若干風(fēng)險模型中期望折現(xiàn)罰金函數(shù)和最優(yōu)分紅的研究
發(fā)布時間：2018-10-13 12:07
【摘要】：風(fēng)險理論是精算學(xué)的重要組成部分.它研究的內(nèi)容主要有兩點：一是公司面臨的風(fēng)險,二是公司的收益.公司的風(fēng)險可以用一些精算量來刻畫,如破產(chǎn)概率、破產(chǎn)前盈余和破產(chǎn)時赤字等,而期望折現(xiàn)罰金函數(shù)將這幾個破產(chǎn)量統(tǒng)一起來,成為風(fēng)險理論中重要的精算量.除了風(fēng)險外,公司還關(guān)心其收益,衡量公司收益最具代表性的量是破產(chǎn)前分紅的總量,如何使公司的收益最大化已成為風(fēng)險理論研究的熱點問題.于是,本文致力于研究更新風(fēng)險模型中的期望折現(xiàn)罰金函數(shù)和對偶風(fēng)險模型中的最優(yōu)分紅問題,具體內(nèi)容如下. 首先,研究兩類更新風(fēng)險模型中的期望折現(xiàn)罰金函數(shù).一類是具有泊松隨機(jī)收益的更新模型,利用拉格朗日插值公式求出了期望折現(xiàn)罰金函數(shù)滿足的更新方程和有理分布索賠下的具體表達(dá)式.另一類是有投資和債務(wù)利率的更新模型,得到了期望折現(xiàn)罰金函數(shù)滿足的積分-微分方程,并用超幾何函數(shù)表示出了絕對破產(chǎn)概率. 其次,在復(fù)合泊松對偶風(fēng)險模型中,研究了帶有注資的最優(yōu)分紅問題,其中考慮了固定交易稅和比例交易稅的影響.這里主要基于兩種情況：一種是若破產(chǎn)發(fā)生,公司將受到一定的懲罰,即考慮破產(chǎn)懲罰；另一種是由于受到外界因素的影響,盈余過程可能被隨機(jī)的終止,即考慮隨機(jī)時間界.在這兩種情況下,我們都得到了破產(chǎn)前紅利與注資成本之差的折現(xiàn)期望的最大值和相應(yīng)的最優(yōu)分紅策略. 最后,在譜正Levy風(fēng)險模型中,研究了三個問題.根據(jù)分紅速率是否有限制,我們考慮兩種最優(yōu)分紅問題,在這兩個問題中研究了帶有注資的最優(yōu)分紅,其中也考慮了交易稅的影響.我們用Levy過程的尺度函數(shù)表示出了破產(chǎn)前紅利與注資成本之差的折現(xiàn)期望的最大值,并得到了相應(yīng)的最優(yōu)分紅策略.第三個問題研究了隨機(jī)離散時間的分紅,亦用Levy過程的尺度函數(shù)表示出了破產(chǎn)前紅利折現(xiàn)期望的最大值,并得到了相應(yīng)的最優(yōu)分紅策略.
[Abstract]:Risk theory is an important part of actuarial science. It has two main contents: the risk that the company faces, and the profit of the company. The risk of a company can be described by some actuarial quantities, such as the probability of bankruptcy, the surplus before bankruptcy and the deficit at the time of bankruptcy, and the expected discounted penalty function unifies these broken yields and becomes an important actuarial quantity in risk theory. Besides risks, companies also care about their earnings. The most representative amount of corporate earnings is the total amount of dividends before bankruptcy. How to maximize the profits of companies has become a hot issue in risk theory. Therefore, this paper is devoted to the study of the expected discounted penalty function in the updated risk model and the optimal dividend in the dual risk model, the details of which are as follows. First, we study the expected discounted penalty function in two kinds of updated risk models. One is the renewal model with Poisson's stochastic income. By using Lagrangian interpolation formula, the renewal equation of the expected discounted penalty function and the concrete expression under the rational distribution claim are obtained. The other is the renewal model with interest rate of investment and debt. The integro-differential equation of expected discounted penalty function is obtained, and the absolute ruin probability is expressed by hypergeometric function. Secondly, in the compound Poisson dual risk model, the optimal dividend problem with capital injection is studied, in which the effects of fixed transaction tax and proportional transaction tax are considered. There are mainly two kinds of cases: one is that if bankruptcy occurs, the company will be punished, that is, considering bankruptcy punishment; the other is due to the influence of external factors, the surplus process may be randomly terminated, that is, consider random time bound. In both cases, we obtain the maximum expected discounted expectation and the corresponding optimal dividend strategy for the difference between the pre-bankruptcy dividend and the capital injection cost. Finally, three problems are studied in the spectral positive Levy risk model. According to whether the rate of dividend is limited or not, we consider two kinds of optimal dividend problem. In these two problems, we study the optimal dividend with capital injection, and consider the effect of transaction tax. We use the scale function of the Levy process to express the maximum value of the discounted expectation of the difference between the dividend before bankruptcy and the cost of capital injection, and obtain the corresponding optimal dividend strategy. In the third problem, the dividend of random discrete time is studied, and the maximum expectation of dividend discounted before bankruptcy is expressed by using the scale function of Levy process, and the corresponding optimal dividend strategy is obtained.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：F840;O211.6
本文編號：2268561
[Abstract]:Risk theory is an important part of actuarial science. It has two main contents: the risk that the company faces, and the profit of the company. The risk of a company can be described by some actuarial quantities, such as the probability of bankruptcy, the surplus before bankruptcy and the deficit at the time of bankruptcy, and the expected discounted penalty function unifies these broken yields and becomes an important actuarial quantity in risk theory. Besides risks, companies also care about their earnings. The most representative amount of corporate earnings is the total amount of dividends before bankruptcy. How to maximize the profits of companies has become a hot issue in risk theory. Therefore, this paper is devoted to the study of the expected discounted penalty function in the updated risk model and the optimal dividend in the dual risk model, the details of which are as follows. First, we study the expected discounted penalty function in two kinds of updated risk models. One is the renewal model with Poisson's stochastic income. By using Lagrangian interpolation formula, the renewal equation of the expected discounted penalty function and the concrete expression under the rational distribution claim are obtained. The other is the renewal model with interest rate of investment and debt. The integro-differential equation of expected discounted penalty function is obtained, and the absolute ruin probability is expressed by hypergeometric function. Secondly, in the compound Poisson dual risk model, the optimal dividend problem with capital injection is studied, in which the effects of fixed transaction tax and proportional transaction tax are considered. There are mainly two kinds of cases: one is that if bankruptcy occurs, the company will be punished, that is, considering bankruptcy punishment; the other is due to the influence of external factors, the surplus process may be randomly terminated, that is, consider random time bound. In both cases, we obtain the maximum expected discounted expectation and the corresponding optimal dividend strategy for the difference between the pre-bankruptcy dividend and the capital injection cost. Finally, three problems are studied in the spectral positive Levy risk model. According to whether the rate of dividend is limited or not, we consider two kinds of optimal dividend problem. In these two problems, we study the optimal dividend with capital injection, and consider the effect of transaction tax. We use the scale function of the Levy process to express the maximum value of the discounted expectation of the difference between the dividend before bankruptcy and the cost of capital injection, and obtain the corresponding optimal dividend strategy. In the third problem, the dividend of random discrete time is studied, and the maximum expectation of dividend discounted before bankruptcy is expressed by using the scale function of Levy process, and the corresponding optimal dividend strategy is obtained.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：F840;O211.6
【參考文獻(xiàn)】
相關(guān)期刊論文 前2條
1 趙翔華,尹傳存;Sparre Andersen風(fēng)險模型的破產(chǎn)問題(英文)[J];應(yīng)用概率統(tǒng)計;2005年04期
2 ;Constant Barrier Strategies in a Two-state Markov-modulated Dual Risk Model[J];Acta Mathematicae Applicatae Sinica(English Series);2011年04期
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