Lévy过程的白噪声分析及应用
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摘要
Levy过程有着丰富的数学结构,是概率论中最为重要的分支之一,在物理、金融等领域有着广泛的应用。Levy过程可以分解为时间变量、布朗运动和纯跳Levy过程的线性组合。Levy白噪声被视为Levy过程的广义时间导数。本文构建了Levy白噪声框架,其中Levy白噪声是指Gauss白噪声和纯跳Levy白噪声复合的二维噪声。并将Levy白噪声框架理论应用于随机微分方程、金融、生物种群和委托代理问题等领域。本文的主要工作包括：
1.系统结合了Gauss白噪声框架理论和纯跳Levy白噪声框架理论,构建了Levy白噪声框架,为求解Levy过程驱动的随机微分方程提供了理论基础。白噪声方法求解随机微分方程依据随机分布空间的特征定理,利用Hermite变换将随机微分方程转化为非随机的普通方程,并求解此确定型方程,利用Hermite反变换将其解转换为随机分布空间中的分布过程,即为原随机微分方程的解。在纯跳Levy白噪声框架中本文用纯跳Levy白噪声表示位势,给出了随机薛定谔方程在随机分布空间的具体解。并论述了随机薛定谔方程的解在弱分布的意义下属于L1(u)空间。在Levy白噪声框架中,本文讨论了由Levy白噪声驱动的随机输运方程,给出了方程在随机分布空间中的具体解。
2.在Levy白噪声框架下,推广Clark-Haussmann-Ocone定理,应用此定理,分别基于完全信息和部分信息,在Levy过程驱动的金融市场中,给出了Malliavin导数表示的欧式期权方差最小复制策略；用Levy白噪声框架理论建立了有界环境中的随机生物种群模型,并引入随机奇异控制研究投资者的最优收获策略。本文应用积分不等式验证定理,将随机控制问题的求解转化为确定型偏微分方程的求解,给出了最优性条件。
3.利用Levy过程的随机控制、复合最优停时—随机控制、随机微分博弈等理论研究一次性支付委托代理问题。其一,在连续扩散环境中,给定线性契约,将经典的Hamilton-Jacobi-Bellman (HJB)方程验证定理推广到弱形式下,用来研究隐蔽行为下的代理人问题,将随机控制问题的求解转化为确定型偏微分方程的求解,给出了最优性条件。其二,将委托代理模型推广到Levy扩散环境中,并允许代理人选择契约的执行时间,给定广义契约,代理人问题被推广为二维的随机控制问题(复合最优停时-随机控制问题)。本文将经典的变分不等式HJB方程验证定理推广到弱形式下,用于求解隐蔽行为下的代理人问题,将复合最优停时-随机控制问题的求解转化为确定型偏微分方程的求解,给出了最优性条件。其三,在Levy扩散环境中,将委托代理问题视为委托人和代理人之间的非零和最优停时-随机控制博弈,其中代理人控制着随机控制过程,委托人选择契约的执行时间。本文证明了非零和最优停时-随机控制微分博弈的变分不等式HJB方程验证定理,将寻找纳什均衡的问题简化为求解一族确定型非线性变分-积分不等式,给出了纳什均衡的最优性条件。纳什均衡的意义在于：通过精心设计执行时间,委托人可以激励代理人做出最优的操作策略；反之,通过操作策略的选择,代理人可以迫使委托人在满足代理人利益最大化的时间执行契约。此定理对于广义的代价函数和一般的状态方程(代理人的操作策略同时影响状态方程的漂移项和扩散项)仍然成立,且适用于求解一般的非零和随机微分博弈,博弈一方控制随机过程,而另一方控制停时。
Levy processes are the most important parts in stochastic analysis, which have abundant mathematic structure. They have been used in a broad range of applications, such as physics and finance. A Levy process can be written as a linear combination of time t. a Brownian motion and a pure jump Levy process. And a Levy white noises are regarded as the derivative of a Levy process. In this thesis, we construct the framework of Levy white noises which are combined by Gaussian white noises and pure jump Levy white noises. Furthermore we study its applications in stochastic differential equations, finance, stochastic population and principal-agent problem. The main results, obtained in this thesis, are summarized as follows:
1. We construct the framework of Levy white noises by combining the framework of Gaussian white noises and that of pure jump Levy white noises. The white noise approach to stochastic differential equation (SDE) driven by a Levy process is based on the characterization theorem of stochastic distribution space in this framework. The main idea of white noise approach can be concluded as follows:the SDE is firstly reduced to the deterministic differential equation (DDE) by Hermite transform, which can be solved. Then the solution of the SDE is obtained by the characterization theorem of stochastic distribution space, converting the solution of the DDE to a distribution. In the framework of pure jump Levy white noises, we give the explicit solution of the stochastic Schrodinger equation (SSE) driven by pure jump Levy white noises. And we prove that the solution of SSE is in L1(v) in sense of weak distribution under some conditions. In the framework of Levy white noises, we apply white noise approach to stochastic transport equation (STE) driven by Levy white noises. The explicit solution of the STE is obtained in stochastic distribution space. Moreover, we get the explicit solution of the stochastic heat equation by the solution of the STE.
2. In the framework of Levy white noises, we give the white noise generalization of the Clark-Haussmann-Ocone theorem for Levy processes. As an application, in a financial market driven by Levy processes, the optimal replicating portfolios for a European option are represented by the explicit functional of Malliavin derivatives under full in-formation and under partial information. On the other hand, we set up by white noise analysis the stochastic population equation in a crowded environment perturbed by Levy processes. And the stochastic singular control is introduced to study the optimal harvesting problem. Based on the verification theorem of Integrovariational Inequali-ties, we reduce the optimal harvesting problem to solving a deterministic differential equations. The solution gives the optimality conditions.
3. We introduce stochastic control, combined optimal stopping-stochastic control and stochastic differential game to study a principal-agent problem with Lump-sum pay-ments. Firstly, for a given linear contract in continuous diffusion setting, we develop the classic verification theorem of HJB equations in weak formulation and present the optimality conditions for the agent's problem in hidden actions. Secondly, the agent is allowed to exercise the contract prior the terminal time. For a given general contract, the agent's problem is formulated as an combined optimal stopping-stochastic control problem. In Levy diffusion setting, the classic verification theorem of Variational-Inequality-HJB equations is generalized in weak formulation to give optimality con-ditions for the agent's optimal exercise time in the case of hidden actions. Finally, when the principal is allowed to choose exercise time in Levy diffusion setting, we formulate the principal-agent problem as a nonzero-sum optimal stopping-stochastic control differential game between the principal and the agent. The principal controls stopping and the agent controls stochastic control. We prove a verification theorem in term of Variational-Inequality-HJB equations to provide the optimality conditions for Nash equilibrium of the game. The interesting feature of Nash equilibrium is that, the principal can induce the agent to make the best efforts, by an appropriate stopping rule design. Conversely, the agent can force the principal to exercise at a time of the agent's choosing, by applying suitable efforts. With the help of the verification the-orem, the characterization of Nash equilibrium is reduced to the solutions of a family of nonlinear variational-integro inequalities. The theorem for the game is applicable to more general nonzero-sum optimal stopping-stochastic control differential game than the specified principal-agent problem studied in this thesis.
1.系统结合了Gauss白噪声框架理论和纯跳Levy白噪声框架理论,构建了Levy白噪声框架,为求解Levy过程驱动的随机微分方程提供了理论基础。白噪声方法求解随机微分方程依据随机分布空间的特征定理,利用Hermite变换将随机微分方程转化为非随机的普通方程,并求解此确定型方程,利用Hermite反变换将其解转换为随机分布空间中的分布过程,即为原随机微分方程的解。在纯跳Levy白噪声框架中本文用纯跳Levy白噪声表示位势,给出了随机薛定谔方程在随机分布空间的具体解。并论述了随机薛定谔方程的解在弱分布的意义下属于L1(u)空间。在Levy白噪声框架中,本文讨论了由Levy白噪声驱动的随机输运方程,给出了方程在随机分布空间中的具体解。
2.在Levy白噪声框架下,推广Clark-Haussmann-Ocone定理,应用此定理,分别基于完全信息和部分信息,在Levy过程驱动的金融市场中,给出了Malliavin导数表示的欧式期权方差最小复制策略；用Levy白噪声框架理论建立了有界环境中的随机生物种群模型,并引入随机奇异控制研究投资者的最优收获策略。本文应用积分不等式验证定理,将随机控制问题的求解转化为确定型偏微分方程的求解,给出了最优性条件。
3.利用Levy过程的随机控制、复合最优停时—随机控制、随机微分博弈等理论研究一次性支付委托代理问题。其一,在连续扩散环境中,给定线性契约,将经典的Hamilton-Jacobi-Bellman (HJB)方程验证定理推广到弱形式下,用来研究隐蔽行为下的代理人问题,将随机控制问题的求解转化为确定型偏微分方程的求解,给出了最优性条件。其二,将委托代理模型推广到Levy扩散环境中,并允许代理人选择契约的执行时间,给定广义契约,代理人问题被推广为二维的随机控制问题(复合最优停时-随机控制问题)。本文将经典的变分不等式HJB方程验证定理推广到弱形式下,用于求解隐蔽行为下的代理人问题,将复合最优停时-随机控制问题的求解转化为确定型偏微分方程的求解,给出了最优性条件。其三,在Levy扩散环境中,将委托代理问题视为委托人和代理人之间的非零和最优停时-随机控制博弈,其中代理人控制着随机控制过程,委托人选择契约的执行时间。本文证明了非零和最优停时-随机控制微分博弈的变分不等式HJB方程验证定理,将寻找纳什均衡的问题简化为求解一族确定型非线性变分-积分不等式,给出了纳什均衡的最优性条件。纳什均衡的意义在于：通过精心设计执行时间,委托人可以激励代理人做出最优的操作策略；反之,通过操作策略的选择,代理人可以迫使委托人在满足代理人利益最大化的时间执行契约。此定理对于广义的代价函数和一般的状态方程(代理人的操作策略同时影响状态方程的漂移项和扩散项)仍然成立,且适用于求解一般的非零和随机微分博弈,博弈一方控制随机过程,而另一方控制停时。
Levy processes are the most important parts in stochastic analysis, which have abundant mathematic structure. They have been used in a broad range of applications, such as physics and finance. A Levy process can be written as a linear combination of time t. a Brownian motion and a pure jump Levy process. And a Levy white noises are regarded as the derivative of a Levy process. In this thesis, we construct the framework of Levy white noises which are combined by Gaussian white noises and pure jump Levy white noises. Furthermore we study its applications in stochastic differential equations, finance, stochastic population and principal-agent problem. The main results, obtained in this thesis, are summarized as follows:
1. We construct the framework of Levy white noises by combining the framework of Gaussian white noises and that of pure jump Levy white noises. The white noise approach to stochastic differential equation (SDE) driven by a Levy process is based on the characterization theorem of stochastic distribution space in this framework. The main idea of white noise approach can be concluded as follows:the SDE is firstly reduced to the deterministic differential equation (DDE) by Hermite transform, which can be solved. Then the solution of the SDE is obtained by the characterization theorem of stochastic distribution space, converting the solution of the DDE to a distribution. In the framework of pure jump Levy white noises, we give the explicit solution of the stochastic Schrodinger equation (SSE) driven by pure jump Levy white noises. And we prove that the solution of SSE is in L1(v) in sense of weak distribution under some conditions. In the framework of Levy white noises, we apply white noise approach to stochastic transport equation (STE) driven by Levy white noises. The explicit solution of the STE is obtained in stochastic distribution space. Moreover, we get the explicit solution of the stochastic heat equation by the solution of the STE.
2. In the framework of Levy white noises, we give the white noise generalization of the Clark-Haussmann-Ocone theorem for Levy processes. As an application, in a financial market driven by Levy processes, the optimal replicating portfolios for a European option are represented by the explicit functional of Malliavin derivatives under full in-formation and under partial information. On the other hand, we set up by white noise analysis the stochastic population equation in a crowded environment perturbed by Levy processes. And the stochastic singular control is introduced to study the optimal harvesting problem. Based on the verification theorem of Integrovariational Inequali-ties, we reduce the optimal harvesting problem to solving a deterministic differential equations. The solution gives the optimality conditions.
3. We introduce stochastic control, combined optimal stopping-stochastic control and stochastic differential game to study a principal-agent problem with Lump-sum pay-ments. Firstly, for a given linear contract in continuous diffusion setting, we develop the classic verification theorem of HJB equations in weak formulation and present the optimality conditions for the agent's problem in hidden actions. Secondly, the agent is allowed to exercise the contract prior the terminal time. For a given general contract, the agent's problem is formulated as an combined optimal stopping-stochastic control problem. In Levy diffusion setting, the classic verification theorem of Variational-Inequality-HJB equations is generalized in weak formulation to give optimality con-ditions for the agent's optimal exercise time in the case of hidden actions. Finally, when the principal is allowed to choose exercise time in Levy diffusion setting, we formulate the principal-agent problem as a nonzero-sum optimal stopping-stochastic control differential game between the principal and the agent. The principal controls stopping and the agent controls stochastic control. We prove a verification theorem in term of Variational-Inequality-HJB equations to provide the optimality conditions for Nash equilibrium of the game. The interesting feature of Nash equilibrium is that, the principal can induce the agent to make the best efforts, by an appropriate stopping rule design. Conversely, the agent can force the principal to exercise at a time of the agent's choosing, by applying suitable efforts. With the help of the verification the-orem, the characterization of Nash equilibrium is reduced to the solutions of a family of nonlinear variational-integro inequalities. The theorem for the game is applicable to more general nonzero-sum optimal stopping-stochastic control differential game than the specified principal-agent problem studied in this thesis.
引文
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