[Paper Review] On obtaining simple identities for overshoots of spectrally negative Lévy processes

Ronnie Loeffen|arXiv (Cornell University)|Oct 20, 2014

Probability and Risk Models18 references7 citations

TL;DR

This paper derives a general analytic identity for the overshoot distribution of spectrally negative Lévy processes over a fixed level by introducing a free auxiliary function extension of the penalty function. The key contribution is a unified framework that simplifies known results and enables direct derivation of simple identities by optimal choice of the extension, applicable to both standard and reflected/refracted processes via scale functions and infinitesimal generators.

ABSTRACT

For a (killed) spectrally negative Lévy process we provide an analytic expression for the distribution of its overshoot over a fixed level in terms of the infinitesimal generator and the scale function of the process. Our identity involves an auxiliary function and the simplicity of the identity depends very much on the choice of this function. In particular, for specific choices one recovers various previous established formulas in the literature. We review several applications and also show that one can get in a similar way identities of overshoots for reflected and refracted spectrally negative Lévy processes.

Motivation & Objective

• To provide a general analytic expression for the overshoot distribution of spectrally negative Lévy processes over a fixed level.
• To unify and simplify previously derived formulas for expected discounted penalty functions by introducing a free auxiliary function extension of the penalty function.
• To demonstrate that optimal choice of this extension yields the simplest possible expressions, recovering known results as special cases.
• To extend the framework to reflected and refracted spectrally negative Lévy processes, enabling similar identities for their overshoots.
• To establish a connection between the infinitesimal generator, scale functions, and the structure of overshoot expectations through a novel smoothing technique.

Proposed method

• Introduces an arbitrary extension $\widetilde{f}$ of the penalty function $f$ defined on $(-\infty, a)$ to $(-\infty, b]$, which is free to choose under regularity conditions.
• Derives a general identity for $\mathbb{E}_x[\mathrm{e}^{-q\tau_a^-}f(X_{\tau_a^-})\mathbf{1}_{\{\tau_a^-<\tau_b^+\}}]$ using the infinitesimal generator and the scale function $W^{(q)}$.
• Employs a mollification technique via convolution with a smooth bump function $\phi_\epsilon$ to construct a smooth approximation $W^{(q)} \star \phi_\epsilon$ of the scale function, ensuring $C^\infty$ regularity.
• Applies the martingale property of $\mathrm{e}^{-q(t\wedge\tau_{a}^{-}\wedge\tau_{b}^{+})}(W^{(q)}\star\phi_\epsilon)(X_{t\wedge\tau_{a}^{-}\wedge\tau_{b}^{+}})$ to derive a PDE-based identity involving the generator and resolvent measure.
• Takes the limit as $\epsilon \downarrow 0$ to recover the identity for the original scale function, leveraging dominated convergence and continuity of $W^{(q)\prime}$ on compact sets.
• Applies the same method to reflected and refracted spectrally negative Lévy processes by adapting the generator and boundary conditions.

Experimental results

Research questions

• RQ1 How can a general identity for overshoot expectations in spectrally negative Lévy processes be derived using the scale function and infinitesimal generator?
• RQ2 Why do certain penalty functions yield particularly simple expressions in existing literature, and what underlies this simplification?
• RQ3 Can a unified framework be constructed that recovers known results as special cases through a free choice of function extension?
• RQ4 How can the same approach be adapted to reflected and refracted spectrally negative Lévy processes?
• RQ5 What role does the mollification of the scale function play in ensuring the validity of the derived identities?

Key findings

• The paper establishes a general identity for the overshoot expectation involving the scale function $W^{(q)}$, the infinitesimal generator, and a free auxiliary function $\widetilde{f}$, which reduces to known formulas upon specific choices of $\widetilde{f}$.
• The optimal choice of $\widetilde{f}$—specifically, $\widetilde{f} = W^{(q)} \star \phi_\epsilon$—leads to the simplest possible expression, recovering classical results in fluctuation theory.
• The identity holds for both bounded and unbounded variation paths, including cases with no Gaussian component, by taking the limit $\epsilon \downarrow 0$.
• The method applies beyond standard processes: analogous identities are derived for reflected and refracted spectrally negative Lévy processes using modified generator dynamics.
• The convergence of the mollified scale function $W^{(q)} \star \phi_\epsilon$ to $W^{(q)}$ and its derivative is rigorously justified using the mean value theorem and dominated convergence.
• The framework explains the structural simplicity of known results by showing that they arise from particular choices of $\widetilde{f}$, making the derivation process systematic and transparent.