A dynamic contagion process and an application to credit risk

We introduce a new point process, the dynamic contagion process, by gener- alising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exoge- nous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piece- wise deterministic Markov process theory developed by Davis (1984), and the extension of the martingale methodology used by Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and the probability generating function of the point process have been de- rived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathemati- cal framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance and insurance. We provide an application of this process to credit risk, and the simulation algorithm for further industrial implementation and statistical analysis

Brownian excursions outside a corridor and two-sided Parisian options

In this paper, we study the excursion time of a Brownian motion with drift outside a corridor by using a four states semi-Markov model. In mathematical finance, these results have an important application in the valuation of double barrier Parisian options. In this paper, we obtain an explicit expression for the Laplace transform of its price

Wave scattering by small impedance particles in a medium

The theory of acoustic wave scattering by many small bodies is developed for bodies with impedance boundary condition. It is shown that if one embeds many small particles in a bounded domain, filled with a known material, then one can create a new material with the properties very different from the properties of the original material. Moreover, these very different properties occur although the total volume of the embedded small particles is negligible compared with the volume of the original material

Electromagnetic wave scattering by many small particles

Scattering of electromagnetic waves by many small particles of arbitrary shapes is reduced rigorously to solving linear algebraic system of equations bypassing the usual usage of integral equations. The matrix elements of this linear algebraic system have physical meaning. They are expressed in terms of the electric and magnetic polarizability tensors. Analytical formulas are given for calculation of these tensors with any desired accuracy for homogeneous bodies of arbitrary shapes. An idea to create a "smart" material by embedding many small particles in a given region is formulated

On the stability of equilibrium for a reformulated foreign trade model of three countries

In this paper, we study the stability of equilibrium for a foreign trade model consisting of three countries. As the gravity equation has been proven an excellent tool of analysis and adequately stable over time and space all over the world, we further enhance the problem to three masses. We use the basic Structure of Heckscher-Ohlin-Samuelson model. The national income equals consumption outlays plus investment plus exports minus imports. The proposed reformulation of the problem focus on two basic concepts: (1) the delay inherited in our economic variables and (2) the interaction effect along the three economies involved. Stability and stabilizability conditions are investigated while numerical examples provide further insight and better understanding. Finally, a generalization of the gravity equation is somehow obtained for the model