Seminaire De Probabilites XXXVIII by Ronald A. Doney (auth.), Michel Émery, Michel Ledoux, Marc

By Ronald A. Doney (auth.), Michel Émery, Michel Ledoux, Marc Yor (eds.)

Besides a chain of six articles on Lévy methods, quantity 38 of the Séminaire de Probabilités includes contributions whose subject matters diversity from research of semi-groups to loose likelihood, through martingale thought, Wiener house and Brownian movement, Gaussian strategies and matrices, diffusions and their functions to PDEs.

As do all past volumes of this sequence, it presents an outline at the present state-of-the-art within the examine on stochastic processes.

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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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