HKU Scholars Hubhttp://hub.hku.hkThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 26 Feb 2021 07:50:13 GMT2021-02-26T07:50:13Z5091- On some Parisian problems in ruin theoryhttp://hdl.handle.net/10722/206448Title: On some Parisian problems in ruin theory
Authors: Wong, Tsun-yu, Jeff; 黃峻儒
Abstract: Traditionally, in the context of ruin theory, most judgements are made on an immediate sense. An example would be the determination of ruin, in which a business is declared broke right away when it attains a negative surplus. Another example would be the decision on dividend payment, in which a business pays dividends whenever the surplus level overshoots certain threshold. Such scheme of decision making is generally being criticized as unrealistic from a practical point of view. The Parisian concept is therefore invoked to handle this issue. This idea is deemed more realistic since it allows certain delay in the execution of decisions. In this thesis, such Parisian concept is utilized on two different aspects.
The first one is to incorporate this concept on defining ruin, leading to the introduction of Parisian ruin time. Under such a setting, a business is considered ruined only when the surplus level stays negative continuously for a prescribed length of time. The case for a fixed delay is considered. Both the renewal risk model and the dual renewal risk model are studied. Under a mild distributional assumption that either the inter arrival time or the claim size is exponentially distributed (while keeping the other arbitrary), the Laplace transform to the Parisian ruin time is derived. Numerical example is performed to confirm the reasonableness of the results. The methodology in obtaining the Laplace transform to the Parisian ruin time is also demonstrated to be useful in deriving the joint distribution to the number of negative surplus causing or without causing Parisian ruin.
The second contribution is to incorporate this concept on the decision for dividend payment. Specifically, a business only pays lump-sum dividends when the surplus level stays above certain threshold continuously for a prescribed length of time. The case for a fixed and an Erlang(n) delay are considered. The dual compound Poisson risk model is studied. Laplace transform to the ordinary ruin time is derived. Numerical examples are performed to illustrate the results.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10722/2064482014-01-01T00:00:00Z
- On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumpshttp://hdl.handle.net/10722/229476Title: On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps
Authors: Wong, JTY; Cheung, ECK
Abstract: This paper studies the Parisian ruin problem first proposed by Dassios and Wu (2008a,b), where the Parisian ruin time is defined to be the first time when the surplus process has stayed below zero continuously for a pre-specified time length $d$. Both the insurance risk process and the dual model will be considered under exponential distributional assumption on the jump sizes while keeping the inter-arrival times arbitrary. In these two models, the Laplace transform of the Parisian ruin time is derived by extending the excursion techniques in Dassios and Wu (2008a) and taking advantage of the memoryless property exponential distributions. Our results are represented in integral forms, which are expressed in terms of the (joint) densities of various ruin-related quantities that are available in the literature or obtainable using the Lagrange's expansion theorem. As a by-product, we also provide the joint distribution of the numbers of periods of negative surplus that are of duration more than $d$ and less than $d$, which can be obtained using some of our intermediate results. The case where the Parisian delay period $d$ is replaced by a random time is also discussed, and it is applied to find the Laplace transform of the occupation time when the surplus is negative. Numerical illustrations concerning an Erlang(2) insurance risk model are given at the end.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10722/2294762015-01-01T00:00:00Z
- On the dual risk model with Parisian implementation delays in dividend paymentshttp://hdl.handle.net/10722/277658Title: On the dual risk model with Parisian implementation delays in dividend payments
Authors: Cheung, Eric C.K.; Wong, Jeff T.Y.
Abstract: © 2016 Elsevier B.V. In this paper, we study the dual compound Poisson risk process, which is suitable for a business that pays expenses at a constant rate over time and earns random amount of income at random times. In contrast to the usual dividend barrier strategy (e.g. Avanzi, Gerber, and Shiu (2007)) in which any overshoot over a pre-specified barrier is paid immediately to the company's shareholders as a dividend, it is assumed that dividend is payable only when the process has stayed above the barrier continuously for a certain amount of time d (known as the ‘Parisian implementation delay’ in Dassios and Wu (2009)). Under such a modification, the Laplace transform of the time of ruin and the expected discounted dividends paid until ruin are derived. Motivated by the ‘Erlangization’ technique (e.g. Asmussen, Avram, and Usabel (2002)) of approximating a fixed time using an Erlang distribution, we also analyze the case where the delay d is replaced by an Erlang random variable. Numerical illustrations are given to study the effect of Parisian implementation delays on ruin-related quantities and to demonstrate the good performance of Erlangization. Interestingly, unlike the traditional barrier strategy, it is found that the optimal dividend barrier maximizing the expected discounted dividends does depend on the initial surplus level.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2776582017-01-01T00:00:00Z
- Poissonian potential measures for Lévy risk modelshttp://hdl.handle.net/10722/277698Title: Poissonian potential measures for Lévy risk models
Authors: Landriault, David; Li, Bin; Wong, Jeff T.Y.; Xu, Di
Abstract: © 2018 Elsevier B.V. This paper studies the potential (or resolvent) measures of spectrally negative Lévy processes killed on exiting (bounded or unbounded) intervals, when the underlying process is observed at the arrival epochs of an independent Poisson process. Explicit representations of these so-called Poissonian potential measures are established in terms of newly defined Poissonian scale functions. Moreover, Poissonian exit measures are explicitly solved by finding a direct relation with Poissonian potential measures. Our results generalize Albrecher et al. (2016) in which Poissonian exit identities are solved. As an application of Poissonian potential measures, we extend the Gerber–Shiu analysis in Baurdoux et al. (2016) to a (more general) Parisian risk model subject to Poissonian observations.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2776982018-01-01T00:00:00Z
- A temporal approach to the Parisian risk modelhttp://hdl.handle.net/10722/277688Title: A temporal approach to the Parisian risk model
Authors: Li, Bin; Willmot, Gordon E.; Wong, Jeff T.Y.
Abstract: Copyright © Applied Probability Trust 2018. In this paper we propose a new approach to study the Parisian ruin problem for spectrally negative Lévy processes. Since our approach is based on a hybrid observation scheme switching between discrete and continuous observations, we call it a temporal approach as opposed to the spatial approximation approach in the literature. Our approach leads to a unified proof for the underlying processes with bounded or unbounded variation paths, and our result generalizes Loeffen et al. (2013).
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2776882018-01-01T00:00:00Z
- On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumpshttp://hdl.handle.net/10722/287718Title: On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps
Authors: Wong, TYJ; Cheung, ECK
Abstract: This paper studies the Parisian ruin problem first proposed by Dassios and Wu (2008a,b), where the Parisian ruin time is defined to be the first time when the surplus process has stayed below zero continuously for a pre-specified time length . Both the insurance risk process and the dual model will be considered under exponential distributional assumption on the jump sizes while keeping the inter-arrival times arbitrary. In these two models, the Laplace transform of the Parisian ruin time is derived by extending the excursion techniques in Dassios and Wu (2008a) and taking advantage of the memoryless property of exponential distributions. Our results are represented in integral forms, which are expressed in terms of the (joint) densities of various ruin-related quantities that are available in the literature or obtainable using the Lagrange’s expansion theorem. As a by-product, we also provide the joint distribution of the numbers of periods of negative surplus that are of duration more than and less than , which can be obtained using some of our intermediate results. The case where the Parisian delay period is replaced by a random time is also discussed, and it is applied to find the Laplace transform of the occupation time when the surplus is negative. Numerical illustrations concerning an Erlang(2) insurance risk model are given at the end.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10722/2877182020-01-01T00:00:00Z
- A temporal approach to the Parisian risk modelhttp://hdl.handle.net/10722/287719Title: A temporal approach to the Parisian risk model
Authors: Bin, L; Gordon, EW; Wong, TYJ
Abstract: In this paper we propose a new approach to study the Parisian ruin problem for spectrally negative Lévy processes. Since our approach is based on a hybrid observation scheme switching between discrete and continuous observations, we call it a temporal approach as opposed to the spatial approximation approach in the literature. Our approach leads to a unified proof for the underlying processes with bounded or unbounded variation paths, and our result generalizes Loeffen et al. (2013).
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10722/2877192020-01-01T00:00:00Z
- On the dual risk model with Parisian implementation delays in dividend paymentshttp://hdl.handle.net/10722/288166Title: On the dual risk model with Parisian implementation delays in dividend payments
Authors: Cheung, ECK; Wong, TYJ
Abstract: In this paper, we study the dual compound Poisson risk process, which is suitable for a business that pays expenses at a constant rate over time and earns random amount of income at random times. In contrast to the usual dividend barrier strategy (e.g. Avanzi, Gerber, and Shiu (2007)) in which any overshoot over a pre-specified barrier is paid immediately to the company’s shareholders as a dividend, it is assumed that dividend is payable only when the process has stayed above the barrier continuously for a certain amount of time d (known as the ‘Parisian implementation delay’ in Dassios and Wu (2009)). Under such a modification, the Laplace transform of the time of ruin and the expected discounted dividends paid until ruin are derived. Motivated by the ‘Erlangization’ technique (e.g. Asmussen, Avram, and Usabel (2002)) of approximating a fixed time using an Erlang distribution, we also analyze the case where the delay d is replaced by an Erlang random variable. Numerical illustrations are given to study the effect of Parisian implementation delays on ruin-related quantities and to demonstrate the good performance of Erlangization. Interestingly, unlike the traditional barrier strategy, it is found that the optimal dividend barrier maximizing the expected discounted dividends does depend on the initial surplus level.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10722/2881662020-01-01T00:00:00Z
- Poissonian potential measures for Lévy risk modelshttp://hdl.handle.net/10722/288167Title: Poissonian potential measures for Lévy risk models
Authors: David, L; Bin, L; Wong, TYJ; Di, X
Abstract: This paper studies the potential (or resolvent) measures of spectrally negative Lévy processes killed on exiting (bounded or unbounded) intervals, when the underlying process is observed at the arrival epochs of an independent Poisson process. Explicit representations of these so-called Poissonian potential measures are established in terms of newly defined Poissonian scale functions. Moreover, Poissonian exit measures are explicitly solved by finding a direct relation with Poissonian potential measures. Our results generalize Albrecher et al. (2016) in which Poissonian exit identities are solved. As an application of Poissonian potential measures, we extend the Gerber–Shiu analysis in Baurdoux et al. (2016) to a (more general) Parisian risk model subject to Poissonian observations.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10722/2881672020-01-01T00:00:00Z