Patricia Saavedra

An optimal investment strategy with maximal risk aversion and its ruin probability

In this paper we study an optimal investment problem of an insurer when the company has the opportunity to invest in a risky asset using stochastic control techniques. A closed form solution is given when the risk preferences are exponential as well as an estimate of the ruin probability when the optimal strategy is used.

Optimal Policies for Payment of Dividends through a Fixed Barrier at Discrete Time.

In this paper a discrete-time reserve process with a fixed bar rier is presented and modelled as a discounted Markov Decision Process. The non-payment of dividends is pe nalized. The minimization of this penalty results in an optimal control problem. This work focuses on d etermining the sequence of premiums that minimize penalty costs, and obtaining a rate for the probability of ruin to ensure a sustainable reserve operation.

Global Bifurcation Diagram for the Kerner-Konhauser Traffic Flow Model

We study traveling wave solutions of the Kerner–Konhauser PDE for traffic flow. By a standard change of variables, the problem is reduced to a dynamical system in the plane with three parameters. In a previous paper [Carrillo et al., 2010] it was shown that under general hypotheses on the fundamental diagram, the dynamical system has a surface of critical points showing either a fold or cusp catastrophe when projected under a two-dimensional plane of parameters named qg–vg. In either case, a one parameter family of Takens–Bogdanov (TB) bifurcation takes place, and therefore local families of Hopf and homoclinic bifurcation arising from each TB point exist. Here, we prove the existence of a degenerate Takens–Bogdanov bifurcation (DTB) which in turn implies the existence of Generalized Hopf or Bautin bifurcations (GH). We describe numerically the global lines of bifurcations continued from the local ones, inside a cuspidal region of the parameter space. In particular, we compute the first Lyapunov exponent, and compare with the GH bifurcation curve. We present some families of stable limit cycles which are taken as initial conditions in the PDE leading to stable traveling waves.

TRAVELING WAVES, CATASTROPHES AND BIFURCATIONS IN A GENERIC SECOND ORDER TRAFFIC FLOW MODEL

We consider the macroscopic, second order model of Kerner–Konhauser for traffic flow given by a system of PDE. Assuming conservation of cars, traveling waves solution of the PDE are reduced to a dynamical system in the plane. We prove that under generic conditions on the so-called fundamental diagram, the surface of critical points has a fold or cusp catastrophe and each fold point gives rise to a Takens–Bogdanov bifurcation. In particular, limit cycles arising from a Hopf bifurcation give place to traveling wave solutions of the PDE.

Clusters in the helbing's improved model

The formation of clusters in Helbings improved model is studied by an iterative method. It is shown that after certain density we will always obtain a density profile which has the structure of a soliton. Its characteristics such as the amplitude and width are determined by the parameters in the model.

Network of M/M/1 Cyclic Polling Systems.

This paper presents a Network of Cyclic Polling Systems that consists of two cyclic polling systems with two queues each when transfer of users from one system to the other is imposed. This system is modelled in discrete time. It is assumed that each system has exponential inter-arrival times and the servers apply an exhaustive policy. Closed form expressions are obtained for the first and second moments of the queue’s lengths for any time.

Markov Decision Processes Applied to the Payment of Dividends of a Reserve Process

Markov decision theory is applied to study the distribution of dividends of a discrete reserve process with a fixed barrier. The non-payment of dividends is penalized through a cost function which implies solving an optimal control problem. Two objective functions are proposed: a discounted cost and an average one. In both cases, the same optimal strategy for the payment of dividends is obtained, which ensures a ruin probability that guarantees a sustainable reserve operation for claims distributed with light or heavy tails.

A Bogdanov–Takens Bifurcation in Generic Continuous Second Order Traffic Flow Models

We consider the continuous model of Kerner–Konhauser for traffic flow given by a second order PDE for the velocity and density. Assuming conservation of cars, traveling waves solution of the PDE are reduced to a dynamical system in the plane. We describe the bifurcations set of critical points and show that there is a curve in the set of parameters consisting of Bogdanov–Takens bifurcation points. In particular there exists Hopf, homoclinic and saddle node bifurcation curves. For each Hopf point a one parameter family of limit cyles exists. Thus we prove the existence of solitons solutions in the form of one bump traveling waves.