A direct approach to the discounted penalty function

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1 784 (2010)], which uses different methods.

Leniency programs, antitrust enforcement and multimarket contact : three essays in industrial organization

The Layout of My Thesis This thesis contains three chapters in Industrial Organization that build on the work outlined above. The first two chapters combine leniency programs with multimarket contact and provide a thorough analysis of the potential effects of Amnesty Plus and Penalty Plus. The third chapter puts the whole discussion on leniency programs into perspective by examining other enforcement tools available to an antitrust authority. The main argument in that last chapter is that a specific instrument can only be as effective as the policy in which it is embedded. It is therefore important for an antitrust authority to know how it best accompanies the introduction or modification of a policy instrument that helps deterrence. INTRODUCTION Chapter 1 examines the efféct of Amnesty Plus and Penalty Plus on the incentives of firms to report cartel activities. The main question is whether the inclusion of these policies in a leniency program undermine the effectiveness of the latter by discouraging the firms to apply for amnesty. The model is static and focus on the ex post incentives of firms to desist from collusion. The results suggest that, because Amnesty Plus and Penalty Plus encourage the reporting of a second cartel after a first detection, a firm, anticipating this, may be reluctant to seek leniency and to report in the first place. However, the effect may also go in the opposite direction, and Amnesty Plus and Penalty Plus may encourage the simultaneous reporting of two cartels. Chapter 2 takes this idea further to the stage of cartel formation. This chapter provides a complete characterization of the potential anticompetitive and procompetitive effects of Amnesty Plus in a infinitely repeated game framework when the firms use their multimarket contact to harshen punishment. I suggest a clear-cut policy rule that prevents potential adverse effects and thereby show that, if policy makers follow this rule, a leniency program with Amnesty Plus performs better than one without. Chapter 3 characterizes the socially optimal enforcement effort of an antitrust authority and shows how this effort changes with the introduction or modification of specific policy instruments. The intuition is that the policy instrument may increase the marginal benefit of conducting investigations. If this effect is strong enough, a more rigorous detection policy becomes socially desirable.

An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

Spatially selective implementation of the adiabatic T2 prep sequence for magnetic resonance angiography of the coronary arteries.

In coronary magnetic resonance angiography, a magnetization-preparation scheme for T2 -weighting (T2 Prep) is widely used to enhance contrast between the coronary blood-pool and the myocardium. This prepulse is commonly applied without spatial selection to minimize flow sensitivity, but the nonselective implementation results in a reduced magnetization of the in-flowing blood and a related penalty in signal-to-noise ratio. It is hypothesized that a spatially selective T2 Prep would leave the magnetization of blood outside the T2 Prep volume unaffected and thereby lower the signal-to-noise ratio penalty. To test this hypothesis, a spatially selective T2 Prep was implemented where the user could freely adjust angulation and position of the T2 Prep slab to avoid covering the ventricular blood-pool and saturating the in-flowing spins. A time gap of 150 ms was further added between the T2 Prep and other prepulses to allow for in-flow of a larger volume of unsaturated spins. Consistent with numerical simulation, the spatially selective T2 Prep increased in vivo human coronary artery signal-to-noise ratio (42.3 ± 2.9 vs. 31.4 ± 2.2, n = 22, P < 0.0001) and contrast-to-noise-ratio (18.6 ± 1.5 vs. 13.9 ± 1.2, P = 0.009) as compared to those of the nonselective T2 Prep. Additionally, a segmental analysis demonstrated that the spatially selective T2 Prep was most beneficial in proximal and mid segments where the in-flowing blood volume was largest compared to the distal segments. Magn Reson Med, 2013. © 2012 Wiley Periodicals, Inc.