Seminaire de Probabilites XIV by Azema J. (Ed)

By Azema J. (Ed)

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Q)(ψ(α) − q) (10) Proof. We begin by noting some facts which will be used in conjunction with the martingale (9). Recall that eq is an exponentially distributed random variable with parameter q > 0 independent of the process X. First note that by an application of Fubini’s theorem, eq E ∞ e−αZs ds = e−qs E e−αZs ds = 0 0 1 E e−αZeq . q Next we recall a well known result, known as the Duality Lemma, which can best be verified with a diagram. That is by defining the process {Xs = X(t−s)− − Xt : 0 s t} as the time reversed L´evy process from the fixed moment, t, the law of X and {−Xs : 0 s t} are the same.

38 Martijn R. Pistorius Step 4: An identity between Laplace transforms. Note that for q, λ > 0 with q > ψ(λ) (or equivalently Φ(q) > λ) one has that q − ψ(λ) −1 ∞ = q −1 E eλXη(q) = 0 ∞ Φ(q) − λ + = Φ (q) ∞ eλx uq (x) dx + e−λx uq (−x) dx 0 e−λx uq (−x) dx. (18) 0 By analytic continuation in λ, the identity (18) remains valid for (λ) > 0 except for λ = Φ(q) and then by continuity for all λ with (λ) 0. Inverting the Laplace transforms in λ leads then to equation (3). Step 5: Wiener–Hopf factorisation.

2004) Some martingales associated to reflected L´evy processes, in this volume. 12. Pistorius, M. R. (2004) A potential-theoretic review of some exit problems of spectrally negative L´evy processes, in this volume. 13. Rogers, L. C. G. (1990) The two-sided exit problem for spectrally positive L´evy processes, Adv. Appl. Probab. 22, 486–487. 14. Sato, K. (1999) L´evy processes and infinitely divisible distributions, Cambridge University Press. 15. N. (1976) Problem of destruction and resolvent of terminating processes with independent increments, Ukranian Math.

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