Multimodal Treatment of Malignant Pleural Mesothelioma: Real-World Experience with 112 Patients.

Malignant pleural mesothelioma (MPM) is a rare pleural cancer associated with asbestos exposure. According to current evidence, the combination of chemotherapy, surgery and radiotherapy improves patients' survival. However, the optimal sequence and weighting of the respective treatment modalities is unclear. In anticipation of the upcoming results of the MARS-2 trial, we sought to determine the relative impact of the respective treatment modalities on complications and overall survival in our own consecutive institutional series of 112 patients. Fifty-seven patients (51%) underwent multimodality therapy with curative intent, while 55 patients (49%) were treated with palliative intent. The median overall survival (OS) of the entire cohort was 16.9 months (95% CI: 13.4−20.4) after diagnosis; 5-year survival was 29% for patients who underwent lung-preserving surgery. In univariate analysis, surgical treatment (p < 0.001), multimodality therapy (p < 0.001), epithelioid subtype (p < 0.001), early tumor stage (p = 0.02) and the absence of arterial hypertension (p = 0.034) were found to be prognostic factors for OS. In multivariate analysis, epithelioid subtype was associated with a survival benefit, whereas the occurrence of complications was associated with worse OS. Multimodality therapy including surgery significantly prolonged the OS of MPM patients compared with multimodal therapy without surgery.

Exception Sets of Intrinsic and Piecewise Lipschitz Functions.

We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in R d , which include Lipschitz submanifolds.

Existence, uniqueness and regularity of the projection onto differentiable manifolds.

We investigate the maximal open domain E ( M ) on which the orthogonal projection map p onto a subset M ⊆ R d can be defined and study essential properties of p. We prove that if M is a C 1 submanifold of R d satisfying a Lipschitz condition on the tangent spaces, then E ( M ) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C 2 or if the topological skeleton of M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C k -submanifold M with k ≥ 2 , the projection map is C k - 1 on E ( M ) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion M ⊆ E ( M ) is that M is a C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with M ⊆ E ( M ) , then M must be C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between E ( M ) and the topological skeleton of M c .

Approximation methods for piecewise deterministic Markov processes and their costs.

In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. We prove a convergence result for our PDMPs approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and compare deterministic and Monte Carlo integration.

Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient.

We prove strong convergence of order [Formula: see text] for arbitrarily small [Formula: see text] of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.

On the length of arcs in labyrinth fractals.

Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length (Cristea and Steinsky in Geom Dedicata 141(1):1-17, 2009; Proc Edinb Math Soc 54(2):329-344, 2011). In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.

Utility indifference pricing of insurance catastrophe derivatives.

We propose a model for an insurance loss index and the claims process of a single insurance company holding a fraction of the total number of contracts that captures both ordinary losses and losses due to catastrophes. In this model we price a catastrophe derivative by the method of utility indifference pricing. The associated stochastic optimization problem is treated by techniques for piecewise deterministic Markov processes. A numerical study illustrates our results.

Fast orthogonal transforms and generation of Brownian paths.

We present a number of fast constructions of discrete Brownian paths that can be used as alternatives to principal component analysis and Brownian bridge for stratified Monte Carlo and quasi-Monte Carlo. By fast we mean that a path of length [Formula: see text] can be generated in [Formula: see text] floating point operations. We highlight some of the connections between the different constructions and we provide some numerical examples.