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

Online instantaneous and targeted feedback for remote learners

Providing rapid but detailed teaching feedback is a significant problem in distance education, especially for large population courses of short duration, when hand-marking is costly and assignments sent through the postal services cannot be turned round sufficiently quickly. Online assignments with automatic teaching feedback are a possible solution, providing the feedback can be well targeted to individual students. This chapter discusses the online assessment of a ‘maths for science’ course, in which meaningful feedback was given in response to student answers on both summative and purely formative exercises. Students were allowed three attempts at each question, with an increasing amount of teaching feedback being given after each attempt, so encouraging students to engage with the feedback, to learn from it, and to correct their answers if necessary. The mark awarded on the summative assessment reflected the amount of help that had been given. The designers’ concerns included producing a fair test of the course’s learning outcomes within the constraints imposed by the online format and with only minimal use of multiple choice, writing questions that might help to uncover common student misunderstandings coupled with feedback that would address these problems, and tying answer-matching to specific feedback comments. Evidence from statistical analysis of submitted work, and from student responses to questionnaires, has provided insights into the impact of this kind of feedback on the student learning experience. While the majority of students were happy with the online nature of the assessment, a significant proportion appeared to value it more for the immediate indication of their overall performance than for the detailed teaching feedback, and some were put off by the technology or their perception of it. Students were considerably more likely to submit the summative assessment if they had previously engaged online with the practice formative exercises

Understanding the Dynamics of Gene Regulatory Systems : Characterisation and Clinical Relevance of cis-Regulatory Polymorphisms

Peer reviewedPublisher PD

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