Small and Large Time Stability of the Time taken for a L\'evy Process to Cross Curved Boundaries

This paper is concerned with the small time behaviour of a L\'{e}vy process $X$. In particular, we investigate the {\it stabilities} of the times, \Tstarb(r) and \Tbarb(r), at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in\R^2: y\le rt^b, t\ge 0\}$ (one-sided exit), or $\{(t,y)\in\R^2: |y|\le rt^b, t\ge 0\}$ (two-sided exit), $0\le b<1$, as r\dto 0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^p$. In many instances these are seen to be equivalent to relative stability of the process $X$ itself. The analogous large time problem is also discussed

Stability of the Exit Time for L\'evy Processes

This paper is concerned with the behaviour of a L\'{e}vy process when it crosses over a positive level, $u$, starting from 0, both as $u$ becomes large and as $u$ becomes small. Our main focus is on the time, $\tau_u$, it takes the process to transit above the level, and in particular, on the {\it stability} of this passage time; thus, essentially, whether or not $\tau_u$ behaves linearly as u\dto 0 or $u\to\infty$. We also consider conditional stability of $\tau_u$ when the process drifts to $-\infty$, a.s. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cram\'er condition

Asymptotic Distributions of the Overshoot and Undershoots for the L\'evy Insurance Risk Process in the Cram\'er and Convolution Equivalent Cases

Recent models of the insurance risk process use a L\'evy process to generalise the traditional Cram\'er-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a L\'{e}vy process which drifts to $-\infty$ and satisfies a Cram\'er or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cram\'er case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the "medium-heavy" tailed convolution equivalent model segues into the "light-tailed" Cram\'er model in a natural way. This suggests a usefully expanded flexibility for modelling the insurance risk process. We illustrate this relationship by comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the L\'evy process belongs to the "GTSC" class

Path decomposition of ruinous behavior for a general L\'{e}vy insurance risk process

We analyze the general L\'{e}vy insurance risk process for L\'{e}vy measures in the convolution equivalence class $\mathcal{S}^{(\alpha)}$, $\alpha>0$, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level $u\to \infty$, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as $u\to \infty$, conditional on ruin occurring under our assumptions. Existing asymptotic results under the $\mathcal{S}^{(\alpha)}$ assumption are synthesized and extended, and proofs are much simplified, by comparison with previous methods specific to the convolution equivalence analyses. Additionally, limiting expressions for penalty functions of the type introduced into actuarial mathematics by Gerber and Shiu are derived as straightforward applications of our main results.Comment: Published in at http://dx.doi.org/10.1214/11-AAP797 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic Distributions of the Overshoot and Undershoots for the Lévy Insurance Risk Process in the Cramér and Convolution Equivalent Cases

Recent models of the insurance risk process use a Levy process to generalise the traditional Cramer-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Levy process which drifts to -infinity and satis es a Cramer or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramer case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the medium-heavy tailed convolution equivalent model segues into the light-tailed Cramer model in a natural way. This suggests a usefully expanded exibility for modelling the insurance risk process. We illustrate this relationship by comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the Levy process belongs to the GTSC class

Lexical neutrality in environmental health research: Reflections on the term walkability.

Neighbourhood environments have important implications for human health. In this piece, we reflect on the environments and health literature and argue that precise use of language is critical for acknowledging the complex and multifaceted influence that neighbourhood environments may have on physical activity and physical activity-related outcomes. Specifically, we argue that the term "neighbourhood walkability", commonly used in the neighbourhoods and health literature, constrains recognition of the breadth of influence that neighbourhood environments might have on a variety of physical activity behaviours. The term draws attention to a single type of physical activity and implies that a universal association exists when in fact the literature is quite mixed. To maintain neutrality in this area of research, we suggest that researchers adopt the term "neighbourhood physical activity environments" for collective measures of neighbourhood attributes that they wish to study in relation to physical activity behaviours or physical activity-related health outcomes

Anomalous Breaking of Anisotropic Scaling Symmetry in the Quantum Lifshitz Model

In this note we investigate the anomalous breaking of anisotropic scaling symmetry in a non-relativistic field theory with dynamical exponent z=2. On general grounds, one can show that there exist two possible "central charges" which characterize the breaking of scale invariance. Using heat kernel methods, we compute these two central charges in the quantum Lifshitz model, a free field theory which is second order in time and fourth order in spatial derivatives. We find that one of the two central charges vanishes. Interestingly, this is also true for strongly coupled non-relativistic field theories with a geometric dual described by a metric and a massive vector field.Comment: 26 pages; major revision (results were unaffected), published versio