随机控制理论在金融和保险中的应用
摘要
保险数学是源自保险业的风险管理而产生的应用数学,而风险理论则是保险数学中最具理论性的重要组成部分。它主要是研究保险公司所关心的几个精算量例如破产概率,破产时,破产前余额,破产赤字等。通过利用随机过程,随机分析的理论和方法,尤其是Gerber,H.U等人将鞅的理论和方法应用到风险理论中,使得该学科得到了迅速的发展,在刻画上面所提到的几个精算量方面取得了丰硕的成果。
随着金融和保险市场发展,对于原有几个精算量的刻画已经不能再满足保险公司的需求。例如对于破产概率,破产赤字,相对于他们的具体表达式,保险公司更关心如何才能使得这些代表风险的量达到最小。为了使他们尽可能的小,保险公司会采取一些相应的措施例如再保险,投资金融市场。这时保险公司所面临的问题就是如何寻找最优的再保险或投资策略使得风险达到最小。保险公司除了关心代表风险的量以外也关心一些代表它的收益和效用的量例如破产前总的分红量和某时刻财富的效用。想尽可能的使得这些量达到最大。所有这些都属于金融保险中的随机最优控制问题。在过去几十年里,通过利用随机控制的理论和方法,尤其是[24]和[3]把随机控制理论应用到风险中,通过Hamilton-Jacobi-Bellman(HJB)方程的方法分别得到了扩散模型下的最优投资(目标是最小破产概率和最大化指数效用)和最优分红策略,使得该领域得到了迅速的发展,并开创了风险理论和随机控制理论相结合的先例。之后有许多文献相继利用HJB方程的方法进行对风险里的最优问题的研究。其中有些也加入了再保险。并取得了很好的结果。
但在许多工作里为了得到明确的最优解,所建立的模型与实际还存在着很大的差距。例如假设分红时没有交易费用和公司的偿付能力的限制;仅仅考虑对于单个风险资产的投资;考虑最优再保险时没有考虑到再保险人的利益;忽略了索赔过程的长程相依性等等。而在实际中这些因素是存在的且不容忽视的。而且这些因素的加入会对以前所得的最优策略产生很大的影响,有的甚至已不再是最优的。因此有必要重新考虑所对应的最优问题和寻找新的最优策略。
以往文献考虑的问题和模型与实际存在差距,其主要一个原因是方法上过于局限于HJB方程。这使得考虑的模型只能限于马尔科夫过程;考虑的问题只能限于(1.1.4)和(1.1.7)的形式;构造的解也必须满足验证定理所要求的条件。而上面提到的大多数问题不再满足这些条件例如分数布朗运动模型下的最优问题,模型不再具有马尔科夫性质;再保险中的博弈问题,不再满足三个限制条件中的任何一个。因此,为了解决这些问题,需要我们寻找新的方法或者在方法上有新的突破。
鉴于上述原因,我的博士毕业论文将致力于下面三个方面。首先是建立与实际更贴近的模型和问题。其次是不局限于HJB方程的方法,根据当前的模型和问题所特有的性质灵活变通,充分发挥各种随机控制理论方法的作用,努力寻找解决问题的路径。最后,为了使最终的结果对实践能起到一个很好的指导作用,将尽可能的对最优问题给出明确解,使得所得最优策略具有可操作性。下面将详细介绍各个章节的内容。
首先在第一章中给出了下面章节中将要涉及到的随机最优控制理论。这些理论主要来自[38]和[89]。
在第二章中,一系列的带交易费用的最优分红问题被考虑。
分红是指公司将部分盈余分给股东或初始准备金的提供者。所以总的分红量从某种意义上反应了一个公司的效益和实力。因此如何选择一个分红策略或采取某种措施(例如再保险和投资)使得破产之前的分红量达到最大一直以来都是金融和保险领域中最热门的研究话题之一。对于这种古典的最优分红问题已经解决的比较完善。包括含有再保险控制的都已经给出了非常明确的结果。结果展示了带漂移布朗运动和复合泊松风险模型下的最优分红策略分别是上限为常值的边界分红策略(barrier strategy)和波段分红策略(band strategy)。但是结果的得到都假设了当进行分红时不需要交一定量的交易费用。而实际中为了避免连续交易。当进行分红时都要求交一部分交易费用。即使它可能很少。但它的出现从根本上会影响到最优分红策略的形式。显然在这种情况下barrierstrategy和band strategy由于都是连续的进行交易,会带来无穷大的交易费用,所以都不再是最优的分红策略。因此需要我们重新选择最优分红策略。由于交易费用的出现,我们对最优分红策略的选择不仅仅要考虑对分红量的选择还要考虑对分红时间的选择,这使得这时的分红问题要比古典分红问题复杂的多。在风险理论里,仅仅[28]给出了带漂移布朗运动风险模型下的最优比例再保险和分红策略。还有很多关于带交易费用的最优分红问题没有解决,例如一直被认为比比例再保险更优(既相对应的最大分红量更大)的超额损失再保险控制下的最优分红问题,复合泊松风险模型下的最优分红问题,带有偿付能力限制和交易费用的最优分红问题。我们将分别在2.1,2.2和2.3节展开对这些问题的讨论。在这一章的最后一节我们放宽了以前文献中所作的限制性的假设,考虑一类更广义的扩散模型下带交易费用的最优分红问题,展现了一个与以前不一样的最优解。
在2.1节我们研究了带交易费用和税收的最优超额损失再保险和分红问题。首先证明了超额损失再保险是比比例再保险好的即所对应的最大的期望折现分红量要大。然后通过解拟变分不等式,给出了最优超额损失再保险和分红策略的明确表达式。超额损失再保险与比例再保险相比不再有比例这么好的形式。[4]曾经考虑过超额损失再保险下的不带交易费用的最优分红问题。我们与[4]的区别首先是对于拟变分不等式解的构造比[4]里面的HJB方程的解的构造要困难。其次是在[4]里面对于HJB方程解的构造依赖于一个辅助函数。我们这里对于拟变分不等式解的构造,没有引入辅助函数而且方法更简单。
2.2节考虑了复合泊松模型下的带交易费用和税收的最优分红问题。复合泊松模型下的最优分红问题一直以来都是一个难点。主要原因是在复合泊松情景下所对应的最优方程不再有边界条件并且有可能不再有连续可微的解。在这一节中我们构造出了当索赔是指数情景下的拟变分不等式的解进而给出了最优策略。而且也给出了分红时间间隔的期望值。结果展现了当分红的税率减少时,应该相应的增加分红的次数同时减少每次分红的量。
在2.3节,假设公司的余额被一个广义的扩散过程所描述。公司的目标是最大化期望折现分红。每次分红仍旧有交易费用和税收被要求。在[93]展现了在一些合理的假设下最优策略是块状边界策略,即有两个边界,当盈余达到上边界时,进行分红,盈余减少到下边界。但是,从偿付能力的角度来看,这些最优边界有可能由于过低而是保险公司所不能接受的。因此我们应该找一个满足偿付能力限制的块状边界策略。类似于在[92]所提出的对于偿付能力的限制,分红策略应该满足在该策略下所对应的有限时间的破产概率不超过一个给定的值。
这里我们首先假定一个块状边界的限制,这个限制的意思是事先给出两个边界,公司仅仅当余额到达给定的上边界时才可以进行分红,而且分红后的盈余不能低于给定的下边界。对于在这个限制下的最优分红问题,我们分为两步进行考虑,首先考虑仅仅有下边界限制的最优分红问题,然后在第一步的基础上考虑有上边界限制的最优分红问题。这两步都借助了相应的拟变分不等式。但在第二步中,拟变分不等式的解不再是连续可微的,这使得It(?)公式不能再被利用,进而验证定理不再成立。我们这里巧妙的利用局部时理论证明了拟变分不等式的解就是最优值函数同时构造了相应的最优策略。首次给出了对于不满足一次连续可微函数的验证定理。
剩下的任务是如何找到最优的两个边界限制使得他们在满足偿付能力限制的情况下对应的最优值函数最大。最优双边界的寻找完全不同于[92]里面对于最优单个边界的寻找。它是极其复杂的。需要偏微分方程方面更高层的工具。我们在2.3节里展现了如何利用Thomas和Crank-Nicolson算法来解决相应的偏微分方程以得到最优双边界限制。在这节的最后,我们还给出了数值解。
在2.4节所考虑的公司的盈余过程是一类更广义的扩散过程。更广义的意思是放宽了以前文献中所作的限制性的假设。这里同样利用的也是拟变分不等式,但是由于没有了一些好的假设条件导致了拟变分不等式的解的性质和形式与以前有很大的不同。这里我们利用一种新的思想把拟变分不等式的解根据他们的性质进行了分类。证明了最优分红策略有三种可能性:(1)块状边界分红策略;(2)块状波段分红策略;(3)最优分红策略不存在。其中最困难的部分是对于块状波段分红策略是最优分红策略的证明。因为在这种情况下,以前所有构造最优值函数和最优策略的方法已经不再有效。为了克服这个困难,我们不再仅仅从拟变分不等式的一个解考虑而是从两个独立解入手。对最优值函数进行了四个区域的构造。并且首次清晰的展现了最优策略是一个块状波段分红策略并给出了它的具体形式的表达。在以前的文献里,当证明最优策略是波段分红策略时,多数是从粘性解出发,因此里面所涉及的边界都没有一个明确的表达。
第三章讨论了最小破产概率和最大化指数效用问题。
在最近几年里,最小化破产概率和最大化指数效用已经被很多研究者考虑过。对于他们之间的关系早在1965年Ferguson在[37]里就给出了猜测。他猜测当效用函数3.1.5中的参数m取适当的值时,两个最优问题所对应的最优策略是一样的。其中假设没有无风险投资。[24]考虑风险盈余过程是一个带漂移的布朗运动和允许保险公司把盈余投资于风险资产和无风险资产。三个最优准则:最大化指数效用,最小化破产概率和最小化折现罚金分别被考虑。给出最优投资的同时验证了Ferguson's的猜测在当前模型下是成立的。另外也展现了当有无风险投资时,猜测是不成立的。即无论m取何值,最大化指数效用和最小化破产概率两个最优问题的最优策略都不同。
再保险是保险公司用来控制他们所面临的风险或提高效用的一个重要的手段。因此我们在3.1节,通过加入再保险控制延拓了Browne的工作。即风险盈余过程是一个带漂移的布朗运动。保险公司被允许把盈余投资于风险资产和无风险资产,并且购买(比例)再保险。在投资和再保险这两种控制下,我们考虑了三个最优准则:最大化指数效用,最小化破产概率和最小化折现罚金。由于比例再保险控制中的比例是在[0,1]之间的。既控制变量是有限制的。因此对于HJB方程解的构造要比[24]复杂。尤其是最后一个准侧。即使这样我们仍旧给出了HJB的解析解,并通过它给出了最优值函数和最优策略。特别的,当没有无风险投资时,相应的结果展现了最大化指数效用和最小化破产概率他们的最优策略是一样的。这就验证了在[37]里面给出的猜测在加入再保险后仍旧成立。同时也展现了当有无风险投资时,猜测不再成立。
对于最优投资问题,之前一直考虑的是保险公司投资单个风险资产。而实际中为了达到预期的目标,保险公司一般是投资到多个风险资产。因此我们在3.2节考虑了投资多个风险资产的最优问题。考虑的风险模型和3.2节一样。同时允许保险公司投资于多个风险资产和购买(比例)再保险。在非卖空的限制下,最大化指数效用和最小化破产概率两个准测被考虑。当投资于单个风险资产时,在没有非卖空限制下得到的最优投资额很自然的满足非卖空的限制。但对于多个风险资产,情况就不一样了。在没有非卖空限制下得到的最优投资策略不一定满足非卖空的限制。因此需要我们对HJB方程进行一定不同方法的处理。在这节中,我们首先处理非卖空带来的影响把HJB转化成一般的可解决方程的形式。然后给出了它的解进而得到了最优值函数和最优策略。同时也验证了Ferguson的猜测在没有无风险投资下仍成立同时当有无风险投资时猜测不成立
3.3节考虑了超额损失再保险和多个风险资产投资两种控制。最优准则与3.2节相同。首先证明了超额损失再保险比比例再保险更优,即超额损失再保险所对应的最小破产概率比比例再保险所对应的小和最大指数效用比比例再保险的大。然后通过HJB方程给出了最优解。同时也验证了Ferguson的猜测在超额损失再保险的情景下仍成立。
第四章对保险里的均方差问题进行了讨论。
投资组合选择简而言之就是把财富分配到不同的资产中,以达到分散风险、确保收益的目的。1952年,Markowitz用方差来量化股票收益的风险,提出了投资组合选择的均值-方差分析方法,揭开了现代金融学研究的序幕。均方差问题是指投资者想找到一个最优的投资策略使得它的期望达到最大同时方差最小。它是一个以投资组合的期望收益(均值)和风险(方差)为目标的双目标决策模型,因此很自然地导出了投资组合选择的均值-方差有效组合、有效前沿等概念。均值-方差投资组合理论不仅是现代投资组合选择理论的先驱工作,也是现代金融学的基石之一。其精髓在于首先对风险进行量化分析,开辟了风险管理的新思路。
最近几年,金融和保险市场已经开始慢慢的结合,保险公司为了增加它的收益或减少它的风险会把部分盈余投资于金融市场。已经有很多文献涉及到了保险理论中的最优投资问题。但大多数文献考虑的最优准则仍旧仅仅限于原有的几个经典精算量例如分红量,破产概率,破产赤字等。均方差准则作为金融学的最重要和最流行的概念之一反而在保险理论中很少被涉及。据我们所知仅仅[120]考虑了均方差准则下的最优投资问题和[27]考虑了均方差准则下的最优再保险/新业务问题。因此,我们在这一章中对保险理论中的均方差问题进行了更深入和广泛的研究。
在4.1节,我们对再保险/新业务和投资两种控制下的均方差问题进行了研究。其中考虑了两种风险模型:复合泊松和带飘移的布朗运动。问题的解决不同于[27],相应的HJB方程不再有古典的解,另外即使给出粘性解,据我们所知仅仅有关于扩散模型下的粘性解验证定理。对于它在复合泊松模型下的有效性还没有得到认证。为了克服这些困难,首先通过两个特殊的Riccati方程构造一个连续可微函数并展现了它是HJB方程的粘性解。对于这个粘性解我们给出了不同于以往的验证定理。而且可以看到它能被应用于一类粘性解。最后通过比较两个风险模型下的结果,发现他们的最优策略是一样的。这个可以启发我们在考虑一些问题时可以用扩散近似来代替复合泊松。这样就可以简化问题。
迄今,人们一直用具有马尔科夫性的随机过程来描述索赔过程。但在大部分情形下,保险公司的索赔过程呈现出长程相依性:给定时刻t后过程的行为,不仅依赖于t时刻的信息,而且还依赖于时刻t以前的历史。因此,最近已经开始有用分数布朗运动来模拟保险公司的索赔过程。但所有文献都是考虑分数布朗运动风险模型的破产问题。由于分数布朗运动很多性质比较难刻画,因此对于它的破产概率一直没有得到明确的结果。对于它的优化问题更是很少有人考虑。这不仅是因为分数布朗运动本身比较难研究,更主要的是它不再具有马氏性,传统的HJB方程方法已经不能再利用。据我们所知,仅仅[63]考虑过特殊的线性模型下的二次规划问题,而且最优策略被限制在马氏策略(即策略在每时刻的取值依赖于当时过程的值)集合里考虑。但我们知道分数布朗运动本身不再具有马氏性,这就意味着真正的最优策略不会再是马氏策略。因此对[63]工作作更进一步的推广和改进,是很有必要的。在4.2节,我们用一个受分数布朗运动干扰的古典风险过程描述了公司的盈余过程。研究了均方差准则所对应的最优投资问题。这里分数布朗运动的Hurst参数H∈(1/2,1)。虽然分数布朗运动不再是Markov过程,不能再利用HJB方程方法。但由于[36]利用Wick乘积定义的关于分数布朗运动的随机积分使得分数布朗运动有了跟标准布朗运动许多类似的性质例如Zero mean,Girsauov Theorem等。同时受到下面5.1节对完全平方方法使用范围的探索结果启发。我们发现分数布朗运动的性质是符合完全平方方法的要求。因此通过利用完全平方的方法我们找到了有效策略和有效前沿。并且当H→1/2+结果被验证收敛到已知的布朗运动情景下的结果。
在第五章,我们研究了受(分数)布朗运动干扰的古典风险模型下的最优问题。
在5.1节,风险过程是一个受布朗运动干扰的复合泊松过程。控制是保险公司向顾客收取的保费。目标是最小化盈余过程与给定轨道的距离与保费折现值的和。我们通过动态规划,完全平房和随机最大值原则三种不同的方法给出了最优控制和最优值函数。这一节的内容,不仅结果对实践有一定的指导作用,更重要的是在理论和方法上的作用。在以前的文献里,基本都是局限于HJB方程的方法。而HJB方程有它的局限性既仅仅使用于马尔科夫过程。因此在考虑模型和问题上都受到了限制。在这节中我们把完全平方和随机最大值原则引入到了保险领域中。他们既跟HJB方法有联系但也有很多的区别,尤其在处理的模型已经不再仅仅限于马尔科夫过程。因此对于我们以后考虑非马尔科夫盈余过程的最优问题有很大的借鉴性。5.2节就是一个很好的例子。我们在这一节研究了受分数布朗运动干扰的古典风险模型下的最优问题。最优准则与5.1节相同。最优策略和最优值函数通过完全平方的方法被清晰的给出。另外4.2节也是受到了5.1节所提供方法的启发。
在第六章研究了一些其他的最优控制问题。
对于再保险已经有很多文献涉及到,例如[24],[58],[4]等。但在所有这些文献里都没有考虑到再保险公司的效用。而实际中,如果只考虑保险公司一方的话,所求的最优再保险策略往往是再保险公司所不能接受的。这就警惕我们在再保险协议里有两方,并且他们的利益是冲突的。因此最优再保险的合约必须显示为一个合理的双方利益的折衷。这其实就是一个双方博弈的问题。用博弈的思想来解决保险定价或风险分配方面的问题,最初是由[21]提出来的,考虑的最优准则是最大化效用。之后,[1],[117]等在风险交换和再保险市场方面也作出了一定的工作。但在所有这些工作中仅仅考虑单期索赔即静态的最优再保险的选择。而实际中,投保人与保险公司以及保险公司和再保险公司在很多情况下都是长期的合作而不是仅仅是针对一次索赔的合作。因此我们在6.1节研究了连续时间模型下的再保险市场上的最大化指数效用问题。连续时间模型下的再保险问题不同于单期的情景,保险公司和再保险公司它们在整个时间段内都是一个博弈的关系。据我们所知,即使在专门的博弈理论中也从没涉及到连续时间模型的例子。因此在借鉴单期的理论和方法的同时,必须在方法上有其他新的突破,才能解决连续情景下的最优再保险问题。为了克服这些困难,我们把问题分成了两步进行处理。首先处理终端财富之间的Pareto最优问题,然后寻找再保险策略去复制Pareto最优终端财富。通过这种方法我们找到了Pareto最优策略。而且结果展现了连续时间情景下的Pareto最优合作再保险是比例再保险。这个结果恰恰与以前的结论相反既当仅仅保险公司效用被考虑时超额损失再保险一直是最优的。最后我们证明了合作再保险的核是非空的。这里核是博弈论非常重要的量,它是所有能被博弈里面所有成员所接受的Pareto最优策略的集合。这一节我们的主要目的是提供一个解决连续时间再保险市场上的Pareto最优问题的方法。其他的很多问题例如分红,破产概率,均方差等都有待研究。
6.2节是关于一个局部信息下的最优控制问题。股票的价格满足一个随机微分方程,其中瞬时的返回率是一个Ornstein-Uhlenbeck过程。这里仅仅股票的价格和利率能被观察。通过利用过滤和动态规划理论,我们给出了指数和对数效用下的最优解。其中对数效用是在不允许投资者进行卖空和借款的限制下考虑的。与[75]相比,我们这里不允许投资者卖空和借款,而且这里所用的HJB方程的方法要比[75]里面的鞅方法简单很多。而且使用HJB方程的方法可以给出所对应值函数的明确解。在这节里考虑的模型是简单的,我们主要的目的一是利用HJB方程的方法解决局部信息问题,它相对于以前大多数文献中的鞅方法要简单很多,二是希望能为考虑保险理论中的局部信息问题提供一些帮助。
最后一节考虑的准则是最小化破产之前到达一个给定目标的期望值。对于一般的投资者,[95]通过最大化波动系数平方除飘移系数得到了最优投资策略。[95]里面的控制变量不受限制和最优策略被展现是比例策略。在这一节中,卖空被限制和比例再保险的比例是在[0,1]之间。这些对控制变量的限制使得[95]里面的方法不能再被利用。我们通过HJB方程的方法给出了最小期望时间和最优策略。而且结果展现了最优投资策略已不再是比例策略。
我的博士毕业论文主要是通过HJB方程,拟变分不等式,偏微分方程,完全平方,Pareto最优,分数布朗运动等理论解决了风险理论中带交易费用和偿付能力限制的最优分红问题,最优多个风险资产,Pareto最优再保险以及在分数布朗运动风险模型下的最优问题。上面以他们所研究的问题对其进行了以章节为单位的分类介绍。下面将从各个章节的创新点和特色之处对他们的内容进行概括性的总结。
1.方法的创新。
(1)2.3节有两个方法上的创新点:(a)关于非连续可微解的验证定理;(b)利用Thomas和Crank-Nicolson算法给出了最优双边界。
(2)2.4节在方法上的两个创新点是:(a)提出了一种新的把拟变分不等式的解进行分类的方法;(b)给出了一种新的构造最优值函数和最优策略的方法。
(3)4.1节给出了一类粘性解的验证定理。
(4)6.1节首次给出了解决连续时间Pareto最优再保险的方法。
(5)6.2节利用HJB方程的方法解决局部信息下的最优问题。以前几乎所有文献都是利用的鞅方法。这里我们也充分展现了HJB方程与鞅方法相比的优点。
2.解方程和构造最优解技巧的创新。
这部分主要是对已有经典工作或优秀工作的更深入的研究和推广。但并不是简单的在同一方法上的计算推广。
(1)2.1节是对文章[28](发表在国际顶级金融杂志Mathematical Finance)的工作一个改进推广。研究了带交易费用和税收的最优超额损失再保险和分红问题。首先证明了超额损失再保险比[28]中的比例再保险更优。这也是我们考虑超额损失再保险的一个原因。超额损失再保险所对应的拟变分不等式比比例再保险复杂的多。在2.1节我们给出了不同的构造解的方法。
(2)2.2节是对于古典风险模型的一个挑战。古典风险模型一直以来都是分红问题的难点。这里对于拟变分不等式的解的构造与2.1节和[28]有很大不同。从2.2节我们可以看到,古典风险模型下拟变分不等式的解的形式更多样化。
(3)[24]的工作被称为保险理论和控制理论相结合的先例。在第三章,我们延拓了Browne的工作考虑了最优再保险和投资问题。比例再保险的介入使得对于相应的HJB方程的解的构造需要更高的技巧性。
(4)6.3节考虑了最小化破产之前到达一个给定目标的期望值。这是对于经典文章[95]工作的一个推广。由于卖空被限制和比例再保险的比例是在[0,1]之间。使得[95]里面的方法不能再被利用。我们通过HJB方程的方法解给出了最优解。这里对于HJB方程解的构造也是极其复杂的。
3.引入新的控制方法进入保险领域。
5.1节利用了HJB方程,随机最大值原则和完全平房三种不同的方法解决最优问题。从中可以看到各种方法的优缺点和他们不同的使用范围。这样可以开拓我们的视眼,考虑不同类型的风险模型和问题。
4.研究非马尔科夫风险余额过程的最优控制问题。
分数布朗运动是具有长程相依性的非马尔科夫过程。虽然它可以用来描述现实中更贴近实际的现象。但对于它的研究一直以来都没有实质性的进展。这主要是因为它的性质比较难刻划。风险中只有很少的文献涉及它的破产概率。对于关于分数布朗运动风险模型下的最优控制问题。更是寥寥无几。我们在4.2和5.2。节研究了受分数布朗运动干扰的复合泊松模型下的最优控制问题。找到了适合于分数布朗运动的最优控制方法。并且给出了最优策略的明确解。
本论文另外一个非常重要的特色就是对于所有的最优问题都给出了非常明确的最优解。
Risk theory is the most theoretical part in insurance mathematics which derives from risk management in insurance.It is mainly about several actuarial variables such as ruin probability,the surplus immediately prior to ruin,the deficit at ruin etc.The applications of stochastic process and stochastic analysis,especially the martingale method,in risk theory make risk theory develop rapidly.Many fruitful achievements in characterizing several actuarial variables mentioned above have been made.As the development of finance and insurance markets,these actuarial variables have no longer met the needs of the insurance company.The insurance company is more and more concerned about how to minimize these actuarial variables which represent the risk instead of their expression.In order to minimize them,the insurance company will take the appropriate measures such as reinsurance,investment in finance market.At this time the problem faced by insurance company is how to find the optimal reinsurance or/and investment strategy to minimize the risk.In addition,to the ruin probability,the insurance company also uses the other measures to measure the risk such as the variance of terminal wealth,VaR(Value-at-Risk), etc.At the same time,the insurance company is also concerned about some other variables which represent their utilities and profits such as the utility of the terminal wealth and the expected discounted dividend payments before ruin.All of these belong to stochastic control problems.In the last few decades,in order to solve these problems,stochastic control theory-has been applied into the insurance.For example,[24]and[3]respectively obtained optimal investment strategy and the optimal dividend strategy under diffusion model by the Hamilton-Jacobi-Bellman(HJB) equation approach.Their works are the pioneer of the combination of the insurance and stochastic control theory.Since then,there have been many papers in which the HJB equation was used to solve the optimal control problems in insurance.However,in order to get a perfect result,most of paper usually exclude the possibility of interference factors.For example,there are not any transaction costs when dividends are paid out;there are not any solvency constrains when the optimal dividend is considered;there are not any attention on the interests of reinsurer when the optimal reinsurance is considered,and so on.In effect,these factors can be not ignored since their existence make the optimal strategies presented not to be optimal.Therefore, it is necessary to consider the corresponding optimal strategy again.In most of previous references,only the HJB equation method was applied.Consequently,the original process have to be Markov process;the problems considered have to be like(1.1.4)和(1.1.7);the constructed solution must satisfy the condition given by verification theorem.However, most of the problems which will be considered in my doctoral dissertation do not satisfy these conditions.For example,the optimal problem under Fractional Brownian motion model and the game problem in reinsurance market.Hence,to solve these problems,we must look for the new methods.On the basis of these reasons,my doctoral dissertation will be devoted to doing the following three aspects:Firstly,I will make the model and problem considered more practical.Secondly,the methods will not be limited on HJB equation,and I will try to find the corresponding new optimal control methods according to the current model and problem.Finally,I will try my best to give the very explicit expressions of optimal solutions.In the following,I will introduce the content of every Chapter and Section in detail.A simply introduction of stochastic control theory which will be used in the following chapters are given in the first chapter.Most of the contents are borrowed from[38]and[89].In Chapter 2,a series of optimal dividend problems are considered.Dividend is payments that the company give to the shareholder or the person who provides the initial surplus.Hence,in a sense,the total dividend payments represent company's benefit.Thus,how to pay dividends to maximize the total dividend payments is always one of the most hot topics in finance and insurance.This classical dividend problem has been solved very well and the explicit optimal solution has been given.These results show that the optimal dividend strategies under Brownian motion with drift and compound Poisson model are barrier strategy and band strategy,respectively.These results are obtained under the assumption that there are not transaction costs when the dividends are paid.But,in practice,to prohibit the continuous trade,some transaction costs are required when the dividend are paid.Even though it may be very few,it will affect the optimal strategy.It is obvious that barrier strategy and band strategy are no longer optimal in that case.Therefore,we need to look for the optimal strategy again. Due to the transaction costs,we have to consider not only the optimal dividend payments but also the optimal dividend time.This makes the optimal dividend problem more difficult than the classical optimal dividend problem.Only[28]gives the optimal proportional reinsurance and dividend strategy under the risk model of Brownian motion with drift. Now,there are still many unsolved optimal dividend problems with transaction costs.For example,the optimal dividend problem with the excess-of-loss reinsurance,the optimal dividend problem under the compound Poisson model,and the optimal dividend problem with solvency constrains.We will consider these problems in Section 2.1,Section 2.2 and Section 2.3.In the final section,we relax the restrictive assumptions made in the previous literature,and consider the optimal dividend problem under an extended family of diffusion processes.A completely different result from the previous results is obtained.In section 2.1,we consider the optimal excess-of-loss reinsurance and dividend strategy subject to transaction cost and taxes.Firstly,it is proved that the excess-of-loss reinsurance is better than the proportional reinsurance,that is,the corresponding total discounted expected dividend payments is larger.Then,by solving the corresponding quasi-variational inequality,we obtain the analytical solutions to the optimal return function and the optimal strategy.In[4],they introduced some associated functions to solve the HJB equation. Comparing with[4],the quasi-variational inequality for our problem is more difficult to solve.Moreover a different and more simple method is employed In Section 2.2,we study optimal dividend problem in the classical risk model when payments are subject to both transaction cost and taxes.The optimal dividend problem in the classical risk model is always one of the most difficult topics.The main reason is that the optimal equation has neither the boundary condition nor continuously differentiable solution.In this section, we construct the solution of quasi-variational inequality when the claims are exponentially distributed.We also find a formula for the expected time between dividends.The results show that,as the dividend tax rate decreases,it is optimal for the shareholders to receive smaller but more frequent dividend payments In Section 2.3,we consider a company where surplus follows a rather general diffusion process and whose objective is to maximize expected discounted dividend payments.With each dividend payment there are transaction costs and taxes and it is shown in[93]that under some reasonable assumptions,optimality is achieved by using a lump sum dividend barrier strategy,i.e.there is a upper barrier(?)~* and a lower barrier u~* so that whenever surplus reaches(?)~*,it is reduced to u~* through a dividend payment.However,these optimal barriers may be unacceptably low from a solvency point of view.It is argued that in that case one should still look for a barrier strategy,but with barriers that satisfy the given constraints.We propose a solvency constraint similar to that in[92];whenever dividends are paid out the probability of ruin within a fixed time T and with the same strategy in the future,should not exceed a predetermined levelε.It is shown how optimality can be achieved under this constraint, and numerical examples are given.
We firstly give lump barrier constrains.This constrains means that there are two barriers: upper barrier and low barrier.The company is allowed to pay dividend only when the reserve exceed the upper barrier and the reserve after dividend is not smaller than the low barrier.We consider the optimal dividend with lump barrier constrains in two steps. We firstly consider the optimal dividend problem with the low barrier constrains.Then, on the basis of the results obtained in the first step,the optimal dividend problem with the upper barrier constrains is considered.The corresponding quasi-variational inequalities are used in both two steps.But,in the second step,the corresponding quasi-variational inequality has no longer continuously differentiable solution.Hence,Ito Formula can not be used,and then the verification theorem does not hold.To tackle this difficulty,we skillfully use the theory of local time to prove that the constructed solution is equal to the optimal value function,and give the optimal strategy.This is the first time to give the verification theorem in non-continuously differentiable case.
In the following,we need to look for a lower barrier u_ε＞0 and an upper barrier u_εthat maximize expected discounted Dividends.Meanwhile,the solvency constraints are satisfied as follows:whenever capital is at u_ε,ruin within a fixed time T by following the lump barrier strategy(u_ε,u_ε) should not exceed a small,predetermined numberε.This problem is more difficult than that in[92]since we must look for a pair(u_ε,u_ε),not just a number u_ε.One issue is to find a fast method to calculate the ruin probability for a given lower and upper barrier,and we will show how we can adapt the Thomas algorithm for solving tridiagonal systems together with the Crank-Nicolson algorithm to solve the relevant partial differential equations.The section ends with numerical examples.
In Section 2.4,we consider a company where surplus follows an extended family of diffusion processes and whose objective is to maximize expected discounted dividend payments. An extended family of diffusion processes means that we relax restrictive assumptions. In this case,the property and form of the solution of quasi-variational inequality are different from that in previous literature.We provide the new idea to classify the solutions according to their properties,and then prove that there are three possibilities for the optimal dividend strategy:(1) the optimal dividend strategy is a lump sum dividend barrier strategy;(2) the optimal dividend strategy is a lump sum dividend band strategy; (3) the optimal dividend strategy does not exist.Among of them,the most difficult one is the case(2).Since,in this case,all of previous methods to construct the optimal value function and the optimal strategy are no longer valid.To tackle this difficulty,we consider two independent solutions instead of only one solution,and construct the optimal value function respectively in four intervals.Moreover,it is the first time to show clearly that the optimal dividend strategy is lump sum dividend band strategy and give the explicit expression of the strategy.
In Chapter 3,the problems of minimizing ruin probability and maximizing the exponential utility are studied.
In recent years,the optimization problems of maximizing the exponential utility and minimizing the probability of ruin have been studied by many authors.There exist relationships between the optimal strategies for maximizing the exponential utility and minimizing the probability of ruin.For the case of an ordinary investor(no external risk process) in the discrete time risk model,[37]found the optimal investment strategy that maximizes the exponential utility with utility function of the form u(x)=-e~(-mx).He conjectured that such a strategy also minimizes the probability of ruin for some value of m under the assumption that the investor was allowed to borrow an unlimited amount of money,and there was no risk-free interest rate.For the investors which face a random risk process correlated with the risky asset process.[24]show that,with zero interest rate,the investment strategy of maximizing the exponential utility also minimizes the probability of ruin for a specific value of m,which validates Ferguson'conjecture in a strong sense, [24]also show that the conjecture did not hold when the interest rate is positive.
Reinsurance is also an important technique for insurance companies seeking to control their risk.To make the utility and ruin analysis even more realistic,we consider incorporating the concept of proportional reinsurance into Browne's model.That is,under these two controls,we consider three optimization problems,namely the problem of maximizing the exponential utility of terminal wealth,the problem of minimizing the probability of ruin,and the problem of minimizing the expected discounted penalty of ruin.By the corresponding Hamilton-Jacobi-Bellman equations,explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risk-free interest rate,the results indicate that the optimal strategies for maximizing the exponential utility and minimizing the probability of ruin are equivalent.This validates Ferguson's conjecture under current model.
For the optimal investment problem,only the single risky asset(stock) is considered in previous literature.Actually,the insurance company,to decrease(increase) their risk (profit),will invest its wealth into multiple risk assets.Hence,we in section 3.2 consider that the insurance company invests its wealth in a financial market consisting of a risk-free asset and n risky assets.The risk model considered is the same as Section 3.1,The insurer is allowed to purchase proportional reinsurance as well as proportional reinsurance.Under the constraint of no-shorting,we consider two optimization problems:the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin.By solving the corresponding Hamilton-Jacobi-Bellman equations,explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risk-free interest rate,the results indicate that Ferguson's conjecture still hold under the current model.
In Section 3.3,following the same framework as section 3.2 we consider the excess-of-loss reinsurance instead of proportional reinsurance.First we show that the excess-of-loss reinsurance strategy is always better than the proportional reinsurance under two objective functions.Then,by solving the corresponding Hamilton-Jacobi-Bellman equations, the closed-form solutions of their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risky-free interest rate,the results indicate that Ferguson's conjecture still hold in excess-of-loss reinsurance case.
In Chapter 4.Mean-Variance problem in insurance is discussed.
The portfolio selection is to allocate the wealth into various assets to disperse risk and ensure the profit.In 1952,Markowitz uses the variance to measure the risk of stock,and provides mean-variance analysis methods for the portfolio selection.This open a prelude to modern finance.Mean-variance problem is that the investors want to find the optimal investment strategy to maximize the mean and minimize the variance.Mean-variance is not only the pioneering work of modern portfolio selection theory but also one of footstone of modern finance.
In recent years,finance and insurance market has started to link.In order to increase its profit or decrease its risk,the insurance company will invest the part of its reserve into financial market.There has been many literatures which considered the optimal investment.But the optimal criteria are still several classical actuarial variables such as dividend payments,the probability of ruin and so on.Mean-variance is rarely considered in insurance field,while it is one of the most important and popular conceptions in finance filed.As far as we know,only[120]consider the optimal investment problem under meanvariance criteria and[27]consider the optimal reinsurance/new business under meanvariance criteria.We will do the further research about mean-variance problem in this chapter.
In Section 4.1,we study optimal reinsurance/new business and investment(no-shorting) strategy for the mean-variance problem in two risk models:a classical risk model and a diffusion model.The problem is firstly reduced to a stochastic linear-quadratic(LQ) control problem with constraints.Then,the efficient frontiers and efficient strategies are derived explicitly by a verification theorem with the viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations,which is different from that given in[129].Furthermore,by comparisons,we find that the optimal solutions are identical under the two risk models Hitherto,people always model the claim processes by stochastic processes with Markovian. However,in most cases,the insurance claims often present long-range dependence: the behavior of the surplus of an insurer after a given time t not only depends on the information at t but also on the whole history up to time t.This phenomenon is not negligible and likely to have an impact on various issues such as solvency,pricing and optimal retention reinsured,etc.Therefore,fractional Brownian motion has been used recently to model the claims that an insurance company may face.Most of literature consider the ruin problem.Since the property of fractional Brownlan motion is difficult to describe, there are always not very explicit results.A fewer authors consider the optimal control under fractional Brownian motion.It is not only because the fractional Brownian motion is difficult to research,but also it is not Markov process,and then the HJB equation method can be not used.As far as we know,only[63]consider linear-quadratic(LQ) problem. However,most of results are obtained under the Markovian controlled system.Therefore, it is valuable in theory and practice to study the optimal control problems under the more extensive environment.In Section 4.2,we consider the optimal investment problem under mean-variance criteria for an insurer.The insurer's risk process is modelled by a classical risk process that is perturbed by a standard fractional Brownian motion with Hurst parameter H∈(1/2,1).Due to non-Markovian of fractional Brownian motion, the well-known HJB equation is not applied.However,stochastic calculus for fractional Brownian Motion introduced in[36]make fractional Brownian Motion have many similar properties to Brownian motion such as Zero mean,Girsanov Theorem.By these properties ,we find the method to solve the optimal problem under a classical risk process that is perturbed by a standard fractional Brownian motion.By virtue of an auxiliary process, the efficient strategy and efficient frontier are obtained.Moreover,when H→1/2+the results converge to the corresponding(known) results for standard Brownian motion.
Section 4.3 considers that the risk process is a compound Poisson process and the insurer can invest in a risk-free asset and multiple risky assets.We obtain the optimal investment policy under mean-variance criteria using the stochastic linear quadratic(LQ) control theory with no-shorting constraint.Then the efficient strategy and efficient frontier are derived explicitly by a verification theorem provided in Section 4.1.
In Chapter 5,we consider optimal control problems when risk processes modeled by (fractional) Brownian motion with drift.
In Section 5.1,we model the surplus process as a compound poisson process perturbed by Brownian motion,and allow the insurer to ask its customers for input to minimize the distance from some prescribed target path and the total discounted cost on a fixed interval. The problem reduces to a version of linear quadratic regulator under jump diffusion process.It is treated by three methods:dynamic programming,completion of square and stochastic maximum principle.The analytic solutions to the optimal control and the corresponding optimal value function are obtained.In this section,we introduce the methods of completion of square and stochastic maximum principle into the insurance filed.These two methods have many differences from the HJB equation method.Especially,the original process is no longer required to be Markov process.Hence,these give some inspiration to solve the optimal problem under Non-Markov process.The following Section 5.2 is a good example.We in this section consider a classical risk model perturbed by a standard fBm with Hurst parameter H∈(1/2,1).The criteria is the same as section 5.1.By using the completion of squares method,the expression of the optimal value function and the corresponding optimal control are derived.In addition,the method in Section 4.2 is also inspired by section 5.1.
In Chapter 6,some other optimal control problems are investigated.
There has been many literatures which consider the optimal reinsurance,such as[24], [58],[4].However,all of them only consider the insurance company.In practice,if we only consider the insurance company,the optimal reinsurance strategy obtained is often unacceptable for the reinsurance company.This alarms us that there are two parties in the reinsurance contract,their interest collide with each other.Therefore,we need to make a reciprocal reinsurance treaty.This is a game problem.[21]firstly use the game theory to solve the optimal reinsurance problem in insurance filed.The optimal criteria is maximizing the expected exponential utility.Since then,[13],[117]make some research for risk exchanges and reinsurance markets.However,in these papers,only the single period model is considered.It will obviously be of interest to extend the single model to multi-period,or a continuous time model.In this sectioin,we will consider a continuous time model.In this case,the interest of the insurance company and the reinsurance company always collide with in the whole time interval,which is different from the case of the single period.Hence,to solve the optimal problem,we need to look for the new and different methods.In this section,we solve the problem in two steps.Firstly we treat the Pareto optimal problem about the terminal wealth.Then we find the reinsurance strategy to copy the Pareto optimal terminal wealth.By this method,we show that the optimal Pareto reinsurance in continuous time is proportional reinsurance.Moreover,it is proved that the core of the cooperative reinsurance game is non-empty.This section intend to provide one method to solve the optimal problem in continuous time reinsurance market. Many other problems are still unsolved such as the optimal dividend,ruin probability, mean-variance.
Section 6.2 deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modeled as an Ornstein-Uhlenbeck process.Here,only the stock price and interest rate can be observable for an investor.It is reduced to a partially observed stochastic control problem.Combining the filtering theory with dynamic programming approach,explicit representations of the optimal value functions and corresponding optimal strategies are derived.Moreover the closed-form solutions are provided in two cases of exponential utility and logarithmic utility.In particular,logarithmic utility is considered under the restriction of short-selling and borrowing.[76]uses martingale method to obtain the optimal strategy.Here,the investor is not allowed to short-sell and borrow,and a simpler method is employed to solve the problem.Moreover,the maximal logarithmic utility is given in the case of a single stock.
In section 6.3,the basic claim process is assumed to follow a Brownian motion with drift.In addition,we allow the insurer to invest in a risky asset and to purchase cheap proportional reinsurance.Under these two controls,we consider the problem of minimizing expected time to reach a given capital before ruin.By using the Hamilton-Jacobi-Bellman equation,explicit expressions for optimal value function and optimal strategy are obtained.
By the HJB equation,Quasi-variational inequalities,partial differential equation,completion of squares,Pareto optimal,fractional Brownian motion theory,my doctoral dissertation will be devoted to solving optimal dividend with transaction costs and solvency constrains,optimal multi-asset investment,Pareto optimal reinsurance strategy and optimal control problem under fractional Brownian motion insurance model.In the following, we will give the innovation and novelty of every section.
1.The innovation in methods
(1) In section 2.3,there are two innovations in methods:(a) the verification theorem about non-continuously differentiable solution;(b) to find a fast method to calculate the ruin probability for a given lower and upper barrier,and show how to adapt the Thomas algorithm for solving tridiagonal systems together with the Crank-Nicoison algorithm to solve the relevant partial differential equations.
(2) There are two innovations in section 2.4:(a) to give a new idea to classify the solutions of quasi-variational inequality;(b) to give a new method to construct the optimal value function and the optimal strategy.
(3) Section 4.1 gives the verification theorem about a class of viscosity solution.
(4) Section 6.1 give the method to solve the Pareto optimal reinsurance in continuous time.
(5) In section 6.2,it is shown that how to use the HJB equation method to solve the optimal problem with partial information.
2.The innovation in skill to solve optimal equation and construct the solution.
This part mainly extends some classical and excellent work,but it is not the simple extension on calculation.
(1) Section 2.1 extend the work of[28](published in Mathematical finance),and consider the optimal excess-of-loss reinsurance and dividend with transaction costs and taxes. The corresponding quasi-variational inequality is more complicated than that under proportional reinsurance.In section 2.1,we give the different method to construct the solution.
(2) In section 2.2,the optimal dividend with transaction costs and taxes is considered under the classical risk process.The construction of the solution is different from Section 2.1 and[28].
(3)[24]is known as the pioneering work on the combination of insurance theory and control theory.In third chapter,we extend[24]'s work by incorporating the reinsurance into the model in[24].In this case,the more technical method to construct the solution is needed.
3.To introduce new control method into insurance filed.
Section 5.1 use three methods of the HJB equation,stochastic maximum principle, completion of square to solve the optimal problem.
4.To consider the optimal control problem under non-markovian process
The fractional Brownian motion is a long-range dependence,non-markovian process. It can model many phenomenon closer to practice.However,since its property is difficult to describe,there are only a few literatures which consider fractional Brownian motion risk model.We in section 4.2 and section 5.2 consider the optimal problem under a classical risk model perturbed by a standard fBm.We find the method to solve the problem and give the explicit solution.
Another innovation is that we give the explicit solution for all of problem.
随着金融和保险市场发展,对于原有几个精算量的刻画已经不能再满足保险公司的需求。例如对于破产概率,破产赤字,相对于他们的具体表达式,保险公司更关心如何才能使得这些代表风险的量达到最小。为了使他们尽可能的小,保险公司会采取一些相应的措施例如再保险,投资金融市场。这时保险公司所面临的问题就是如何寻找最优的再保险或投资策略使得风险达到最小。保险公司除了关心代表风险的量以外也关心一些代表它的收益和效用的量例如破产前总的分红量和某时刻财富的效用。想尽可能的使得这些量达到最大。所有这些都属于金融保险中的随机最优控制问题。在过去几十年里,通过利用随机控制的理论和方法,尤其是[24]和[3]把随机控制理论应用到风险中,通过Hamilton-Jacobi-Bellman(HJB)方程的方法分别得到了扩散模型下的最优投资(目标是最小破产概率和最大化指数效用)和最优分红策略,使得该领域得到了迅速的发展,并开创了风险理论和随机控制理论相结合的先例。之后有许多文献相继利用HJB方程的方法进行对风险里的最优问题的研究。其中有些也加入了再保险。并取得了很好的结果。
但在许多工作里为了得到明确的最优解,所建立的模型与实际还存在着很大的差距。例如假设分红时没有交易费用和公司的偿付能力的限制;仅仅考虑对于单个风险资产的投资;考虑最优再保险时没有考虑到再保险人的利益;忽略了索赔过程的长程相依性等等。而在实际中这些因素是存在的且不容忽视的。而且这些因素的加入会对以前所得的最优策略产生很大的影响,有的甚至已不再是最优的。因此有必要重新考虑所对应的最优问题和寻找新的最优策略。
以往文献考虑的问题和模型与实际存在差距,其主要一个原因是方法上过于局限于HJB方程。这使得考虑的模型只能限于马尔科夫过程;考虑的问题只能限于(1.1.4)和(1.1.7)的形式;构造的解也必须满足验证定理所要求的条件。而上面提到的大多数问题不再满足这些条件例如分数布朗运动模型下的最优问题,模型不再具有马尔科夫性质;再保险中的博弈问题,不再满足三个限制条件中的任何一个。因此,为了解决这些问题,需要我们寻找新的方法或者在方法上有新的突破。
鉴于上述原因,我的博士毕业论文将致力于下面三个方面。首先是建立与实际更贴近的模型和问题。其次是不局限于HJB方程的方法,根据当前的模型和问题所特有的性质灵活变通,充分发挥各种随机控制理论方法的作用,努力寻找解决问题的路径。最后,为了使最终的结果对实践能起到一个很好的指导作用,将尽可能的对最优问题给出明确解,使得所得最优策略具有可操作性。下面将详细介绍各个章节的内容。
首先在第一章中给出了下面章节中将要涉及到的随机最优控制理论。这些理论主要来自[38]和[89]。
在第二章中,一系列的带交易费用的最优分红问题被考虑。
分红是指公司将部分盈余分给股东或初始准备金的提供者。所以总的分红量从某种意义上反应了一个公司的效益和实力。因此如何选择一个分红策略或采取某种措施(例如再保险和投资)使得破产之前的分红量达到最大一直以来都是金融和保险领域中最热门的研究话题之一。对于这种古典的最优分红问题已经解决的比较完善。包括含有再保险控制的都已经给出了非常明确的结果。结果展示了带漂移布朗运动和复合泊松风险模型下的最优分红策略分别是上限为常值的边界分红策略(barrier strategy)和波段分红策略(band strategy)。但是结果的得到都假设了当进行分红时不需要交一定量的交易费用。而实际中为了避免连续交易。当进行分红时都要求交一部分交易费用。即使它可能很少。但它的出现从根本上会影响到最优分红策略的形式。显然在这种情况下barrierstrategy和band strategy由于都是连续的进行交易,会带来无穷大的交易费用,所以都不再是最优的分红策略。因此需要我们重新选择最优分红策略。由于交易费用的出现,我们对最优分红策略的选择不仅仅要考虑对分红量的选择还要考虑对分红时间的选择,这使得这时的分红问题要比古典分红问题复杂的多。在风险理论里,仅仅[28]给出了带漂移布朗运动风险模型下的最优比例再保险和分红策略。还有很多关于带交易费用的最优分红问题没有解决,例如一直被认为比比例再保险更优(既相对应的最大分红量更大)的超额损失再保险控制下的最优分红问题,复合泊松风险模型下的最优分红问题,带有偿付能力限制和交易费用的最优分红问题。我们将分别在2.1,2.2和2.3节展开对这些问题的讨论。在这一章的最后一节我们放宽了以前文献中所作的限制性的假设,考虑一类更广义的扩散模型下带交易费用的最优分红问题,展现了一个与以前不一样的最优解。
在2.1节我们研究了带交易费用和税收的最优超额损失再保险和分红问题。首先证明了超额损失再保险是比比例再保险好的即所对应的最大的期望折现分红量要大。然后通过解拟变分不等式,给出了最优超额损失再保险和分红策略的明确表达式。超额损失再保险与比例再保险相比不再有比例这么好的形式。[4]曾经考虑过超额损失再保险下的不带交易费用的最优分红问题。我们与[4]的区别首先是对于拟变分不等式解的构造比[4]里面的HJB方程的解的构造要困难。其次是在[4]里面对于HJB方程解的构造依赖于一个辅助函数。我们这里对于拟变分不等式解的构造,没有引入辅助函数而且方法更简单。
2.2节考虑了复合泊松模型下的带交易费用和税收的最优分红问题。复合泊松模型下的最优分红问题一直以来都是一个难点。主要原因是在复合泊松情景下所对应的最优方程不再有边界条件并且有可能不再有连续可微的解。在这一节中我们构造出了当索赔是指数情景下的拟变分不等式的解进而给出了最优策略。而且也给出了分红时间间隔的期望值。结果展现了当分红的税率减少时,应该相应的增加分红的次数同时减少每次分红的量。
在2.3节,假设公司的余额被一个广义的扩散过程所描述。公司的目标是最大化期望折现分红。每次分红仍旧有交易费用和税收被要求。在[93]展现了在一些合理的假设下最优策略是块状边界策略,即有两个边界,当盈余达到上边界时,进行分红,盈余减少到下边界。但是,从偿付能力的角度来看,这些最优边界有可能由于过低而是保险公司所不能接受的。因此我们应该找一个满足偿付能力限制的块状边界策略。类似于在[92]所提出的对于偿付能力的限制,分红策略应该满足在该策略下所对应的有限时间的破产概率不超过一个给定的值。
这里我们首先假定一个块状边界的限制,这个限制的意思是事先给出两个边界,公司仅仅当余额到达给定的上边界时才可以进行分红,而且分红后的盈余不能低于给定的下边界。对于在这个限制下的最优分红问题,我们分为两步进行考虑,首先考虑仅仅有下边界限制的最优分红问题,然后在第一步的基础上考虑有上边界限制的最优分红问题。这两步都借助了相应的拟变分不等式。但在第二步中,拟变分不等式的解不再是连续可微的,这使得It(?)公式不能再被利用,进而验证定理不再成立。我们这里巧妙的利用局部时理论证明了拟变分不等式的解就是最优值函数同时构造了相应的最优策略。首次给出了对于不满足一次连续可微函数的验证定理。
剩下的任务是如何找到最优的两个边界限制使得他们在满足偿付能力限制的情况下对应的最优值函数最大。最优双边界的寻找完全不同于[92]里面对于最优单个边界的寻找。它是极其复杂的。需要偏微分方程方面更高层的工具。我们在2.3节里展现了如何利用Thomas和Crank-Nicolson算法来解决相应的偏微分方程以得到最优双边界限制。在这节的最后,我们还给出了数值解。
在2.4节所考虑的公司的盈余过程是一类更广义的扩散过程。更广义的意思是放宽了以前文献中所作的限制性的假设。这里同样利用的也是拟变分不等式,但是由于没有了一些好的假设条件导致了拟变分不等式的解的性质和形式与以前有很大的不同。这里我们利用一种新的思想把拟变分不等式的解根据他们的性质进行了分类。证明了最优分红策略有三种可能性:(1)块状边界分红策略;(2)块状波段分红策略;(3)最优分红策略不存在。其中最困难的部分是对于块状波段分红策略是最优分红策略的证明。因为在这种情况下,以前所有构造最优值函数和最优策略的方法已经不再有效。为了克服这个困难,我们不再仅仅从拟变分不等式的一个解考虑而是从两个独立解入手。对最优值函数进行了四个区域的构造。并且首次清晰的展现了最优策略是一个块状波段分红策略并给出了它的具体形式的表达。在以前的文献里,当证明最优策略是波段分红策略时,多数是从粘性解出发,因此里面所涉及的边界都没有一个明确的表达。
第三章讨论了最小破产概率和最大化指数效用问题。
在最近几年里,最小化破产概率和最大化指数效用已经被很多研究者考虑过。对于他们之间的关系早在1965年Ferguson在[37]里就给出了猜测。他猜测当效用函数3.1.5中的参数m取适当的值时,两个最优问题所对应的最优策略是一样的。其中假设没有无风险投资。[24]考虑风险盈余过程是一个带漂移的布朗运动和允许保险公司把盈余投资于风险资产和无风险资产。三个最优准则:最大化指数效用,最小化破产概率和最小化折现罚金分别被考虑。给出最优投资的同时验证了Ferguson's的猜测在当前模型下是成立的。另外也展现了当有无风险投资时,猜测是不成立的。即无论m取何值,最大化指数效用和最小化破产概率两个最优问题的最优策略都不同。
再保险是保险公司用来控制他们所面临的风险或提高效用的一个重要的手段。因此我们在3.1节,通过加入再保险控制延拓了Browne的工作。即风险盈余过程是一个带漂移的布朗运动。保险公司被允许把盈余投资于风险资产和无风险资产,并且购买(比例)再保险。在投资和再保险这两种控制下,我们考虑了三个最优准则:最大化指数效用,最小化破产概率和最小化折现罚金。由于比例再保险控制中的比例是在[0,1]之间的。既控制变量是有限制的。因此对于HJB方程解的构造要比[24]复杂。尤其是最后一个准侧。即使这样我们仍旧给出了HJB的解析解,并通过它给出了最优值函数和最优策略。特别的,当没有无风险投资时,相应的结果展现了最大化指数效用和最小化破产概率他们的最优策略是一样的。这就验证了在[37]里面给出的猜测在加入再保险后仍旧成立。同时也展现了当有无风险投资时,猜测不再成立。
对于最优投资问题,之前一直考虑的是保险公司投资单个风险资产。而实际中为了达到预期的目标,保险公司一般是投资到多个风险资产。因此我们在3.2节考虑了投资多个风险资产的最优问题。考虑的风险模型和3.2节一样。同时允许保险公司投资于多个风险资产和购买(比例)再保险。在非卖空的限制下,最大化指数效用和最小化破产概率两个准测被考虑。当投资于单个风险资产时,在没有非卖空限制下得到的最优投资额很自然的满足非卖空的限制。但对于多个风险资产,情况就不一样了。在没有非卖空限制下得到的最优投资策略不一定满足非卖空的限制。因此需要我们对HJB方程进行一定不同方法的处理。在这节中,我们首先处理非卖空带来的影响把HJB转化成一般的可解决方程的形式。然后给出了它的解进而得到了最优值函数和最优策略。同时也验证了Ferguson的猜测在没有无风险投资下仍成立同时当有无风险投资时猜测不成立
3.3节考虑了超额损失再保险和多个风险资产投资两种控制。最优准则与3.2节相同。首先证明了超额损失再保险比比例再保险更优,即超额损失再保险所对应的最小破产概率比比例再保险所对应的小和最大指数效用比比例再保险的大。然后通过HJB方程给出了最优解。同时也验证了Ferguson的猜测在超额损失再保险的情景下仍成立。
第四章对保险里的均方差问题进行了讨论。
投资组合选择简而言之就是把财富分配到不同的资产中,以达到分散风险、确保收益的目的。1952年,Markowitz用方差来量化股票收益的风险,提出了投资组合选择的均值-方差分析方法,揭开了现代金融学研究的序幕。均方差问题是指投资者想找到一个最优的投资策略使得它的期望达到最大同时方差最小。它是一个以投资组合的期望收益(均值)和风险(方差)为目标的双目标决策模型,因此很自然地导出了投资组合选择的均值-方差有效组合、有效前沿等概念。均值-方差投资组合理论不仅是现代投资组合选择理论的先驱工作,也是现代金融学的基石之一。其精髓在于首先对风险进行量化分析,开辟了风险管理的新思路。
最近几年,金融和保险市场已经开始慢慢的结合,保险公司为了增加它的收益或减少它的风险会把部分盈余投资于金融市场。已经有很多文献涉及到了保险理论中的最优投资问题。但大多数文献考虑的最优准则仍旧仅仅限于原有的几个经典精算量例如分红量,破产概率,破产赤字等。均方差准则作为金融学的最重要和最流行的概念之一反而在保险理论中很少被涉及。据我们所知仅仅[120]考虑了均方差准则下的最优投资问题和[27]考虑了均方差准则下的最优再保险/新业务问题。因此,我们在这一章中对保险理论中的均方差问题进行了更深入和广泛的研究。
在4.1节,我们对再保险/新业务和投资两种控制下的均方差问题进行了研究。其中考虑了两种风险模型:复合泊松和带飘移的布朗运动。问题的解决不同于[27],相应的HJB方程不再有古典的解,另外即使给出粘性解,据我们所知仅仅有关于扩散模型下的粘性解验证定理。对于它在复合泊松模型下的有效性还没有得到认证。为了克服这些困难,首先通过两个特殊的Riccati方程构造一个连续可微函数并展现了它是HJB方程的粘性解。对于这个粘性解我们给出了不同于以往的验证定理。而且可以看到它能被应用于一类粘性解。最后通过比较两个风险模型下的结果,发现他们的最优策略是一样的。这个可以启发我们在考虑一些问题时可以用扩散近似来代替复合泊松。这样就可以简化问题。
迄今,人们一直用具有马尔科夫性的随机过程来描述索赔过程。但在大部分情形下,保险公司的索赔过程呈现出长程相依性:给定时刻t后过程的行为,不仅依赖于t时刻的信息,而且还依赖于时刻t以前的历史。因此,最近已经开始有用分数布朗运动来模拟保险公司的索赔过程。但所有文献都是考虑分数布朗运动风险模型的破产问题。由于分数布朗运动很多性质比较难刻画,因此对于它的破产概率一直没有得到明确的结果。对于它的优化问题更是很少有人考虑。这不仅是因为分数布朗运动本身比较难研究,更主要的是它不再具有马氏性,传统的HJB方程方法已经不能再利用。据我们所知,仅仅[63]考虑过特殊的线性模型下的二次规划问题,而且最优策略被限制在马氏策略(即策略在每时刻的取值依赖于当时过程的值)集合里考虑。但我们知道分数布朗运动本身不再具有马氏性,这就意味着真正的最优策略不会再是马氏策略。因此对[63]工作作更进一步的推广和改进,是很有必要的。在4.2节,我们用一个受分数布朗运动干扰的古典风险过程描述了公司的盈余过程。研究了均方差准则所对应的最优投资问题。这里分数布朗运动的Hurst参数H∈(1/2,1)。虽然分数布朗运动不再是Markov过程,不能再利用HJB方程方法。但由于[36]利用Wick乘积定义的关于分数布朗运动的随机积分使得分数布朗运动有了跟标准布朗运动许多类似的性质例如Zero mean,Girsauov Theorem等。同时受到下面5.1节对完全平方方法使用范围的探索结果启发。我们发现分数布朗运动的性质是符合完全平方方法的要求。因此通过利用完全平方的方法我们找到了有效策略和有效前沿。并且当H→1/2+结果被验证收敛到已知的布朗运动情景下的结果。
在第五章,我们研究了受(分数)布朗运动干扰的古典风险模型下的最优问题。
在5.1节,风险过程是一个受布朗运动干扰的复合泊松过程。控制是保险公司向顾客收取的保费。目标是最小化盈余过程与给定轨道的距离与保费折现值的和。我们通过动态规划,完全平房和随机最大值原则三种不同的方法给出了最优控制和最优值函数。这一节的内容,不仅结果对实践有一定的指导作用,更重要的是在理论和方法上的作用。在以前的文献里,基本都是局限于HJB方程的方法。而HJB方程有它的局限性既仅仅使用于马尔科夫过程。因此在考虑模型和问题上都受到了限制。在这节中我们把完全平方和随机最大值原则引入到了保险领域中。他们既跟HJB方法有联系但也有很多的区别,尤其在处理的模型已经不再仅仅限于马尔科夫过程。因此对于我们以后考虑非马尔科夫盈余过程的最优问题有很大的借鉴性。5.2节就是一个很好的例子。我们在这一节研究了受分数布朗运动干扰的古典风险模型下的最优问题。最优准则与5.1节相同。最优策略和最优值函数通过完全平方的方法被清晰的给出。另外4.2节也是受到了5.1节所提供方法的启发。
在第六章研究了一些其他的最优控制问题。
对于再保险已经有很多文献涉及到,例如[24],[58],[4]等。但在所有这些文献里都没有考虑到再保险公司的效用。而实际中,如果只考虑保险公司一方的话,所求的最优再保险策略往往是再保险公司所不能接受的。这就警惕我们在再保险协议里有两方,并且他们的利益是冲突的。因此最优再保险的合约必须显示为一个合理的双方利益的折衷。这其实就是一个双方博弈的问题。用博弈的思想来解决保险定价或风险分配方面的问题,最初是由[21]提出来的,考虑的最优准则是最大化效用。之后,[1],[117]等在风险交换和再保险市场方面也作出了一定的工作。但在所有这些工作中仅仅考虑单期索赔即静态的最优再保险的选择。而实际中,投保人与保险公司以及保险公司和再保险公司在很多情况下都是长期的合作而不是仅仅是针对一次索赔的合作。因此我们在6.1节研究了连续时间模型下的再保险市场上的最大化指数效用问题。连续时间模型下的再保险问题不同于单期的情景,保险公司和再保险公司它们在整个时间段内都是一个博弈的关系。据我们所知,即使在专门的博弈理论中也从没涉及到连续时间模型的例子。因此在借鉴单期的理论和方法的同时,必须在方法上有其他新的突破,才能解决连续情景下的最优再保险问题。为了克服这些困难,我们把问题分成了两步进行处理。首先处理终端财富之间的Pareto最优问题,然后寻找再保险策略去复制Pareto最优终端财富。通过这种方法我们找到了Pareto最优策略。而且结果展现了连续时间情景下的Pareto最优合作再保险是比例再保险。这个结果恰恰与以前的结论相反既当仅仅保险公司效用被考虑时超额损失再保险一直是最优的。最后我们证明了合作再保险的核是非空的。这里核是博弈论非常重要的量,它是所有能被博弈里面所有成员所接受的Pareto最优策略的集合。这一节我们的主要目的是提供一个解决连续时间再保险市场上的Pareto最优问题的方法。其他的很多问题例如分红,破产概率,均方差等都有待研究。
6.2节是关于一个局部信息下的最优控制问题。股票的价格满足一个随机微分方程,其中瞬时的返回率是一个Ornstein-Uhlenbeck过程。这里仅仅股票的价格和利率能被观察。通过利用过滤和动态规划理论,我们给出了指数和对数效用下的最优解。其中对数效用是在不允许投资者进行卖空和借款的限制下考虑的。与[75]相比,我们这里不允许投资者卖空和借款,而且这里所用的HJB方程的方法要比[75]里面的鞅方法简单很多。而且使用HJB方程的方法可以给出所对应值函数的明确解。在这节里考虑的模型是简单的,我们主要的目的一是利用HJB方程的方法解决局部信息问题,它相对于以前大多数文献中的鞅方法要简单很多,二是希望能为考虑保险理论中的局部信息问题提供一些帮助。
最后一节考虑的准则是最小化破产之前到达一个给定目标的期望值。对于一般的投资者,[95]通过最大化波动系数平方除飘移系数得到了最优投资策略。[95]里面的控制变量不受限制和最优策略被展现是比例策略。在这一节中,卖空被限制和比例再保险的比例是在[0,1]之间。这些对控制变量的限制使得[95]里面的方法不能再被利用。我们通过HJB方程的方法给出了最小期望时间和最优策略。而且结果展现了最优投资策略已不再是比例策略。
我的博士毕业论文主要是通过HJB方程,拟变分不等式,偏微分方程,完全平方,Pareto最优,分数布朗运动等理论解决了风险理论中带交易费用和偿付能力限制的最优分红问题,最优多个风险资产,Pareto最优再保险以及在分数布朗运动风险模型下的最优问题。上面以他们所研究的问题对其进行了以章节为单位的分类介绍。下面将从各个章节的创新点和特色之处对他们的内容进行概括性的总结。
1.方法的创新。
(1)2.3节有两个方法上的创新点:(a)关于非连续可微解的验证定理;(b)利用Thomas和Crank-Nicolson算法给出了最优双边界。
(2)2.4节在方法上的两个创新点是:(a)提出了一种新的把拟变分不等式的解进行分类的方法;(b)给出了一种新的构造最优值函数和最优策略的方法。
(3)4.1节给出了一类粘性解的验证定理。
(4)6.1节首次给出了解决连续时间Pareto最优再保险的方法。
(5)6.2节利用HJB方程的方法解决局部信息下的最优问题。以前几乎所有文献都是利用的鞅方法。这里我们也充分展现了HJB方程与鞅方法相比的优点。
2.解方程和构造最优解技巧的创新。
这部分主要是对已有经典工作或优秀工作的更深入的研究和推广。但并不是简单的在同一方法上的计算推广。
(1)2.1节是对文章[28](发表在国际顶级金融杂志Mathematical Finance)的工作一个改进推广。研究了带交易费用和税收的最优超额损失再保险和分红问题。首先证明了超额损失再保险比[28]中的比例再保险更优。这也是我们考虑超额损失再保险的一个原因。超额损失再保险所对应的拟变分不等式比比例再保险复杂的多。在2.1节我们给出了不同的构造解的方法。
(2)2.2节是对于古典风险模型的一个挑战。古典风险模型一直以来都是分红问题的难点。这里对于拟变分不等式的解的构造与2.1节和[28]有很大不同。从2.2节我们可以看到,古典风险模型下拟变分不等式的解的形式更多样化。
(3)[24]的工作被称为保险理论和控制理论相结合的先例。在第三章,我们延拓了Browne的工作考虑了最优再保险和投资问题。比例再保险的介入使得对于相应的HJB方程的解的构造需要更高的技巧性。
(4)6.3节考虑了最小化破产之前到达一个给定目标的期望值。这是对于经典文章[95]工作的一个推广。由于卖空被限制和比例再保险的比例是在[0,1]之间。使得[95]里面的方法不能再被利用。我们通过HJB方程的方法解给出了最优解。这里对于HJB方程解的构造也是极其复杂的。
3.引入新的控制方法进入保险领域。
5.1节利用了HJB方程,随机最大值原则和完全平房三种不同的方法解决最优问题。从中可以看到各种方法的优缺点和他们不同的使用范围。这样可以开拓我们的视眼,考虑不同类型的风险模型和问题。
4.研究非马尔科夫风险余额过程的最优控制问题。
分数布朗运动是具有长程相依性的非马尔科夫过程。虽然它可以用来描述现实中更贴近实际的现象。但对于它的研究一直以来都没有实质性的进展。这主要是因为它的性质比较难刻划。风险中只有很少的文献涉及它的破产概率。对于关于分数布朗运动风险模型下的最优控制问题。更是寥寥无几。我们在4.2和5.2。节研究了受分数布朗运动干扰的复合泊松模型下的最优控制问题。找到了适合于分数布朗运动的最优控制方法。并且给出了最优策略的明确解。
本论文另外一个非常重要的特色就是对于所有的最优问题都给出了非常明确的最优解。
Risk theory is the most theoretical part in insurance mathematics which derives from risk management in insurance.It is mainly about several actuarial variables such as ruin probability,the surplus immediately prior to ruin,the deficit at ruin etc.The applications of stochastic process and stochastic analysis,especially the martingale method,in risk theory make risk theory develop rapidly.Many fruitful achievements in characterizing several actuarial variables mentioned above have been made.As the development of finance and insurance markets,these actuarial variables have no longer met the needs of the insurance company.The insurance company is more and more concerned about how to minimize these actuarial variables which represent the risk instead of their expression.In order to minimize them,the insurance company will take the appropriate measures such as reinsurance,investment in finance market.At this time the problem faced by insurance company is how to find the optimal reinsurance or/and investment strategy to minimize the risk.In addition,to the ruin probability,the insurance company also uses the other measures to measure the risk such as the variance of terminal wealth,VaR(Value-at-Risk), etc.At the same time,the insurance company is also concerned about some other variables which represent their utilities and profits such as the utility of the terminal wealth and the expected discounted dividend payments before ruin.All of these belong to stochastic control problems.In the last few decades,in order to solve these problems,stochastic control theory-has been applied into the insurance.For example,[24]and[3]respectively obtained optimal investment strategy and the optimal dividend strategy under diffusion model by the Hamilton-Jacobi-Bellman(HJB) equation approach.Their works are the pioneer of the combination of the insurance and stochastic control theory.Since then,there have been many papers in which the HJB equation was used to solve the optimal control problems in insurance.However,in order to get a perfect result,most of paper usually exclude the possibility of interference factors.For example,there are not any transaction costs when dividends are paid out;there are not any solvency constrains when the optimal dividend is considered;there are not any attention on the interests of reinsurer when the optimal reinsurance is considered,and so on.In effect,these factors can be not ignored since their existence make the optimal strategies presented not to be optimal.Therefore, it is necessary to consider the corresponding optimal strategy again.In most of previous references,only the HJB equation method was applied.Consequently,the original process have to be Markov process;the problems considered have to be like(1.1.4)和(1.1.7);the constructed solution must satisfy the condition given by verification theorem.However, most of the problems which will be considered in my doctoral dissertation do not satisfy these conditions.For example,the optimal problem under Fractional Brownian motion model and the game problem in reinsurance market.Hence,to solve these problems,we must look for the new methods.On the basis of these reasons,my doctoral dissertation will be devoted to doing the following three aspects:Firstly,I will make the model and problem considered more practical.Secondly,the methods will not be limited on HJB equation,and I will try to find the corresponding new optimal control methods according to the current model and problem.Finally,I will try my best to give the very explicit expressions of optimal solutions.In the following,I will introduce the content of every Chapter and Section in detail.A simply introduction of stochastic control theory which will be used in the following chapters are given in the first chapter.Most of the contents are borrowed from[38]and[89].In Chapter 2,a series of optimal dividend problems are considered.Dividend is payments that the company give to the shareholder or the person who provides the initial surplus.Hence,in a sense,the total dividend payments represent company's benefit.Thus,how to pay dividends to maximize the total dividend payments is always one of the most hot topics in finance and insurance.This classical dividend problem has been solved very well and the explicit optimal solution has been given.These results show that the optimal dividend strategies under Brownian motion with drift and compound Poisson model are barrier strategy and band strategy,respectively.These results are obtained under the assumption that there are not transaction costs when the dividends are paid.But,in practice,to prohibit the continuous trade,some transaction costs are required when the dividend are paid.Even though it may be very few,it will affect the optimal strategy.It is obvious that barrier strategy and band strategy are no longer optimal in that case.Therefore,we need to look for the optimal strategy again. Due to the transaction costs,we have to consider not only the optimal dividend payments but also the optimal dividend time.This makes the optimal dividend problem more difficult than the classical optimal dividend problem.Only[28]gives the optimal proportional reinsurance and dividend strategy under the risk model of Brownian motion with drift. Now,there are still many unsolved optimal dividend problems with transaction costs.For example,the optimal dividend problem with the excess-of-loss reinsurance,the optimal dividend problem under the compound Poisson model,and the optimal dividend problem with solvency constrains.We will consider these problems in Section 2.1,Section 2.2 and Section 2.3.In the final section,we relax the restrictive assumptions made in the previous literature,and consider the optimal dividend problem under an extended family of diffusion processes.A completely different result from the previous results is obtained.In section 2.1,we consider the optimal excess-of-loss reinsurance and dividend strategy subject to transaction cost and taxes.Firstly,it is proved that the excess-of-loss reinsurance is better than the proportional reinsurance,that is,the corresponding total discounted expected dividend payments is larger.Then,by solving the corresponding quasi-variational inequality,we obtain the analytical solutions to the optimal return function and the optimal strategy.In[4],they introduced some associated functions to solve the HJB equation. Comparing with[4],the quasi-variational inequality for our problem is more difficult to solve.Moreover a different and more simple method is employed In Section 2.2,we study optimal dividend problem in the classical risk model when payments are subject to both transaction cost and taxes.The optimal dividend problem in the classical risk model is always one of the most difficult topics.The main reason is that the optimal equation has neither the boundary condition nor continuously differentiable solution.In this section, we construct the solution of quasi-variational inequality when the claims are exponentially distributed.We also find a formula for the expected time between dividends.The results show that,as the dividend tax rate decreases,it is optimal for the shareholders to receive smaller but more frequent dividend payments In Section 2.3,we consider a company where surplus follows a rather general diffusion process and whose objective is to maximize expected discounted dividend payments.With each dividend payment there are transaction costs and taxes and it is shown in[93]that under some reasonable assumptions,optimality is achieved by using a lump sum dividend barrier strategy,i.e.there is a upper barrier(?)~* and a lower barrier u~* so that whenever surplus reaches(?)~*,it is reduced to u~* through a dividend payment.However,these optimal barriers may be unacceptably low from a solvency point of view.It is argued that in that case one should still look for a barrier strategy,but with barriers that satisfy the given constraints.We propose a solvency constraint similar to that in[92];whenever dividends are paid out the probability of ruin within a fixed time T and with the same strategy in the future,should not exceed a predetermined levelε.It is shown how optimality can be achieved under this constraint, and numerical examples are given.
We firstly give lump barrier constrains.This constrains means that there are two barriers: upper barrier and low barrier.The company is allowed to pay dividend only when the reserve exceed the upper barrier and the reserve after dividend is not smaller than the low barrier.We consider the optimal dividend with lump barrier constrains in two steps. We firstly consider the optimal dividend problem with the low barrier constrains.Then, on the basis of the results obtained in the first step,the optimal dividend problem with the upper barrier constrains is considered.The corresponding quasi-variational inequalities are used in both two steps.But,in the second step,the corresponding quasi-variational inequality has no longer continuously differentiable solution.Hence,Ito Formula can not be used,and then the verification theorem does not hold.To tackle this difficulty,we skillfully use the theory of local time to prove that the constructed solution is equal to the optimal value function,and give the optimal strategy.This is the first time to give the verification theorem in non-continuously differentiable case.
In the following,we need to look for a lower barrier u_ε＞0 and an upper barrier u_εthat maximize expected discounted Dividends.Meanwhile,the solvency constraints are satisfied as follows:whenever capital is at u_ε,ruin within a fixed time T by following the lump barrier strategy(u_ε,u_ε) should not exceed a small,predetermined numberε.This problem is more difficult than that in[92]since we must look for a pair(u_ε,u_ε),not just a number u_ε.One issue is to find a fast method to calculate the ruin probability for a given lower and upper barrier,and we will show how we can adapt the Thomas algorithm for solving tridiagonal systems together with the Crank-Nicolson algorithm to solve the relevant partial differential equations.The section ends with numerical examples.
In Section 2.4,we consider a company where surplus follows an extended family of diffusion processes and whose objective is to maximize expected discounted dividend payments. An extended family of diffusion processes means that we relax restrictive assumptions. In this case,the property and form of the solution of quasi-variational inequality are different from that in previous literature.We provide the new idea to classify the solutions according to their properties,and then prove that there are three possibilities for the optimal dividend strategy:(1) the optimal dividend strategy is a lump sum dividend barrier strategy;(2) the optimal dividend strategy is a lump sum dividend band strategy; (3) the optimal dividend strategy does not exist.Among of them,the most difficult one is the case(2).Since,in this case,all of previous methods to construct the optimal value function and the optimal strategy are no longer valid.To tackle this difficulty,we consider two independent solutions instead of only one solution,and construct the optimal value function respectively in four intervals.Moreover,it is the first time to show clearly that the optimal dividend strategy is lump sum dividend band strategy and give the explicit expression of the strategy.
In Chapter 3,the problems of minimizing ruin probability and maximizing the exponential utility are studied.
In recent years,the optimization problems of maximizing the exponential utility and minimizing the probability of ruin have been studied by many authors.There exist relationships between the optimal strategies for maximizing the exponential utility and minimizing the probability of ruin.For the case of an ordinary investor(no external risk process) in the discrete time risk model,[37]found the optimal investment strategy that maximizes the exponential utility with utility function of the form u(x)=-e~(-mx).He conjectured that such a strategy also minimizes the probability of ruin for some value of m under the assumption that the investor was allowed to borrow an unlimited amount of money,and there was no risk-free interest rate.For the investors which face a random risk process correlated with the risky asset process.[24]show that,with zero interest rate,the investment strategy of maximizing the exponential utility also minimizes the probability of ruin for a specific value of m,which validates Ferguson'conjecture in a strong sense, [24]also show that the conjecture did not hold when the interest rate is positive.
Reinsurance is also an important technique for insurance companies seeking to control their risk.To make the utility and ruin analysis even more realistic,we consider incorporating the concept of proportional reinsurance into Browne's model.That is,under these two controls,we consider three optimization problems,namely the problem of maximizing the exponential utility of terminal wealth,the problem of minimizing the probability of ruin,and the problem of minimizing the expected discounted penalty of ruin.By the corresponding Hamilton-Jacobi-Bellman equations,explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risk-free interest rate,the results indicate that the optimal strategies for maximizing the exponential utility and minimizing the probability of ruin are equivalent.This validates Ferguson's conjecture under current model.
For the optimal investment problem,only the single risky asset(stock) is considered in previous literature.Actually,the insurance company,to decrease(increase) their risk (profit),will invest its wealth into multiple risk assets.Hence,we in section 3.2 consider that the insurance company invests its wealth in a financial market consisting of a risk-free asset and n risky assets.The risk model considered is the same as Section 3.1,The insurer is allowed to purchase proportional reinsurance as well as proportional reinsurance.Under the constraint of no-shorting,we consider two optimization problems:the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin.By solving the corresponding Hamilton-Jacobi-Bellman equations,explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risk-free interest rate,the results indicate that Ferguson's conjecture still hold under the current model.
In Section 3.3,following the same framework as section 3.2 we consider the excess-of-loss reinsurance instead of proportional reinsurance.First we show that the excess-of-loss reinsurance strategy is always better than the proportional reinsurance under two objective functions.Then,by solving the corresponding Hamilton-Jacobi-Bellman equations, the closed-form solutions of their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risky-free interest rate,the results indicate that Ferguson's conjecture still hold in excess-of-loss reinsurance case.
In Chapter 4.Mean-Variance problem in insurance is discussed.
The portfolio selection is to allocate the wealth into various assets to disperse risk and ensure the profit.In 1952,Markowitz uses the variance to measure the risk of stock,and provides mean-variance analysis methods for the portfolio selection.This open a prelude to modern finance.Mean-variance problem is that the investors want to find the optimal investment strategy to maximize the mean and minimize the variance.Mean-variance is not only the pioneering work of modern portfolio selection theory but also one of footstone of modern finance.
In recent years,finance and insurance market has started to link.In order to increase its profit or decrease its risk,the insurance company will invest the part of its reserve into financial market.There has been many literatures which considered the optimal investment.But the optimal criteria are still several classical actuarial variables such as dividend payments,the probability of ruin and so on.Mean-variance is rarely considered in insurance field,while it is one of the most important and popular conceptions in finance filed.As far as we know,only[120]consider the optimal investment problem under meanvariance criteria and[27]consider the optimal reinsurance/new business under meanvariance criteria.We will do the further research about mean-variance problem in this chapter.
In Section 4.1,we study optimal reinsurance/new business and investment(no-shorting) strategy for the mean-variance problem in two risk models:a classical risk model and a diffusion model.The problem is firstly reduced to a stochastic linear-quadratic(LQ) control problem with constraints.Then,the efficient frontiers and efficient strategies are derived explicitly by a verification theorem with the viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations,which is different from that given in[129].Furthermore,by comparisons,we find that the optimal solutions are identical under the two risk models Hitherto,people always model the claim processes by stochastic processes with Markovian. However,in most cases,the insurance claims often present long-range dependence: the behavior of the surplus of an insurer after a given time t not only depends on the information at t but also on the whole history up to time t.This phenomenon is not negligible and likely to have an impact on various issues such as solvency,pricing and optimal retention reinsured,etc.Therefore,fractional Brownian motion has been used recently to model the claims that an insurance company may face.Most of literature consider the ruin problem.Since the property of fractional Brownlan motion is difficult to describe, there are always not very explicit results.A fewer authors consider the optimal control under fractional Brownian motion.It is not only because the fractional Brownian motion is difficult to research,but also it is not Markov process,and then the HJB equation method can be not used.As far as we know,only[63]consider linear-quadratic(LQ) problem. However,most of results are obtained under the Markovian controlled system.Therefore, it is valuable in theory and practice to study the optimal control problems under the more extensive environment.In Section 4.2,we consider the optimal investment problem under mean-variance criteria for an insurer.The insurer's risk process is modelled by a classical risk process that is perturbed by a standard fractional Brownian motion with Hurst parameter H∈(1/2,1).Due to non-Markovian of fractional Brownian motion, the well-known HJB equation is not applied.However,stochastic calculus for fractional Brownian Motion introduced in[36]make fractional Brownian Motion have many similar properties to Brownian motion such as Zero mean,Girsanov Theorem.By these properties ,we find the method to solve the optimal problem under a classical risk process that is perturbed by a standard fractional Brownian motion.By virtue of an auxiliary process, the efficient strategy and efficient frontier are obtained.Moreover,when H→1/2+the results converge to the corresponding(known) results for standard Brownian motion.
Section 4.3 considers that the risk process is a compound Poisson process and the insurer can invest in a risk-free asset and multiple risky assets.We obtain the optimal investment policy under mean-variance criteria using the stochastic linear quadratic(LQ) control theory with no-shorting constraint.Then the efficient strategy and efficient frontier are derived explicitly by a verification theorem provided in Section 4.1.
In Chapter 5,we consider optimal control problems when risk processes modeled by (fractional) Brownian motion with drift.
In Section 5.1,we model the surplus process as a compound poisson process perturbed by Brownian motion,and allow the insurer to ask its customers for input to minimize the distance from some prescribed target path and the total discounted cost on a fixed interval. The problem reduces to a version of linear quadratic regulator under jump diffusion process.It is treated by three methods:dynamic programming,completion of square and stochastic maximum principle.The analytic solutions to the optimal control and the corresponding optimal value function are obtained.In this section,we introduce the methods of completion of square and stochastic maximum principle into the insurance filed.These two methods have many differences from the HJB equation method.Especially,the original process is no longer required to be Markov process.Hence,these give some inspiration to solve the optimal problem under Non-Markov process.The following Section 5.2 is a good example.We in this section consider a classical risk model perturbed by a standard fBm with Hurst parameter H∈(1/2,1).The criteria is the same as section 5.1.By using the completion of squares method,the expression of the optimal value function and the corresponding optimal control are derived.In addition,the method in Section 4.2 is also inspired by section 5.1.
In Chapter 6,some other optimal control problems are investigated.
There has been many literatures which consider the optimal reinsurance,such as[24], [58],[4].However,all of them only consider the insurance company.In practice,if we only consider the insurance company,the optimal reinsurance strategy obtained is often unacceptable for the reinsurance company.This alarms us that there are two parties in the reinsurance contract,their interest collide with each other.Therefore,we need to make a reciprocal reinsurance treaty.This is a game problem.[21]firstly use the game theory to solve the optimal reinsurance problem in insurance filed.The optimal criteria is maximizing the expected exponential utility.Since then,[13],[117]make some research for risk exchanges and reinsurance markets.However,in these papers,only the single period model is considered.It will obviously be of interest to extend the single model to multi-period,or a continuous time model.In this sectioin,we will consider a continuous time model.In this case,the interest of the insurance company and the reinsurance company always collide with in the whole time interval,which is different from the case of the single period.Hence,to solve the optimal problem,we need to look for the new and different methods.In this section,we solve the problem in two steps.Firstly we treat the Pareto optimal problem about the terminal wealth.Then we find the reinsurance strategy to copy the Pareto optimal terminal wealth.By this method,we show that the optimal Pareto reinsurance in continuous time is proportional reinsurance.Moreover,it is proved that the core of the cooperative reinsurance game is non-empty.This section intend to provide one method to solve the optimal problem in continuous time reinsurance market. Many other problems are still unsolved such as the optimal dividend,ruin probability, mean-variance.
Section 6.2 deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modeled as an Ornstein-Uhlenbeck process.Here,only the stock price and interest rate can be observable for an investor.It is reduced to a partially observed stochastic control problem.Combining the filtering theory with dynamic programming approach,explicit representations of the optimal value functions and corresponding optimal strategies are derived.Moreover the closed-form solutions are provided in two cases of exponential utility and logarithmic utility.In particular,logarithmic utility is considered under the restriction of short-selling and borrowing.[76]uses martingale method to obtain the optimal strategy.Here,the investor is not allowed to short-sell and borrow,and a simpler method is employed to solve the problem.Moreover,the maximal logarithmic utility is given in the case of a single stock.
In section 6.3,the basic claim process is assumed to follow a Brownian motion with drift.In addition,we allow the insurer to invest in a risky asset and to purchase cheap proportional reinsurance.Under these two controls,we consider the problem of minimizing expected time to reach a given capital before ruin.By using the Hamilton-Jacobi-Bellman equation,explicit expressions for optimal value function and optimal strategy are obtained.
By the HJB equation,Quasi-variational inequalities,partial differential equation,completion of squares,Pareto optimal,fractional Brownian motion theory,my doctoral dissertation will be devoted to solving optimal dividend with transaction costs and solvency constrains,optimal multi-asset investment,Pareto optimal reinsurance strategy and optimal control problem under fractional Brownian motion insurance model.In the following, we will give the innovation and novelty of every section.
1.The innovation in methods
(1) In section 2.3,there are two innovations in methods:(a) the verification theorem about non-continuously differentiable solution;(b) to find a fast method to calculate the ruin probability for a given lower and upper barrier,and show how to adapt the Thomas algorithm for solving tridiagonal systems together with the Crank-Nicoison algorithm to solve the relevant partial differential equations.
(2) There are two innovations in section 2.4:(a) to give a new idea to classify the solutions of quasi-variational inequality;(b) to give a new method to construct the optimal value function and the optimal strategy.
(3) Section 4.1 gives the verification theorem about a class of viscosity solution.
(4) Section 6.1 give the method to solve the Pareto optimal reinsurance in continuous time.
(5) In section 6.2,it is shown that how to use the HJB equation method to solve the optimal problem with partial information.
2.The innovation in skill to solve optimal equation and construct the solution.
This part mainly extends some classical and excellent work,but it is not the simple extension on calculation.
(1) Section 2.1 extend the work of[28](published in Mathematical finance),and consider the optimal excess-of-loss reinsurance and dividend with transaction costs and taxes. The corresponding quasi-variational inequality is more complicated than that under proportional reinsurance.In section 2.1,we give the different method to construct the solution.
(2) In section 2.2,the optimal dividend with transaction costs and taxes is considered under the classical risk process.The construction of the solution is different from Section 2.1 and[28].
(3)[24]is known as the pioneering work on the combination of insurance theory and control theory.In third chapter,we extend[24]'s work by incorporating the reinsurance into the model in[24].In this case,the more technical method to construct the solution is needed.
3.To introduce new control method into insurance filed.
Section 5.1 use three methods of the HJB equation,stochastic maximum principle, completion of square to solve the optimal problem.
4.To consider the optimal control problem under non-markovian process
The fractional Brownian motion is a long-range dependence,non-markovian process. It can model many phenomenon closer to practice.However,since its property is difficult to describe,there are only a few literatures which consider fractional Brownian motion risk model.We in section 4.2 and section 5.2 consider the optimal problem under a classical risk model perturbed by a standard fBm.We find the method to solve the problem and give the explicit solution.
Another innovation is that we give the explicit solution for all of problem.
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