本文選題：CEV模型 + 最優(yōu)化策略��； 參考：《河北師范大學(xué)》2014年博士論文

【摘要】：常方差彈性系數(shù)(CEV)模型是幾何布朗運(yùn)動(dòng)(GBM)模型的一個(gè)推廣.它最早常用于計(jì)算期權(quán)等資產(chǎn)的定價(jià),敏感性分析和隱含波動(dòng)率等問題,近年來, CEV模型開始用于最優(yōu)化投資問題.本文主要討論了有限時(shí)間水平下CEV模型在保險(xiǎn)和金融中兩大方面的應(yīng)用,一是一般投資人的最優(yōu)消費(fèi)和投資問題;一是保險(xiǎn)人的最優(yōu)再保險(xiǎn)和投資問題. 在投資行為中,財(cái)務(wù)顧問通常推薦年輕的投資人相對(duì)于年老的投資人來說應(yīng)該投入較多的資產(chǎn)到風(fēng)險(xiǎn)資產(chǎn)中,這是因?yàn)槟贻p的投資人擁有更長(zhǎng)的時(shí)間來進(jìn)行投資行為.這說明投資者所面臨不的剩余時(shí)間不同,他們的投資策略就不應(yīng)相同.為了切合實(shí)際,我們考慮的是有限時(shí)間水平.在有限時(shí)間水平下,從不同的時(shí)刻出發(fā),剩余時(shí)間也不同,這樣所得到的策略不再像無限時(shí)間水平下那樣與時(shí)間無關(guān). 投資與消費(fèi)是社會(huì)總需求的重要組成部分,投資與消費(fèi)關(guān)系又是經(jīng)濟(jì)理論和實(shí)證研究中最重要、最復(fù)雜的關(guān)系之一.對(duì)投資人來說,他把資產(chǎn)進(jìn)行投資的目的就是增強(qiáng)自己的消費(fèi)能力.本文在第三章研究了最優(yōu)化消費(fèi)和投資問題,投資人把總資產(chǎn)的一部分投資到無風(fēng)險(xiǎn)資產(chǎn)(存入銀行或購買債券),剩下的部分投入到風(fēng)險(xiǎn)資產(chǎn)(股票或基金),同時(shí)投資者還進(jìn)行消費(fèi),且以最大化直到固定終端時(shí)刻的期望指數(shù)消費(fèi)效用(指數(shù)效用函數(shù)是唯一一個(gè)滿足“零效益”原則的效用函數(shù))為目的,尋求最優(yōu)化消費(fèi)和投資策略.本章首先利用動(dòng)態(tài)規(guī)劃的方法建立了本問題的HJB方程,然后通過冪變換和變量替換等手段,求得了最優(yōu)消費(fèi)和投資策略的明確表達(dá)式. 保險(xiǎn)人的最優(yōu)再保險(xiǎn)和投資問題是保險(xiǎn)問題和金融活動(dòng)的一個(gè)很好的結(jié)合,具有很強(qiáng)的實(shí)用性和研究?jī)r(jià)值.再保險(xiǎn)是保險(xiǎn)公司規(guī)避風(fēng)險(xiǎn)的一個(gè)有力手段,投資則是保險(xiǎn)公司提高收益的重要保障.本文的第四章考慮了保險(xiǎn)公司的超額損失再保險(xiǎn)和投資策略.保險(xiǎn)公司收取保費(fèi),就要承擔(dān)索賠.為了避免大額索賠出現(xiàn)的危險(xiǎn),于是進(jìn)行再保險(xiǎn);同時(shí)為了確保資金的保值與增值,需要把手中的資金進(jìn)行一種無風(fēng)險(xiǎn)資產(chǎn)和一種風(fēng)險(xiǎn)資產(chǎn)的投資分配.針對(duì)最大化固定終端時(shí)刻指數(shù)效用的目標(biāo),利用動(dòng)態(tài)規(guī)劃的方法求出了相應(yīng)的HJB方程,然后通過求解HJB方程,得到了最優(yōu)的超額損失再保險(xiǎn)和投資策略. 第五章和第六章研究的是比例再保險(xiǎn)和投資行為的結(jié)合.其中第五章關(guān)心的是均值-方差問題,也就是在達(dá)到預(yù)期效益的同時(shí)要確保風(fēng)險(xiǎn)最小化.這是一個(gè)以期望收益和風(fēng)險(xiǎn)（即均值和方差）為雙目標(biāo)的最優(yōu)化問題.本章采用的路線是首先通過拉格朗日乘數(shù)的引入把兩個(gè)目標(biāo)融合在一起,化為一個(gè)單目標(biāo)問題,并建立其HJB方程,然后求解HJB方程,得到了帶有拉格朗日乘數(shù)的最優(yōu)策略,最后借由Lagrange對(duì)偶定理,求出了原問題的有效邊界和有效策略. 在前面幾章的研究中,為了突出問題的重點(diǎn),都假設(shè)了投資者只投資到一種無風(fēng)險(xiǎn)資產(chǎn)和一種風(fēng)險(xiǎn)資產(chǎn).第六章為了更加貼近實(shí)際,不再只考慮單一風(fēng)險(xiǎn)資產(chǎn),而是投資于多種風(fēng)險(xiǎn)資產(chǎn),同時(shí)結(jié)合比例再保險(xiǎn),目標(biāo)依然是最大化固定終端時(shí)刻的期望指數(shù)效用.本章在建立了相應(yīng)的HJB方程的基礎(chǔ)上,借由冪變換和變量替換的技巧,得到了其最優(yōu)比例再保險(xiǎn)和投資策略的明確表達(dá)式.
[Abstract]:The constant difference elasticity coefficient (CEV) model is a generalization of the geometric Brown motion (GBM) model. It is first used to calculate the pricing, sensitivity analysis and implicit volatility of options and other assets. In recent years, the CEV model has been used to optimize the investment problem. This paper mainly discusses the CEV model under the finite time level in the insurance and finance. The two major applications are the optimal consumption and investment of the general investors. The first is the insurer's optimal reinsurance and investment.
In investment behavior, financial advisers usually recommend that young investors should invest more assets in risky assets than older investors because young investors have longer time to invest. This indicates that investors are faced with different remaining time and their investment strategies are not appropriate. In the same way, in order to be practical, we consider a finite time level. At a limited time, the remaining time is different from different moments, so the strategy is no longer independent of time as it is at an infinite time level.
Investment and consumption are an important part of the total social demand. The relationship between investment and consumption is one of the most important and most complex relationships in economic theory and empirical research. For investors, the purpose of investing the assets is to enhance their consumption ability. In the third chapter, the problem of optimal consumption and investment and investment are studied in this paper. A person investing part of a total asset into a riskless asset (deposited in a bank or buying a bond), and the remaining part is invested in a risky asset (stock or fund), and the investor is also consumed and maximized the expected index consumption utility until the fixed terminal moment (the index utility function is the only one that meets the "zero benefit" principle. In this chapter, the HJB equation of this problem is established by dynamic programming, and then the explicit expression of the optimal consumption and investment strategy is obtained by means of power transformation and variable substitution.
The problem of the best reinsurance and investment of the insurer is a good combination of insurance and financial activities. It has strong practicability and research value. Reinsurance is a powerful means for insurance companies to avoid risk. Investment is an important guarantee for the insurance company to improve its income. The fourth chapter of this article has considered the excess loss of insurance companies. In order to avoid the risk of large amount of claims, the insurance company will reinsurance, and in order to ensure the value and value of the capital, it is necessary to allocate the funds in the hands of a riskless asset and a kind of venture capital. The goal of exponential utility is to obtain the corresponding HJB equation by using the method of dynamic programming, and then the optimal excess loss reinsurance and investment strategy are obtained by solving the HJB equation.
The fifth chapter and the sixth chapter deals with the combination of proportional reinsurance and investment behavior. The fifth chapter is concerned with the mean variance problem, which is to minimize the risk while achieving the expected benefit. This is an optimization problem with expected returns and risks (mean and variance). The route adopted in this chapter is the first. First, the two targets are fused together by the introduction of the Lagrange multiplier and transformed into a single objective problem, and its HJB equation is established. Then the HJB equation is solved, and the optimal strategy with the Lagrange multiplier is obtained. Finally, the effective boundary and effective strategy of the original problem are obtained by the Lagrange duality theorem.
In the study of the previous chapters, in order to highlight the focus of the problem, we assume that investors only invest in a riskless asset and a risk asset. The sixth chapter, in order to be closer to the reality, not only consider a single risk asset, but also invest in a variety of risky assets, and combine the proportional reinsurance, the goal is to maximize the fixed terminal. In this chapter, based on the establishment of the corresponding HJB equation, this chapter obtains the explicit expression of the optimal proportional reinsurance and investment strategy by using the technique of power transformation and variable substitution.

【學(xué)位授予單位】：河北師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2014
【分類號(hào)】：F830;F840;O211.6;F224

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