We consider the super-replication problem for a class of exotic options known as life-contingent options within the framework of the Black–Scholes market model. The option is allowed to be exercised if the death of the option holder occurs before the expiry date, otherwise there is a compensation payoff at the expiry date. We show that there exists a minimal super-replication portfolio and determine the associated initial investment. We then give a characterisation of when replication of the option is possible. Finally, we give an example of an explicit super-replicating hedge for a simple life-contingent option.

The increasingly widespread application of drones and the emergence of urban air mobility leads to a challenging question in airspace modernisation: how to create a safe and scalable air traffic management system that can handle the expected density of operations. Increasing the number of vehicles in a given airspace volume and enabling routine operations are essential for these services to be economically viable. However, a higher density of operations increases risks, poses a great challenge for coordination and necessitates the development of a novel solution for traffic management. This paper contributes to the research towards such a strategy and the field of airspace management by calculating and analysing the conflict probability in an en-route, free-flight scenario for autonomous vehicles. Analytical methods are used to determine the directional dependence of conflict probabilities for exponential and normal prescribed speed probability distributions. The notions of geometric and speed conflict are introduced and distinguished throughout the calculations of the paper. The effect of truncating the set of available flight speeds is also investigated. The sensitivity of the calculated results to speed and heading perturbations is studied within the analytical framework and verified by numerical simulations. Results enable a fresh approach to conflict detection and resolution through distribution shaping and are intended to be used in an integrated, stochastic coordination framework.

The aim of this paper is to provide preliminary results on a traffic coordination framework based on stochastic task allocation. General trends and the predicted advent of personal aerial vehicles increase traffic rapidly, but current air traffic management methods admittedly cannot scale appropriately. A hierarchical system is proposed to overcome the problem, the middle layer of which is elaborated in this paper. This layer aims to enable stochastic control of traffic behaviour using a single parameter, which is achieved by applying distributed stochastic task allocation. The task allocation algorithm is used to allocate speeds to vehicles in a scalable way. By regulating the speed distribution of vehicles the conflict rates remain manageable. Multi-agent simulation results show that it is possible to control ensemble dynamics and together with that traffic safety and throughput via a single parameter. Using transient simulations the dynamic performance of the system is analysed. It is shown that the traffic conflict reduction problem can be transformed into a control design problem. The performance of a simple controller is also evaluated. It was shown that by applying the controller, quicker transients can be achieved for the mean speed of the system.

In the field of Life Sciences, it is very common to deal with extremely complex systems, from both analytical and computational points of view, due to the unavoidable coupling of different interacting structures. As an example, angiogenesis has revealed to be an highly complex, and extremely interesting biomedical problem, due to the strong coupling between the kinetic parameters of the relevant branching – growth – anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. In this paper, an original revisited conceptual stochastic model of tumour-driven angiogenesis has been proposed, for which it has been shown that it is possible to reduce complexity by taking advantage of the intrinsic multiscale structure of the system; one may keep the stochasticity of the dynamics of the vessel tips at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. While in previous papers, only an heuristic justification of this approach had been offered; in this paper, a rigorous proof is given of the so called ‘propagation of chaos’, which leads to a mean field approximation of the stochastic relevant measures associated with the vessel dynamics, and consequently of the underlying tumour angiogenic factor (TAF) field. As a side, though important result, the non-extinction of the random process of tips has been proven during any finite time interval.

A data-driven model reduction strategy is presented for the representation of random polycrystal microstructures. Given a set of microstructure snapshots that satisfy certain statistical constraints such as given low-order moments of the grain size distribution, using a non-linear manifold learning approach, we identify the intrinsic low-dimensionality of the microstructure manifold. In addition to grain size, a linear dimensionality reduction technique (Karhunun-Loéve Expansion) is used to reduce the texture representation. The space of viable microstructures is mapped to a low-dimensional region thus facilitating the analysis and design of polycrystal microstructures. This methodology allows us to sample microstructure features in the reduced-order space thus making it a highly efficient, low-dimensional surrogate for representing microstructures (grain size and texture). We demonstrate the model reduction approach by computing the variability of homogenized thermal properties using sparse grid collocation in the reduced-order space that describes the grain size and orientation variability.

This special volume of the ESAIM Journal, Mathematical Modelling and Numerical Analysis,contains a collection of articles on probabilistic interpretations of some classes of nonlinear integro-differential equations.The selected contributions deal with a wide range of topics in applied probability theory and stochastic analysis, with applications in a variety of scientific disciplines, includingphysics, biology, fluid mechanics, molecular chemistry, financial mathematics and bayesian statistics. In this preface, we provide a brief presentation of the main contributions presented in this special volume. We have also included an introduction to classic probabilistic methods and a presentation of the more recent particle methods, with a synthetic picture of their mathematical foundations and their range of applications.

Fatigue crack propagation data of a batch of AISI 4340 steel specimens are released in the present paper. The statistical nature of the data is specially emphasized, and a probabilistic fracture mechanics model is introduced to analyze the data. The stochastic differential equation associated with the model is then solved. The solution gives us the crack exceedance probability as well as the probability distribution of the random time to reach a specified crack size. These quantities are useful in the reliability assessment of structures made of the tested material. Comparing the analytical result with the experimental result, it is found that the proposed probabilistic fracture mechanics model can reasonably explain the experimental data. For those data that cannot be fitted well by the proposed model, methods of improvement are proposed in the present paper as well.

We study a classical stochastic control problem arising in financial economics: to maximize expected logarithmic utility from terminal wealth and/or consumption. The novel feature of our work is that the portfolio is allowed to anticipate the future, i.e. the terminal values of the prices, or of the driving Brownian motion, are known to the investor, either exactly or with some uncertainty. Results on the finiteness of the value of the control problem are obtained in various setups, using techniques from the so-called enlargement of filtrations. When the value of the problem is finite, we compute it explicitly and exhibit an optimal portfolio in closed form.

This is a largely expository paper on approximate methods of solving random ordinary differential equations, with an emphasis on direct numerical methods. Two methods are discussed in some detail and several others are mentioned briefly.