopping of pure-jump Lévy processes, recasting the associated variational inequality as a fixed-point problem driven by the resolvent of a non-local generator. The resolvent admits a Wiener-Hopf factorization $q – \psi(\xi) = \phi^{+}_{q}(\xi)\,\phi^{-}_{q}(\xi)$, and we organise the resulting calculus into a cascade of explicit operator equations whose solutions yield closed-form expressions for the value function and the optimal stopping boundary. As a showcase we solve the timing of irreversible green-capacity investment under a Kou jump-diffusion carbon price, calibrated to EU ETS data, and show that the cascade threshold is stable at $€67/$tCO$_2$ across the policy-relevant hurdle-rate range, while the McDonald-Siegel diffusion-only baseline produces thresholds ranging from $€100$ to $€350/$tCO$_2$ over the same range — an instability that masks the option value of waiting under jump-asymmetric noise. The construction is constructive: every step reduces to a one-sided Fourier integral computable in $O(N \log N)$, giving a numerical scheme whose convergence rate matches the Hölder regularity of the payoff.