An explicit Skorokhod embedding for spectrally negative Levy processes

We present an explicit solution to the Skorokhod embedding problem for spectrally negative L\'evy processes. Given a process $X$ and a target measure $\mu$ satisfying an explicit admissibility condition we define functions \f_\pm such that the stopping time T = \inf\{t>0: X_t \in \{-\f_-(L_t), \f_+(L_t)\}\} induces $X_T\sim \mu$. We also treat versions of $T$ which take into account the sign of the excursion straddling time $t$. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (1992) and we compute explicitly their general quantities in our setup. Our method relies on some new explicit calculations relating scale functions and the It\^o excursion measure of $X$. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.Comment: This is the final version of the paper that has been accepted for publication in J. Theor. Probab. In this new version several typos were corrected and Lemma 6(iii) [now Lemma 5(iii)] was modified. The original publication is available at http://www.springerlink.co

Asymptotic independence of three statistics of maximal segmental scores

Let $\xi_1,\xi_2,\ldots$ be an iid sequence with negative mean. The $(m,n)$-segment is the subsequence $\xi_{m+1},\ldots,\xi_n$ and its \textit{score} is given by $\max\{\sum_{m+1}^n\xi_i,0\}$. Let $R_n$ be the largest score of any segment ending at time $n$, $R^*_n$ the largest score of any segment in the sequence $\xi_{1},\ldots,\xi_n$, and $O_x$ the overshoot of the score over a level $x$ at the first epoch the score of such a size arises. We show that, under the Cram\'er assumption on $\xi_1$, asymptotic independence of the statistics $R_n$, $R_n^* -y$ and $O_{x+y}$ holds as $\min\{n,y,x\}\to\infty$. Furthermore, we establish a novel Spitzer-type identity characterising the limit law $O_\infty$ in terms of the laws of $(1,n)$-scores. As corollary we obtain: (1) a novel factorization of the exponential distribution as a convolution of $O_\infty$ and the stationary distribution of $R$; (2) if $y=\gamma^{-1}\log n$ (where $\gamma$ is the Cram\'er coefficient), our results, together with the classical theorem of Iglehart \cite{Iglehart}, yield the existence and explicit form of the joint weak limit of $(R_n, R_n^* -y,O_{x+y})$.Comment: 13 pages, no figure