By Professor Ronald A. Doney (auth.), Jean Picard (eds.)

Lévy techniques, i.e. strategies in non-stop time with desk bound and autonomous increments, are named after Paul Lévy, who made the relationship with infinitely divisible distributions and defined their constitution. They shape a versatile type of versions, which were utilized to the examine of garage techniques, coverage chance, queues, turbulence, laser cooling, ... and naturally finance, the place the function that they contain examples having "heavy tails" is especially vital. Their pattern course behaviour poses various tricky and interesting difficulties. Such difficulties, and likewise a few comparable distributional difficulties, are addressed intimately in those notes that mirror the content material of the path given by way of R. Doney in St. Flour in 2005.

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**Extra info for Fluctuation Theory for Lévy Processes: Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005**

**Example text**

5) are diﬀerentiable. 4) and assume δ − > 0; then, according to Theorem 11 of Chapter 4, U− admits a density u− which is bounded and continuous on (0, ∞) and has u− (0+) > 0. 11) x and I claim this implies that µ+ is diﬀerentiable on (0, ∞). To see this take x > 0 ﬁxed and write 1 {µ (x) − µ+ (x + h)} h + ∞ 1 u− (y − x)Π + (y)dy − = h x = u− (y − x − h)Π + (y)dy x+h x+h 1 h − ∞ u− (y − x)Π + (y)dy x 1 h ∞ (u− (y − x) − u− (y − x − h)) Π + (y)dy. x+h Clearly the ﬁrst term here converges to u− (0+)Π + (x) as h ↓ 0, and the following shows that the second term also converges.

This question was ﬁrst addressed in Millar [77], where the concept of creep was introduced, although actually Millar called it continuous upward passage. Some partial answers were given in Rogers [85], where the name “creeping” was ﬁrst introduced, but the complete solution is due to Vigon [99], [100]. s. ) Of course a as r ↓ 0. (Here Or necessary and suﬃcient condition for this to hold as r → ∞ is that ∞ m+ = EH+ (1) = δ + + µ+ (x)dx < ∞, 0 and similarly one can ask how we can recognise when this happens from the characteristics of X.

40 4 Ladder Processes and the Wiener–Hopf Factorisation However this inequality is also valid for the random walk deﬁned by Sn = n Sn − 1 Yr 1{Yr ∈(−K,0]} , which has B + = B + , so that EB + (Y − ; Y − ≥ K) ≥ 1/2. Since K is arbitrary we conclude that EB + (Y − ) = ∞. We then see that always at least one of EB + (Y − ) and EB − (Y + ) is inﬁnite and when Sn oscillates both are. 16) shows that when Sn drifts to ∞ we have EV (D1 ) < ∞, which again by the Erickson bound means that EB ∗ (D1 ) < ∞.