分数布朗运动驱动的随机方程及其在期权定价中的应用
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摘要
与经典的布朗运动相比,分数布朗运动所具有的增量间相关的性质使得分数布朗运动能用来描述或者能更确切的描述一些经典的随机分析方法所不能描述的现象,因此,分数布朗运动在金融、气象、交通、水文、互联网、生态学等领域有着重要的应用。
本文主要研究由分数布朗运动驱动的随机方程及其在金融期权定价中的应用,在综述已有的相关研究成果的基础上,从倒向随机微分方程、随机Volterra方程、期权定价以及随机人口动力系统等四个方面进行了研究。
在绪论中,介绍了分数布朗运动的随机计算的研究历史与进展,特别是对由分数布朗运动驱动的随机方程(如随机微分方程、随机发展方程、倒向随机微分方程、随机Volterra方程等)以及分数Black-Scholes模型的研究历史及研究热点进行了较为详尽的综述。
在第二章中,我们介绍了本文研究所需要的分数布朗运动的定义、性质以及随机积分理论的主要成果。
在第三章中,我们研究了由分数布朗运动驱动的倒向随机微分方程的解的存在性与唯一性问题。首先,在假设生成元关于y满足Lipschitz连续但关于z仅满足一致连续条件下,利用拟条件期望的单调性质以及Tanaka公式得到了该类方程解的存在性与唯一性结果。其次,在假设生成元满足随机Lipschitz条件下,利用方程的解的不等式估计性质,构造压缩映射证明了这类方程解的存在性与唯一性。另外,我们还应用关于分数布朗运动的Girsanov定理,获得了两个倒向随机微分方程弱解存在性的等价结果。
在第四章中,我们研究了分数噪声驱动的随机Volterra方程的解的存在性与唯一性问题。首先,通过研究确定性线性Volterra方程预解算子的性质,得到关于分数布朗运动的随机卷积的正则性结果,应用关于耗散非线性确定性Volterra方程广义适度解的存在性与唯一性结果,获得了分数噪声驱动的耗散非线性随机Volterra方程广义适度解的存在性与唯一性。其次,我们引入了一个与分数布朗运动性质相似的增量间任意相关的不正则高斯过程,得到了不正则高斯噪声驱动的线性随机Volterra方程的弱解的存在性与唯一性,并将所得到的结果应用于对随机热方程弱解的存在性与唯一性研究。
在第五章中,我们研究了分数Black-Scholes模型下的金融期权定价问题,首先,在研究一般性分数Black-Scholes模型下的金融期权定价原理的基础上,通过研究基于不同的计价单位变换的测度变换结果,得到了一个新的研究期权定价模型的方法,在此方法上,我们研究了标准买权、交换期权、分数随机利率情形下的买权等常见期权的定价公式,我们采用的方法在计算中具有便利性优点。
在第六章中,我们研究了分数噪声驱动的年龄依赖型人口动力系统模型的解的存在性与唯一性问题。
最后,我们对本文的结果进行了总结,并提出了一些还需要进一步研究问题。
By comparing with the classic standard Brownian motion, fractional Brownian motion shares a proposition which is the correlation between increments, so fractional Brownian motion can be applied to describe exactly some phenomenon which can't be described by classic stochastic analysis method, so fractional Brownian motion has important applications in many fields such as finance, meteorological phenomena, traffic, hydrology, international net and ecology et.
In this dissertation, based on the summarization of some research results correlated with our topics, we studied stochastic equations driven by fractional Brownian motion and its applications in option pricing theory. That is to say, we studied the following four topics:backward stochastic differential equations, stochastic Volterra equations, option pricing theory and stochastic population dynamic system.
In introduction, we introduced the research history and advance of the stochastic calculus to fractional Brownian motion, particularly, we gave a detailed summarization of the research history and hot topics in stochastic equations driven by fractional Brownian motion such as stochastic differential equations, stochastic evolution equations, backward stochastic differential equations, stochastic Volterra equations and fractional Black-Scholes model.
In Chapter2, we introduced the main results in studying fractional Brownian motion such as the definition, properties and stochastic integral theory, which were used in our research topics.
In Chapter3, we studied the existence and uniqueness of solution to backward stochastic differential equations driven by fractional Brownian motion. Firstly, under the assumption that the generator of these equations is Lipschitz continuous in y but uniformly continuous in z, we obtained a existence and uniqueness result by using the monotone proposition of quasi-conditional expectation and Tanaka formula. Secondly, under the assumption of the generator satisfied stochastic Lipschitz condition, we also obtained a existence and uniqueness result by constructing contraction mapping based on some priori inequality estimates. In addition, we obtained a equivalence result between existence of the weak solutions to two backward stochastic differential equations by using Girsanov theorem for fractional Brownian motion.
In Chapter4, we studied the existence and uniqueness problem of the solution to stochastic Volterra equations driven by fractional noise. Firstly, we obtained a regularity result to stochastic convolution process with respect to fractional Brownian motion by using the propositions of resolvent operator to deterministic linear Volterra equations, then we obtained the existence and uniqueness of generalized mild solution to stochastic Volterra equations with fractional Brownian noise and dissipative nonlinearity by using the results that were obtained in deterministic Volterra equations with dissipative nonlinearity. Secondly, we introduced a irregular Gaussian process with arbitrary correlation between increments, which shared some similar propositions by comparing with fractional Brownian motion, then, a existence and uniqueness result of weak solution to linear stochastic Volterra equations with additive irregular Gaussian noise was obtained, and we also used the result to study the weak to stochastic heat equations.
In Chapter5, we studied the option pricing problem in the fractional Black-Scholes environment. Firstly, based on the general option pricing theory for fractional Black-Scholes model, by selecting different assets as numeraire, we obtained a result of measure transformation and a new option pricing method. Using the method we had obtained, we obtained a option pricing formula for standard call option, exchange option, a option pricing formula was also obtained under the assumption of fractional stochastic rate process. The method we used in this chapter have convenient advantage in computing.
In Chapter6, we studied the existence and uniqueness of solution to stochastic age-dependent population dynamic system with additive fractional noise.
Finally, we summarized the main results of this dissertation and pointed out some issues remaining unsolved.
本文主要研究由分数布朗运动驱动的随机方程及其在金融期权定价中的应用,在综述已有的相关研究成果的基础上,从倒向随机微分方程、随机Volterra方程、期权定价以及随机人口动力系统等四个方面进行了研究。
在绪论中,介绍了分数布朗运动的随机计算的研究历史与进展,特别是对由分数布朗运动驱动的随机方程(如随机微分方程、随机发展方程、倒向随机微分方程、随机Volterra方程等)以及分数Black-Scholes模型的研究历史及研究热点进行了较为详尽的综述。
在第二章中,我们介绍了本文研究所需要的分数布朗运动的定义、性质以及随机积分理论的主要成果。
在第三章中,我们研究了由分数布朗运动驱动的倒向随机微分方程的解的存在性与唯一性问题。首先,在假设生成元关于y满足Lipschitz连续但关于z仅满足一致连续条件下,利用拟条件期望的单调性质以及Tanaka公式得到了该类方程解的存在性与唯一性结果。其次,在假设生成元满足随机Lipschitz条件下,利用方程的解的不等式估计性质,构造压缩映射证明了这类方程解的存在性与唯一性。另外,我们还应用关于分数布朗运动的Girsanov定理,获得了两个倒向随机微分方程弱解存在性的等价结果。
在第四章中,我们研究了分数噪声驱动的随机Volterra方程的解的存在性与唯一性问题。首先,通过研究确定性线性Volterra方程预解算子的性质,得到关于分数布朗运动的随机卷积的正则性结果,应用关于耗散非线性确定性Volterra方程广义适度解的存在性与唯一性结果,获得了分数噪声驱动的耗散非线性随机Volterra方程广义适度解的存在性与唯一性。其次,我们引入了一个与分数布朗运动性质相似的增量间任意相关的不正则高斯过程,得到了不正则高斯噪声驱动的线性随机Volterra方程的弱解的存在性与唯一性,并将所得到的结果应用于对随机热方程弱解的存在性与唯一性研究。
在第五章中,我们研究了分数Black-Scholes模型下的金融期权定价问题,首先,在研究一般性分数Black-Scholes模型下的金融期权定价原理的基础上,通过研究基于不同的计价单位变换的测度变换结果,得到了一个新的研究期权定价模型的方法,在此方法上,我们研究了标准买权、交换期权、分数随机利率情形下的买权等常见期权的定价公式,我们采用的方法在计算中具有便利性优点。
在第六章中,我们研究了分数噪声驱动的年龄依赖型人口动力系统模型的解的存在性与唯一性问题。
最后,我们对本文的结果进行了总结,并提出了一些还需要进一步研究问题。
By comparing with the classic standard Brownian motion, fractional Brownian motion shares a proposition which is the correlation between increments, so fractional Brownian motion can be applied to describe exactly some phenomenon which can't be described by classic stochastic analysis method, so fractional Brownian motion has important applications in many fields such as finance, meteorological phenomena, traffic, hydrology, international net and ecology et.
In this dissertation, based on the summarization of some research results correlated with our topics, we studied stochastic equations driven by fractional Brownian motion and its applications in option pricing theory. That is to say, we studied the following four topics:backward stochastic differential equations, stochastic Volterra equations, option pricing theory and stochastic population dynamic system.
In introduction, we introduced the research history and advance of the stochastic calculus to fractional Brownian motion, particularly, we gave a detailed summarization of the research history and hot topics in stochastic equations driven by fractional Brownian motion such as stochastic differential equations, stochastic evolution equations, backward stochastic differential equations, stochastic Volterra equations and fractional Black-Scholes model.
In Chapter2, we introduced the main results in studying fractional Brownian motion such as the definition, properties and stochastic integral theory, which were used in our research topics.
In Chapter3, we studied the existence and uniqueness of solution to backward stochastic differential equations driven by fractional Brownian motion. Firstly, under the assumption that the generator of these equations is Lipschitz continuous in y but uniformly continuous in z, we obtained a existence and uniqueness result by using the monotone proposition of quasi-conditional expectation and Tanaka formula. Secondly, under the assumption of the generator satisfied stochastic Lipschitz condition, we also obtained a existence and uniqueness result by constructing contraction mapping based on some priori inequality estimates. In addition, we obtained a equivalence result between existence of the weak solutions to two backward stochastic differential equations by using Girsanov theorem for fractional Brownian motion.
In Chapter4, we studied the existence and uniqueness problem of the solution to stochastic Volterra equations driven by fractional noise. Firstly, we obtained a regularity result to stochastic convolution process with respect to fractional Brownian motion by using the propositions of resolvent operator to deterministic linear Volterra equations, then we obtained the existence and uniqueness of generalized mild solution to stochastic Volterra equations with fractional Brownian noise and dissipative nonlinearity by using the results that were obtained in deterministic Volterra equations with dissipative nonlinearity. Secondly, we introduced a irregular Gaussian process with arbitrary correlation between increments, which shared some similar propositions by comparing with fractional Brownian motion, then, a existence and uniqueness result of weak solution to linear stochastic Volterra equations with additive irregular Gaussian noise was obtained, and we also used the result to study the weak to stochastic heat equations.
In Chapter5, we studied the option pricing problem in the fractional Black-Scholes environment. Firstly, based on the general option pricing theory for fractional Black-Scholes model, by selecting different assets as numeraire, we obtained a result of measure transformation and a new option pricing method. Using the method we had obtained, we obtained a option pricing formula for standard call option, exchange option, a option pricing formula was also obtained under the assumption of fractional stochastic rate process. The method we used in this chapter have convenient advantage in computing.
In Chapter6, we studied the existence and uniqueness of solution to stochastic age-dependent population dynamic system with additive fractional noise.
Finally, we summarized the main results of this dissertation and pointed out some issues remaining unsolved.
引文
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