本文選題：均值-方差準(zhǔn)則 + 時(shí)間一致性��； 參考：《蘭州大學(xué)》2014年博士論文

【摘要】：隨機(jī)最優(yōu)控制或動(dòng)態(tài)規(guī)劃方法是解決經(jīng)濟(jì)金融中許多動(dòng)態(tài)優(yōu)化問(wèn)題的有力工具.隨著經(jīng)濟(jì)金融理論的不斷豐富和發(fā)展,出現(xiàn)了許多的時(shí)間不一致性隨機(jī)控制問(wèn)題,所謂的時(shí)間不一致性是指不滿足Bellman最優(yōu)性原理,以至于動(dòng)態(tài)規(guī)劃方法也不能應(yīng)用.因此,研究時(shí)間不一致性隨機(jī)控制問(wèn)題,特別是研究它的時(shí)間一致性控制或策略就顯得十分必要.常見(jiàn)的時(shí)間不一致性隨機(jī)控制問(wèn)題有著名的動(dòng)態(tài)均值-方差模型,雙曲折現(xiàn)的最優(yōu)投資消費(fèi)問(wèn)題等.另外,許多的經(jīng)濟(jì)金融模型都需要考慮一些現(xiàn)實(shí)的限制,因此,帶約束的隨機(jī)優(yōu)化問(wèn)題也是一個(gè)非常重要和具有挑戰(zhàn)性的研究領(lǐng)域.本文主要考慮了兩個(gè)時(shí)間不一致隨機(jī)控制問(wèn)題和一個(gè)約束隨機(jī)優(yōu)化問(wèn)題.首先研究了時(shí)間一致性投資再保險(xiǎn)策略.其次討論了時(shí)間一致性分紅策略,最后討論了帶償付能力約束的分紅優(yōu)化問(wèn)題. 第一章首先介紹了時(shí)間不一致性隨機(jī)控制問(wèn)題及約束隨機(jī)優(yōu)化問(wèn)題產(chǎn)生的背景；其次介紹了本文的主要工作；最后,列出了一些解決時(shí)間不一致隨機(jī)控制問(wèn)題的預(yù)備知識(shí). 第二章考慮了風(fēng)險(xiǎn)厭惡依賴于狀態(tài)時(shí)的時(shí)間一致性投資再保險(xiǎn)策略.假設(shè)盈余過(guò)程為擴(kuò)散過(guò)程,保險(xiǎn)公司可以購(gòu)買比例再保險(xiǎn)并且在金融市場(chǎng)投資.金融市場(chǎng)由一個(gè)無(wú)風(fēng)險(xiǎn)資產(chǎn)和多種風(fēng)險(xiǎn)資產(chǎn)組成,其中風(fēng)險(xiǎn)資產(chǎn)的價(jià)格服從幾何布朗運(yùn)動(dòng).在此情形下,我們分別考慮兩個(gè)優(yōu)化問(wèn)題,其中一個(gè)是投資-再保險(xiǎn)問(wèn)題,另外一個(gè)是只有投資的情形.特別地,當(dāng)考慮風(fēng)險(xiǎn)厭惡動(dòng)態(tài)地依賴于當(dāng)前財(cái)富時(shí),模型更符合現(xiàn)實(shí).我們使用由Bjork和Murgoci(2010)所發(fā)展的方法,通過(guò)相應(yīng)的擴(kuò)展HJB方程導(dǎo)出了這兩個(gè)問(wèn)題的時(shí)間一致性策略.結(jié)果表明我們的時(shí)間一致性策略也依賴于當(dāng)前財(cái)富,這比當(dāng)風(fēng)險(xiǎn)厭惡為常數(shù)的情形更合理. 第三章研究了對(duì)偶模型下具有非指數(shù)折現(xiàn)函數(shù)的時(shí)間一致性分紅策略.一個(gè)公司要分紅給股東,折現(xiàn)函數(shù)是非指數(shù)的,并且公司的財(cái)富過(guò)程用一個(gè)對(duì)偶模型來(lái)描述.我們的目的是尋找一個(gè)分紅策略來(lái)最大化到公司破產(chǎn)為止支付給股東的紅利的期望折現(xiàn)值.非指數(shù)折現(xiàn)函數(shù)會(huì)導(dǎo)致這個(gè)問(wèn)題是時(shí)間不一致的.但是我們要尋找時(shí)間一致性策略,把我們的問(wèn)題看做一個(gè)非合作博弈,導(dǎo)出的均衡策略就是時(shí)間一致的.我們給出一個(gè)擴(kuò)展的Hamilton-Jacobi-Bellman方程系統(tǒng)和驗(yàn)證定理來(lái)導(dǎo)出均衡策略和均衡值函數(shù).對(duì)一種偽指數(shù)函數(shù)的情形,我們給出了均衡策略和均衡值函數(shù)的解析表達(dá)式.另外,給出了數(shù)值結(jié)果來(lái)例證我們的結(jié)果并且分析參數(shù)對(duì)結(jié)果的影響. 第四章研究了跳擴(kuò)散模型下帶償付能力約束的分紅優(yōu)化問(wèn)題.假設(shè)保險(xiǎn)公司的盈余遵從跳擴(kuò)散模型,考慮公司在償付能力約束或破產(chǎn)概率約束下的最優(yōu)分紅問(wèn)題.由已有的文獻(xiàn)知道,當(dāng)不考慮破產(chǎn)概率約束時(shí),跳擴(kuò)散模型下的最優(yōu)分紅策略為障礙分紅策略.附加了破產(chǎn)概率約束后,分紅優(yōu)化問(wèn)題變得比較復(fù)雜,甚至很難求解.我們利用隨機(jī)分析和偏微分積分方程的方法和技巧來(lái)考慮一個(gè)約束分紅優(yōu)化問(wèn)題,通過(guò)分析破產(chǎn)概率的一些性質(zhì),給出了分紅優(yōu)化問(wèn)題的最優(yōu)策略及最優(yōu)值函數(shù).
[Abstract]:Stochastic optimal control or dynamic programming is a powerful tool for solving many dynamic optimization problems in economy and finance. With the continuous enrichment and development of economic and financial theory, there are many problems of time inconsistency stochastic control. The so-called time inconsistency is that it is not satisfied with the Bellman optimality principle, so that the dynamic programming method is not satisfied. Therefore, it is necessary to study the time consistency stochastic control problem, especially the time consistency control or strategy. The common time inconsistencies stochastic control problem has the famous dynamic mean variance model, the hyperbolic discounted optimal investment consumption problem and so on. In addition, many economic and financial models. We need to consider some practical constraints, therefore, the stochastic optimization problem with constraints is also a very important and challenging field of research. This paper mainly considers two time inconsistent random control problems and a constrained stochastic optimization problem. First, the time induced reinsurance strategy is studied. Secondly, the time is discussed. Consistency dividend strategy. Finally, the dividend optimization problem with solvency constraints is discussed.
In the first chapter, the background of time inconsistency stochastic control problem and constrained stochastic optimization problem is introduced. Secondly, the main work of this paper is introduced. Finally, some preparatory knowledge to solve the problem of time inconsistent random control are listed.
In the second chapter, the time consistent investment reinsurance strategy is considered when the risk aversion depends on the state. It is assumed that the surplus process is a diffusion process, and the insurance company can buy proportional reinsurance and invest in the financial market. The financial market consists of a riskless asset and a variety of risky assets, in which the price of the risk assets obeys geometric Brown. In this case, we consider two optimization problems, one is the investment reinsurance problem, and the other is the only case of investment. In particular, when the risk aversion is dynamically dependent on the current wealth, the model is more realistic. We use the method developed by Bjork and Murgoci (2010), through the corresponding expansion of H The JB equation derives the time consistency strategy for these two problems. The results show that our time consistency strategy also depends on the current wealth, which is more reasonable than the case of risk aversion as a constant.
In the third chapter, we study the time consistency dividend strategy with non exponential discounted function under dual model. A company should pay dividends to shareholders, the discounted function is non exponential, and the company's wealth process is described with a dual model. The purpose is to find a dividend policy to maximize the company's bankruptcy and pay the shareholders. The non exponential discounted function will cause the problem to be inconsistent with time. But we have to look for the time consistency strategy and consider our problem as a non cooperative game. The derived equilibrium strategy is time consistent. We give an extended Hamilton-Jacobi-Bellman equation system and validation. The equilibrium strategy and the equilibrium value function are derived. For the case of a pseudo exponential function, we give the equilibrium strategy and the analytic expression of the equilibrium value function. In addition, the numerical results are given to illustrate our results and to analyze the effect of the parameters on the results.
The fourth chapter studies the problem of dividend optimization with the solvency constraint under the jump diffusion model. It is assumed that the surplus of the insurance company follows the jump diffusion model and considers the company's optimal bonus problem under the solvency constraint or the ruin probability constraint. The strategy is obstacle bonus strategy. When the ruin probability constraint is added, the problem of dividend optimization becomes more complex and difficult to solve. We use the method and technique of the stochastic analysis and partial differential integral equation to consider a constrained dividend optimization problem. By analyzing some properties of the ruin probability, the optimal problem of the dividend optimization is given. Strategy and optimal value function.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2014
【分類號(hào)】：F840.31;F224

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 ;Deterministic Time-inconsistent Optimal Control Problems - an Essentially Cooperative Approach[J];Acta Mathematicae Applicatae Sinica(English Series);2012年01期

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本文編號(hào)：2113894