本文選題：投資組合 + 隨機(jī)LQ控制　； 參考：《西安工程大學(xué)》2012年碩士論文

【摘要】：由Markowitz提出的證券投資組合模型是現(xiàn)代投資理論的基石，論文主要內(nèi)容是對(duì)帶有Markov調(diào)制參數(shù)的證券投資組合選擇模型的研究，論文首先研究了帶有Markov調(diào)制參數(shù)的離散時(shí)間LQG完全狀態(tài)信息情況，其次，給出了帶有Markov調(diào)制參數(shù)的隨機(jī)LQ控制的最優(yōu)控制策略及其應(yīng)用，此外，論文研究了金融市場(chǎng)上處于競(jìng)爭(zhēng)狀態(tài)下小投資者在有利與不利條件下證券投資組合的選擇模型，給出了其最優(yōu)投資組合策略。最后，論文還研究了帶有Markov調(diào)制參數(shù)的條件破產(chǎn)概率問題，給出了條件破產(chǎn)概率滿足的一個(gè)偏微分方程，并在隨機(jī)利率條件下作了進(jìn)一步深入研究。 全文主要內(nèi)容共分為六章： (1)第一章介紹了帶有Markov調(diào)制參數(shù)的投資組合選擇問題的研究意義及研究現(xiàn)狀，并簡單介紹了本論文的所解決的問題和主要研究內(nèi)容。 (2)第二章主要介紹了本論文研究過程中應(yīng)用的主要工具，涉及的重要引理和定義。 (3)第三章研究內(nèi)容是帶有Markov調(diào)制參數(shù)的離散時(shí)間LQG完全狀態(tài)信息情況，論文將Markov體制轉(zhuǎn)換引入離散時(shí)間線性二次型高斯（LQG）完全狀態(tài)信息情況原模型中，主要是考慮到現(xiàn)實(shí)生活中宏觀經(jīng)濟(jì)條件下體制轉(zhuǎn)換會(huì)對(duì)狀態(tài)因素產(chǎn)生影響，引入有限Markov鏈更具有現(xiàn)實(shí)意義，最后論文利用貝爾曼動(dòng)態(tài)規(guī)劃法給出其最優(yōu)控制策略。 (4)第四章研究的是帶有Markov調(diào)制參數(shù)的隨機(jī)LQ框架，在隨機(jī)LQ控制模型中考慮狀態(tài)因素的影響，論文將該模型推廣到系統(tǒng)狀態(tài)為跳躍-擴(kuò)散過程的隨機(jī)LQ控制，并引入跳-擴(kuò)散的隨機(jī)Riccati方程及連續(xù)時(shí)間Markov鏈，然后應(yīng)用隨機(jī)變分法求得問題的最優(yōu)反饋控制策略。最后論文運(yùn)用該模型去處理了金融中借貸利率不等和非自融條件下的最優(yōu)投資組合問題和套期保值問題，分別得到了他們的有效投資組合策略。 (5)第五章在如下的金融市場(chǎng)上研究了投資策略的選擇問題：設(shè)金融市場(chǎng)中有一種無風(fēng)險(xiǎn)證券（假設(shè)為債券），兩種風(fēng)險(xiǎn)證券（假設(shè)為股票），投資者A,B均只能在一種風(fēng)險(xiǎn)證券與無風(fēng)險(xiǎn)證券上進(jìn)行投資，以兩個(gè)投資者財(cái)富之比為對(duì)策狀態(tài)變量，構(gòu)成零和隨機(jī)對(duì)策。論文在風(fēng)險(xiǎn)價(jià)格表達(dá)式中引入了Markov鏈，研究了投資者在有利與不利條件下證券組合選擇的競(jìng)爭(zhēng)模型,最終得到了其最優(yōu)投資組合策略。 (6)第六章研究了條件破產(chǎn)概率問題，論文將Markov鏈引入模型，，并將盈余過程推廣到跳-擴(kuò)散過程，應(yīng)用It'o公式和鞅方法得到了有限條件破產(chǎn)概率滿足的一個(gè)偏微分方程，然后論文在隨機(jī)利率條件下做了進(jìn)一步研究。
[Abstract]:The portfolio model proposed by Markowitz is the cornerstone of modern investment theory. The main content of this paper is to study the portfolio selection model with Markov modulation parameters. Firstly, the complete state information of discrete time LQG with Markov modulation parameters is studied. Secondly, the optimal control strategy of stochastic LQ control with Markov modulation parameters and its application are given. This paper studies the portfolio selection model of small investors in the competitive state of financial market under favorable and unfavorable conditions, and gives the optimal portfolio strategy. Finally, the conditional ruin probability problem with Markov modulation parameters is studied, and a partial differential equation of conditional ruin probability is given, which is further studied under the condition of stochastic interest rate. The main contents of this paper are divided into six chapters: (1) Chapter 1 introduces the significance and research status of portfolio selection with Markov modulation parameters. And briefly introduced the problems solved in this paper and the main research content. (2) the second chapter mainly introduces the main tools used in the research process of this paper. (3) in Chapter 3, the discrete time LQG complete state information with Markov modulation parameters is studied. In this paper, Markov system transformation is introduced into the original model of discrete time linear quadratic Gao Si (LQG) complete state information. It is more practical to introduce finite Markov chain. In the end, the optimal control strategy is given by using Berman dynamic programming method. (4) in chapter 4, the stochastic LQ frame with Markov modulation parameters is studied. Considering the influence of state factors in the stochastic LQ control model, the model is extended to the stochastic LQ control in which the state of the system is a hopping diffusion process, and the stochastic Riccati equation of hopping diffusion and the continuous time Markov chain are introduced. Then the optimal feedback control strategy of the problem is obtained by using the stochastic variational method. Finally, this paper uses the model to deal with the optimal portfolio problem and hedging problem under the condition of unequal lending rate and non-self-financing. (5) in chapter 5, we study the choice of investment strategies in the following financial markets: let there be a risk-free security in the financial market (assuming bonds), two kinds of investment strategies are obtained. (5) in the fifth chapter, we study the choice of investment strategies in the following financial markets: let there be a risk-free security in the financial market (assuming bonds). Risk securities (assuming stocks), investors AWAB can only invest in a risky and risk-free securities, Taking the ratio of the two investors' wealth as the game state variable, the zero sum stochastic game is formed. In this paper, Markov chain is introduced into the expression of risk price, and the competitive model of portfolio selection for investors under favorable and unfavorable conditions is studied. Finally, the optimal portfolio strategy is obtained. (6) in Chapter 6, the conditional ruin probability problem is studied. Markov chain is introduced into the model, and the surplus process is extended to the jump-diffusion process. In this paper, we obtain a partial differential equation with finite ruin probability by using ITO formula and martingale method, and then we study it further under the condition of stochastic interest rate.
【學(xué)位授予單位】：西安工程大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2012
【分類號(hào)】：F830.91;F224

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