金融决策问题的在线策略及其竞争分析
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摘要
理论计算机学科兴起的在线学习算法的特点是不对未来输入序列作任何统计假设和仅根据已获得信息进行决策，这为许多具有在线特点的金融决策问题的研究提供了新思路和方法。该文首先论述了设备资产租赁、单一风险资产投资和多种风险资产投资等金融决策问题的研究背景和意义；回顾了决策理论的起源与发展并论述了国内外研究现状。在此基础上，结合实际情况，应用传统竞争比分析方法进一步研究了设备资产在线租赁策略和风险策略。探讨了单一风险资产的在线投资策略，分别给出了基于风险厌恶型的保守策略和基于预期的风险策略。提出了基于线性学习函数的多种风险资产在线投资策略并分析了其竞争性能；将弱集成在线学习算法应用到多种风险资产在线投资，给出了具体在线决策策略。设备资产在线租赁、单一风险资产在线投资和多种风险资产在线投资的研究方法既有区别又有明显的关联。它们的输入都具有在线性，都根据已获得的历史数据进行决策，都需要选择一个标准来衡量所给在线策略的竞争性能。设备资产在线租赁是最简单最基本的风险决策问题之一，可将它的研究方法推广应用到单一风险资产在线投资和多种风险资产在线投资，而单一风险资产在线投资又是多种风险资产在线投资的一种特例。具有创新性的研究成果主要包括以下几个方面：
1.研究了受市场供求关系变化、自由市场灵活性和间隔使用设备等因素影响的在线租赁问题，建立了对应的在线租赁模型，用竞争分析方法得到了相应的在线租赁策略及其竞争性能分析。
设备的购买价格和租赁费用往往随着需求量变动而发生改变；同时随着科技的发展，设备的品牌层出不尽；并且市场利率会对阶段性使用设备的在线策略产生影响。基于此，本文研究了设备价格离散可变的在线租赁策略及其竞争性能，探讨了多种设备获得方式的可折旧设备租赁的在线随机选择策略，给出了受市场利率影响的两阶段在线租赁策略的竞争性能分析。
2.针对大型设备的可折旧性和二手市场的存在性，建立了可折旧设备在线租赁模型，得到了它的确定性策略、随机性策略和基于预期的风险策略。
关于设备租赁问题的大多数研究都是建立在设备一旦购买便永久可以使用的基础上，而并没有对可折旧设备在线租赁问题的竞争策略进行系统的研究，本文在可折旧设备在线租赁模型建立的基础上，首先给出了它的确定性策略和随机性策略，进一步给出了它在确定性预期下的风险策略和在概率预期下的风险策略，最后并将确定性预期下的风险策略运用到了两阶段在线设备更新问题中。结果表明折旧的引入使得确定性策略和随机性策略的竞争性能有所提高，市场利率的引入使得最优确定性竞争比有所减少，却使得最优随机性竞争比有所增大；风险策略的最优约束竞争比减少幅度与风险容忍度成正比；结果也表明了概率预期下的风险策略扩展了确定性预期下的风险策略。
3.研究了风险资产收益率线性变动和对数变动情形下的单一风险资产投资问题，建立了阶段性预期下单一风险资产在线投资的风险补偿模型，分别得到了单一风险资产在线投资的保守策略和风险策略。
实际风险资产收益率波动的形式多样，从风险厌恶角度本文分别给出了每期风险资产收益率变化服从线性波动和对数波动的保守投资策略。数值算例说明了基于风险厌恶型的单一风险资产在线投资策略更适合于波动平稳的风险资产收益率序列。已有的风险策略是基于风险资产收益率变化点的预期，本文给出了阶段性预期下单一风险资产投资的风险策略及其竞争性能分析，数值算例同时也表明了阶段性预期下最优约束竞争比的平均改善幅度与投资者的风险容忍度密切相关并且成正比。
4.研究了多种风险资产的在线投资问题，建立了基于线性学习函数的在线投资组合策略，得到了一类泛证券投资组合策略；并建立了基于弱集成学习算法的多种风险资产在线投资模型，得到了竞争性能较好的在线投资策略。
提出了基于线性学习函数的多种风险资产在线投资策略，其中线性函数的系数是一个与风险资产收益率有关的区间中点。用相对熵函数定义两个投资组合向量之间的距离，证明了所给出的在线投资组合是泛证券投资组合。当线性系数取值于区间的任意一点时，得到了一类泛证券投资组合。探讨了弱集成算法在多种风险资产在线投资组合选择中的应用。首先将弱集成学习算法应用到投资于单个风险资产的专家策略，得到了多种风险资产在线投资策略WAAS。考虑到泛证券投资组合策略是相对于最优定常再调整策略而言的，进一步将弱集成算法应用到定常再调整策略，得到了多种风险资产在线投资策略WAAC。理论和数值算例都说明了WAAS策略的收益与表现最好的风险资产的收益相当；WAAC策略的收益与最优定常再调整策略的收益相当。
The online algorithm developed in theoretical computer science makes no statis-tical assumption about the future inputs and makes decision only based on availabledata. These properties provide new ideas and methods for the research of many financialdecision-making problems whose decision strategies preserve online property. This thesisfirst states the background and significance of the research about equipment asset leas-ing, single risk asset investment and various risk assets investment; and presents a reviewabout the origin and development of decision-making theory and the related research’sstatus. Based on that, this thesis, considering actual situation, further applies the tra-ditional competitive analysis to study the online leasing strategy and risk strategies ofthe equipment asset. This thesis explores the online investment strategy of single riskasset, and provides the conservative strategy that bases on risk aversion and the riskstrategy that bases on forecast, respectively. Based on linear learning function, this the-sis presents online investment strategy of various risk assets and provides its competitiveperformance analysis. This thesis applies the weak aggregating algorithm to online in-vestment of various risk assets and obtains the specific strategies. There exists diferencebetween the research methods of online leasing of equipment asset, online investment ofsingle risk asset and online investment of various risk assets. However, the relationshipis also obvious since their inputs are both obtained in online manner, their decisions areboth decided according to historical data and they both need to choose a benchmark toevaluate the competitive performance of the presented online strategies. The online leas-ing of equipment asset is one of the simplest and basic risk decision-making problems. Itsresearch methods can be generalized to online investment of single risk asset and variousrisk assets, and online investment of single risk asset is a special case of online investmentof various risk assets. The main creative results of the research are as follows:
1. Study the online leasing problems that are afected by the change of supply-demand’s relationship, the free market’s flexibility and the discontinuous usage and con-struct the corresponding online leasing models and obtain their competitive performanceanalysis by using competitive analysis.
The purchase cost and rent cost of an equipment usually change with the demand; inthe same time, as the development of technology, the brands of equipment are various; andthe interest rate will afect the online strategy of discontinuous use of equipment. Basedon these situations, this thesis studies the online leasing strategy with discrete change of equipment’s price and its competitive performance, explores the online randomizedselection strategy of the depreciable equipment with many obtaining’s methods, andprovides the two period online leasing strategy that is afected by interest rate, and itscompetitive performance analysis.
2. As the large scale equipment is depreciable and the secondary market exists,this thesis constructs the online leasing model of depreciable equipment and obtains itsdeterministic strategy, randomized strategy and risk strategies that base on forecast.
Most of previous research on equipment leasing problems base on that the equipmentcan be used for ever once it is bought, and did not provide a system study on onlineleasing strategy of depreciable equipment. This thesis on the basis of the model of onlineleasing of depreciable equipment, first provides its deterministic strategy and randomizedstrategy; then gives its risk strategies based on certain forecast and probability forecast,respectively; Lastly, this thesis apply the risk strategy that obtained under certain forecastto two period online replacement problem. The results show that the introduction ofdepreciation has enabled the competitive performances of both deterministic strategyand randomized strategy get improved, and the introduction of interest rate made thethe competitive ratio of deterministic strategy get decreased while that of randomized getincreased; the optimal competitive ratio of risk strategy is decreased, and the extent towhich is related to risk tolerance; in the same time, the results also show the risk strategyunder probability forecast has generalized the one under certain forecast.
3. This thesis studies the online investment of single risk asset under the situationsthat the return rate are linear-growth and log-growth; constructs the risk-reward modelwith period forecast for online investment of one risk asset. This paper then obtains theconservative strategy and risk strategy for online investment of single risk asset.
The fluctuation of real return rate is various. Based on the risk aversion, this thesispresents the conservative strategies for the return rates that are linear-growth and log-growth, respectively. Numerical analysis shows the conservative strategy that bases onrisk aversion is more suitable for the stable return rate series. The existed risk strategyis based on the forecast of one point of the return rate series; this thesis based on theperiod forecast provides risk strategy for online investment of single risk asset and itscompetitive performance. The numerical analysis results show the average improvementof optimal restricted competitive ratio is closely related to the investor’s risk toleranceand is proportional to the risk tolerance.
4. This paper studies the online investment of various risk assets and constructs theonline portfolio selection strategy that bases on linear learning function, and constructsonline investment model for various risk assets based on the weak aggregating algorithmand obtains online invest strategy that has better competitive performance.
Based on the on-line learning of linear function, this thesis presents on-line invest- ment strategy for various risk assets, where the linear coefcient is the middle of aninterval. And the interval is related to risk assets’ returns. Using the relative entropy asthe distance function of two portfolios, this paper proves that the provided on-line portfo-lio is universal. The linear factor can be any point of the interval and a class of universalportfolio is obtained. This thesis explores the application of weak aggregating algorithmto online investment of various risk assets. First, apply it to the experts’ strategies thatinvest on single risk asset and obtain the portfolio selection algorithm WAAS. Considerthe definition of universal portfolio is based on BCRP, so this thesis further applies theweak aggregating algorithm to CRP, and obtains the portfolio selection algorithm WAAC.Theoretical and numerical results illustrate the return of WAAS strategy is as good asthat of the best risk asset, and the return of the WAAC is as good as that of the BCRP.
1.研究了受市场供求关系变化、自由市场灵活性和间隔使用设备等因素影响的在线租赁问题，建立了对应的在线租赁模型，用竞争分析方法得到了相应的在线租赁策略及其竞争性能分析。
设备的购买价格和租赁费用往往随着需求量变动而发生改变；同时随着科技的发展，设备的品牌层出不尽；并且市场利率会对阶段性使用设备的在线策略产生影响。基于此，本文研究了设备价格离散可变的在线租赁策略及其竞争性能，探讨了多种设备获得方式的可折旧设备租赁的在线随机选择策略，给出了受市场利率影响的两阶段在线租赁策略的竞争性能分析。
2.针对大型设备的可折旧性和二手市场的存在性，建立了可折旧设备在线租赁模型，得到了它的确定性策略、随机性策略和基于预期的风险策略。
关于设备租赁问题的大多数研究都是建立在设备一旦购买便永久可以使用的基础上，而并没有对可折旧设备在线租赁问题的竞争策略进行系统的研究，本文在可折旧设备在线租赁模型建立的基础上，首先给出了它的确定性策略和随机性策略，进一步给出了它在确定性预期下的风险策略和在概率预期下的风险策略，最后并将确定性预期下的风险策略运用到了两阶段在线设备更新问题中。结果表明折旧的引入使得确定性策略和随机性策略的竞争性能有所提高，市场利率的引入使得最优确定性竞争比有所减少，却使得最优随机性竞争比有所增大；风险策略的最优约束竞争比减少幅度与风险容忍度成正比；结果也表明了概率预期下的风险策略扩展了确定性预期下的风险策略。
3.研究了风险资产收益率线性变动和对数变动情形下的单一风险资产投资问题，建立了阶段性预期下单一风险资产在线投资的风险补偿模型，分别得到了单一风险资产在线投资的保守策略和风险策略。
实际风险资产收益率波动的形式多样，从风险厌恶角度本文分别给出了每期风险资产收益率变化服从线性波动和对数波动的保守投资策略。数值算例说明了基于风险厌恶型的单一风险资产在线投资策略更适合于波动平稳的风险资产收益率序列。已有的风险策略是基于风险资产收益率变化点的预期，本文给出了阶段性预期下单一风险资产投资的风险策略及其竞争性能分析，数值算例同时也表明了阶段性预期下最优约束竞争比的平均改善幅度与投资者的风险容忍度密切相关并且成正比。
4.研究了多种风险资产的在线投资问题，建立了基于线性学习函数的在线投资组合策略，得到了一类泛证券投资组合策略；并建立了基于弱集成学习算法的多种风险资产在线投资模型，得到了竞争性能较好的在线投资策略。
提出了基于线性学习函数的多种风险资产在线投资策略，其中线性函数的系数是一个与风险资产收益率有关的区间中点。用相对熵函数定义两个投资组合向量之间的距离，证明了所给出的在线投资组合是泛证券投资组合。当线性系数取值于区间的任意一点时，得到了一类泛证券投资组合。探讨了弱集成算法在多种风险资产在线投资组合选择中的应用。首先将弱集成学习算法应用到投资于单个风险资产的专家策略，得到了多种风险资产在线投资策略WAAS。考虑到泛证券投资组合策略是相对于最优定常再调整策略而言的，进一步将弱集成算法应用到定常再调整策略，得到了多种风险资产在线投资策略WAAC。理论和数值算例都说明了WAAS策略的收益与表现最好的风险资产的收益相当；WAAC策略的收益与最优定常再调整策略的收益相当。
The online algorithm developed in theoretical computer science makes no statis-tical assumption about the future inputs and makes decision only based on availabledata. These properties provide new ideas and methods for the research of many financialdecision-making problems whose decision strategies preserve online property. This thesisfirst states the background and significance of the research about equipment asset leas-ing, single risk asset investment and various risk assets investment; and presents a reviewabout the origin and development of decision-making theory and the related research’sstatus. Based on that, this thesis, considering actual situation, further applies the tra-ditional competitive analysis to study the online leasing strategy and risk strategies ofthe equipment asset. This thesis explores the online investment strategy of single riskasset, and provides the conservative strategy that bases on risk aversion and the riskstrategy that bases on forecast, respectively. Based on linear learning function, this the-sis presents online investment strategy of various risk assets and provides its competitiveperformance analysis. This thesis applies the weak aggregating algorithm to online in-vestment of various risk assets and obtains the specific strategies. There exists diferencebetween the research methods of online leasing of equipment asset, online investment ofsingle risk asset and online investment of various risk assets. However, the relationshipis also obvious since their inputs are both obtained in online manner, their decisions areboth decided according to historical data and they both need to choose a benchmark toevaluate the competitive performance of the presented online strategies. The online leas-ing of equipment asset is one of the simplest and basic risk decision-making problems. Itsresearch methods can be generalized to online investment of single risk asset and variousrisk assets, and online investment of single risk asset is a special case of online investmentof various risk assets. The main creative results of the research are as follows:
1. Study the online leasing problems that are afected by the change of supply-demand’s relationship, the free market’s flexibility and the discontinuous usage and con-struct the corresponding online leasing models and obtain their competitive performanceanalysis by using competitive analysis.
The purchase cost and rent cost of an equipment usually change with the demand; inthe same time, as the development of technology, the brands of equipment are various; andthe interest rate will afect the online strategy of discontinuous use of equipment. Basedon these situations, this thesis studies the online leasing strategy with discrete change of equipment’s price and its competitive performance, explores the online randomizedselection strategy of the depreciable equipment with many obtaining’s methods, andprovides the two period online leasing strategy that is afected by interest rate, and itscompetitive performance analysis.
2. As the large scale equipment is depreciable and the secondary market exists,this thesis constructs the online leasing model of depreciable equipment and obtains itsdeterministic strategy, randomized strategy and risk strategies that base on forecast.
Most of previous research on equipment leasing problems base on that the equipmentcan be used for ever once it is bought, and did not provide a system study on onlineleasing strategy of depreciable equipment. This thesis on the basis of the model of onlineleasing of depreciable equipment, first provides its deterministic strategy and randomizedstrategy; then gives its risk strategies based on certain forecast and probability forecast,respectively; Lastly, this thesis apply the risk strategy that obtained under certain forecastto two period online replacement problem. The results show that the introduction ofdepreciation has enabled the competitive performances of both deterministic strategyand randomized strategy get improved, and the introduction of interest rate made thethe competitive ratio of deterministic strategy get decreased while that of randomized getincreased; the optimal competitive ratio of risk strategy is decreased, and the extent towhich is related to risk tolerance; in the same time, the results also show the risk strategyunder probability forecast has generalized the one under certain forecast.
3. This thesis studies the online investment of single risk asset under the situationsthat the return rate are linear-growth and log-growth; constructs the risk-reward modelwith period forecast for online investment of one risk asset. This paper then obtains theconservative strategy and risk strategy for online investment of single risk asset.
The fluctuation of real return rate is various. Based on the risk aversion, this thesispresents the conservative strategies for the return rates that are linear-growth and log-growth, respectively. Numerical analysis shows the conservative strategy that bases onrisk aversion is more suitable for the stable return rate series. The existed risk strategyis based on the forecast of one point of the return rate series; this thesis based on theperiod forecast provides risk strategy for online investment of single risk asset and itscompetitive performance. The numerical analysis results show the average improvementof optimal restricted competitive ratio is closely related to the investor’s risk toleranceand is proportional to the risk tolerance.
4. This paper studies the online investment of various risk assets and constructs theonline portfolio selection strategy that bases on linear learning function, and constructsonline investment model for various risk assets based on the weak aggregating algorithmand obtains online invest strategy that has better competitive performance.
Based on the on-line learning of linear function, this thesis presents on-line invest- ment strategy for various risk assets, where the linear coefcient is the middle of aninterval. And the interval is related to risk assets’ returns. Using the relative entropy asthe distance function of two portfolios, this paper proves that the provided on-line portfo-lio is universal. The linear factor can be any point of the interval and a class of universalportfolio is obtained. This thesis explores the application of weak aggregating algorithmto online investment of various risk assets. First, apply it to the experts’ strategies thatinvest on single risk asset and obtain the portfolio selection algorithm WAAS. Considerthe definition of universal portfolio is based on BCRP, so this thesis further applies theweak aggregating algorithm to CRP, and obtains the portfolio selection algorithm WAAC.Theoretical and numerical results illustrate the return of WAAS strategy is as good asthat of the best risk asset, and the return of the WAAC is as good as that of the BCRP.
引文
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