22 research outputs found
Risk theory and optimal control of Lévy driven processes
Esta tesis contiene tres artÃculos de investigación con aportes originales. El primer artÃculo, que coincide con el CapÃtulo 2, ha sido publicado (Diko and Usabel (17)) en Insurance: Mathematics and Economics, una revista de reconocimiento internacional incluÃda en JCR. En el citado capÃtulo se propone un método numérico que permite evaluar la función de utilidad en un marco de proceso de Poisson compuesto con cambio de régimen. Esto supone que los parámetros del modelo de Poisson compuesto pueden variar en el tiempo, gobernados por un proceso de Markov subyacente. Este modelo es una generalización de los procesos que se analizan en la literatura relevante hasta el momento, por tanto el aporte de este capÃtulo consiste tanto en el desarrollo de un modelo nuevo, capaz de reflejar un entorno económico variable, como en el método de cálculo de cuantÃas de interés relacionadas con este. Estas incluyen entre otras la probabilidad de la ruina, supervivencia o el déficit medio al producirse la ruina. El CapÃtulo 3 expone el tratamiento genérico de un problema de control estocástico en el marco de procesos generales de difusión de Lévy. Este tipo de problemas es conocido por su difficultad a la hora de obtener soluciones concretas, ya que las equaciones diferenciales o integro-diferenciales que caracterizan la solución no admiten tratamiento analÃtico exacto. Habitualmente se aplican métodos numéricos de discretización de tiempo. En esta tesis, se desarrolla un método de solución alternativo que consiste en Erlangizar (dividir en intervalos aleatorios exponenciales) el horizonte temporal establecido con lo que se consigue simplificar la complejidad de las equaciones diferenciales involucradas. Esta transformación lleva a una metodologÃa de aproximación iterativa aplicable a un gran abanico de problemas del area de finanzas y seguros. Los resultados de este capÃulo están en el proceso de revisión en Mathematical Finance, una de las revistas de finanzas estocásticas más importantes en el mundo. Por último, el CapÃulo 4 ofrece una aplicación de la metodologÃa presentada anteriormente en el marco de solvencia de una compañÃa de seguros. En este contexto se plantea un problema de decisión sobre la composición de la cartera de inversión optima con el fin de maximizar la utilidad esperada de una cartera sometida a un proceso de riesgo. Aplicando el algoritmo iterativo del CapÃtulo 3 se calculan las cuantÃas de interés y se demuestra la rápida convergencia y buenas propiedades del método propuesto. El contenido de este capÃtulo también representa un aporte original y está actualmente bajo revisión en la revista ASTIN Bulletin, referente principal en el campo de investigación actuarial. En conlusi on, los tres aportes de investigaci on original presentados en esta tesis permiten una aplicación de métodos numéricos para obtener resultados concretos en situaciones que hasta ahora no han sido tratadas en la literaturaChebyshev approximation in risk processes. Optimal control of Lévy diffusions. Risk theory and optimal investmen
On finite-time ruin probabilities with reinsurance cycles influenced by large claims
Market cycles play a great role in reinsurance. Cycle transitions are not independent from the claim arrival process : a large claim or a high number of claims may accelerate cycle transitions. To take this into account, a semi-Markovian risk model is proposed and analyzed. A refined Erlangization method is developed to compute the finite-time ruin probability of a reinsurance company. As this model needs the claim amounts to be Phase-type distributed, we explain how to fit mixtures of Erlang distributions to long-tailed distributions. Numerical applications and comparisons to results obtained from simulation methods are given. The impact of dependency between claim amounts and phase changes is studied.
On finite-time ruin probabilities with reinsurance cycles influenced by large claims
International audienceMarket cycles play a great role in reinsurance. Cycle transitions are not independent from the claim arrival process : a large claim or a high number of claims may accelerate cycle transitions. To take this into account, a semi-Markovian risk model is proposed and analyzed. A refined Erlangization method is developed to compute the finite-time ruin probability of a reinsurance company. As this model needs the claim amounts to be Phase-type distributed, we explain how to fit mixtures of Erlang distributions to long-tailed distributions. Numerical applications and comparisons to results obtained from simulation methods are given. The impact of dependency between claim amounts and phase changes is studied
Approximations for time-dependent distributions in Markovian fluid models
In this paper we study the distribution of the level at time of
Markovian fluid queues and Markovian continuous time random walks, the maximum
(and minimum) level over , and their joint distributions. We
approximate by a random variable with Erlang distribution and we
use an alternative way, with respect to the usual Laplace transform approach,
to compute the distributions. We present probabilistic interpretation of the
equations and provide a numerical illustration
The Markov Additive risk process under an Erlangized dividend barrier strategy
In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (2011). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang() distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (2011), it is assumed that the event of ruin is monitored continuously (Avanzi et al. (2013) and Zhang (2014)), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.postprin
The finite/infinite horizon ruin problem with multi-threshold premiums: a Markov fluid queue approach
We present a new numerical method to obtain the finite- and infinite-horizon ruin probabilities for a general continuous-time risk problem. We assume the claim arrivals are modeled by the versatile Markovian arrival process, the claim sizes are PH-distributed, and the premium rate is allowed to depend on the instantaneous risk reserve in a piecewise-constant manner driven by a number of thresholds, i.e., multi-threshold premiums. We introduce a novel sample path technique by which the ruin problems are shown to reduce to the steady-state solution of a certain multi-regime Markov fluid queue. We propose to use the already existing numerically efficient and stable numerical algorithms for such Markov fluid queues. Numerical results are presented to validate the effectiveness of the proposed method regarding the computation of the finite- and infinite-horizon ruin probabilities for risk models including those with relatively large number of thresholds. © 2016, Springer Science+Business Media New York
Flow Level QoE of Video Streaming in Wireless Networks
The Quality of Experience (QoE) of streaming service is often degraded by
frequent playback interruptions. To mitigate the interruptions, the media
player prefetches streaming contents before starting playback, at a cost of
delay. We study the QoE of streaming from the perspective of flow dynamics.
First, a framework is developed for QoE when streaming users join the network
randomly and leave after downloading completion. We compute the distribution of
prefetching delay using partial differential equations (PDEs), and the
probability generating function of playout buffer starvations using ordinary
differential equations (ODEs) for CBR streaming. Second, we extend our
framework to characterize the throughput variation caused by opportunistic
scheduling at the base station, and the playback variation of VBR streaming.
Our study reveals that the flow dynamics is the fundamental reason of playback
starvation. The QoE of streaming service is dominated by the first moments such
as the average throughput of opportunistic scheduling and the mean playback
rate. While the variances of throughput and playback rate have very limited
impact on starvation behavior.Comment: 14 page
On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency
postprin
On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions
In this paper, we further extend the insurance risk model in Albrecher et al. (2011b), who proposed to only intervene in the compound Poisson risk process at the discrete time points where the event of ruin is checked and dividend decisions are made. In practice, an insurance company typically balances its books (and monitors its solvency) more frequently than deciding on dividend payments. This motivates us to propose a generalization in which ruin is monitored at whereas dividend decisions are only made at for some positive integer . Assuming that the intervals between the time points are Erlang() distributed, the Erlangization technique (e.g. Asmussen et al. (2002)) allows us to model the more realistic situation with the books balanced e.g. monthly and dividend decisions made e.g. quarterly or semi-annually. Under a dividend barrier strategy with the above randomized interventions, we derive the expected discounted dividends paid until ruin. Numerical examples about dividend maximization with respect to the barrier and/or the value of are given.postprin