Fredholm Methods for Optimal Stopping under Lévy Noise

We develop a Fredholm-operator framework for optimal stopping of Lévy processes with exponential tails, built on the Wiener–Hopf factorisation of the resolvent. The resolvent symbol factorises as $q – \psi(\xi) = \phi^{+}_{q}(\xi)\,\phi^{-}_{q}(\xi)$, where $\phi^{\pm}_{q}$ are the reciprocal transforms of the supremum and the infimum of the process at an independent exponential time. For payoffs that increase with the state, the variational inequality is solved by a two-level cascade. A Fredholm convolution equation of the first kind, whose kernel is the law of the supremum, gives an averaging function. The stopping region is the half-line above the root of that function, and the value is the supremum-average of its positive part. Under Kou double-exponential jumps both levels are explicit, and the investment threshold for the payoff $(P – I)^+$ is $P^\ast = I\,\mathbb{E}[e^{S}] = I\,\alpha_1\alpha_2(\eta_+ – 1)/(\eta_+(\alpha_1 – 1)(\alpha_2 – 1))$, where $\alpha_1, \alpha_2$ are the two positive roots of $\kappa(r) = q$. We fit the Kou model by exact maximum likelihood to 1,567 EU ETS primary-auction clearing prices from 2018-01-08 to 2025-05-20 (EEX auction results as published by the Instrat Foundation). A probability-integral-transform test does not reject the Kou fit (Kolmogorov–Smirnov $p = 0.98$) and rejects a Gaussian fit to the same prices ($p = 0.005$). With the price expected to grow at the euro risk-free rate of 2.73 %, the threshold multiple $P^\ast/I$ falls from 4.53 at a 7 % hurdle rate to 2.29 at 18 %. The McDonald–Siegel threshold of a geometric Brownian motion fitted to the same prices lies 0.2–0.3 % above the Kou threshold at every hurdle rate. On these data the jump-aware threshold is therefore not stable across hurdle rates: the jumps move it by less than half a per cent, and the hurdle rate moves it by a factor of 1.98. Each threshold is checked by a grid solution of the variational inequality and by Monte Carlo simulation of the stopping rule.