金融市场被看作国民经济的“晴雨表”和“气象台”，它通常是由投资者、金融资产、金融中介机构以及政府监管机构组成的运行复杂、结构紧密的动态复杂系统。传统的金融学理论假设金融市场是完全有效的，任意大的资产头寸可以在当前市场价格进行交易，然而实际情况并非如此，大额交易往往比小额交易具有更糟糕的执行价格。除此之外，大额交易还会进一步加剧金融资产价格的波动，引起金融市场不稳定，增大投资者的交易执行风险。为了减少大额交易带来的不利影响，投资者需要根据自身风险偏好设计交易执行策略，将大额交易订单拆分为多个子订单并逐次交易。传统的最优交易执行策略研究往往使用扩散模型来刻画金融资产价格的动力学特征，然而由于金融市场的复杂性，新信息的到达会对金融资产价格造成影响，使其在演变过程中发生跳跃现象，扩散模型不能满足这类特点。另外，最优交易执行策略的本质是复杂系统的最优控制，目前的研究大多建立在线性系统的基础之上，这是由于传统的动态规划算法在求解非线性系统最优控制时会引起维数灾难。本论文以价格模型的跳跃和非线性特征作为切入点，对最优交易执行策略和控制算法展开深入研究，主要工作如下：
针对传统的扩散模型无法刻画金融资产价格的跳跃现象，以及收益率分布呈现出尖峰厚尾的特征，本文首先将金融资产价格模型扩展为离散跳–扩散过程，模型中的跳部分用复合泊松过程描述。在 Almgren–Chriss 流动性模型基础上，以最大化期望变现金融资产的现金流为目标，在库存终值约束条件下，通过动态规划方法进行逆向递推从而得到了最优变现策略的递推关系式。进一步，在优化模型中加入剩余库存的执行风险，在均值–在险价值准则下得到了最优变现策略的递推关系式。最后，通过仿真交易对最优变现策略的性能进行了评价，并说明该变现策略具有更小的交易执行成本。

由于离散时间模型存在计算过程复杂、计算强度大和不易处理交易执行期间的风险度量等问题，本文进一步在连续时间模型下，假设金融市场中的交易可以连续进行，金融资产的价格过程由加性高斯噪声和泊松噪声共同组成，在 Almgren–Chriss 流动性模型基础上，分别讨论了四种不同风险准则下的最优变现策略问题，其中包括：期望总财富、常绝对风险厌恶 (Constant Absolutely Risk Aversion, CARA) 效用函数、均值–二次变差以及均值–在险价值。接下来通过动态规划方法将其转化为最优控制问题的哈密顿–雅可比–贝尔曼偏积分微分方程 (Hamilton–Jacobi–Bellman Partial Integro–differential Equation, HJB–PIDE)，利用启发式方法分别在期望总财富、均值–二次变差以及均值–在险价值准则下得到了最优变现策略的解析表达式，而对于 CARA 效用函数，我们证明了最优变现策略是关于时间的确定性函数，基于函数逼近思想得到了近似变现策略表达式。最后，通过数值仿真对推导的结果以及参数敏感性给予了说明。

针对加性噪声导致金融资产价格为负的不足，本文假设金融资产价格服从几何跳–扩散过程，考虑交易速率约束与交易费率，在 Almgren–Chriss 流动性模型基础上，最大化期望变现金融资产的现金流，此时最优变现问题 HJB–PIDE 的经典解不存在，我们从弱解角度讨论 HJB–PIDE 的性质，使用最优控制中的粘性解理论建立最优值函数与HJB–PIDE 之间的关系，通过动态规划原理 (Dynamic Programming Principle, DPP) 证明了最优值函数是 HJB–PIDE 的粘性解，进一步根据比较原理得到了 HJB–PIDE 粘性解的唯一性以及给出了最优值函数满足凸性所需要的条件。

由于金融资产价格的非线性与不确定性特征，比如价格均值回复性、价格波动弹
性以及价格波动长记忆性等，单纯的使用线性系统来描述其特征显然不能满足实际情况，因此我们针对非线性与不确定性提出了一类非线性随机系统的智能控制算法。在无限时间最优控制问题中，我们首先设计了一种基于自适应动态规划 (Adaptive Dynamic Programming, ADP) 的在线策略迭代算法，并给出了随机系统中算法的收敛性证明。其次采用单评价神经网络来逼近随机系统的最优值函数，并且激活函数需要满足关于输入变量的二阶混合偏导数连续，与传统执行–评价双网络结构相比，该算法在降低计算量的同时减少权重更新误差。最后利用 Lyapunov 直接法证明了神经网络权重估计误差的动态特征一致最终有界。在有限时间最优控制问题中，由于性能指标泛函存在终端约束，最优控制问题的 HJB 方程增加了时变特点。针对这种特点，我们提出了一种基于最小平方误差的离线学习算法。采用具有时变权重的单评价神经网络来逼近未知连续的值函数，在
不使用策略迭代算法下在紧集上通过最小平方近似将非线性时变 HJB 方程转化为关于权重向量的非线性常微分方程组，并利用时间反向积分方法求解。我们证明了随着线性无关激活函数的增加，近似的时变 HJB 方程、近似值函数以及近似控制律将一致收敛到真实值。对于上述算法，我们首先通过仿真模拟研究算法的有效性，随后将算法应用到了金融市场中的最优交易执行问题，分别针对股票库存跟踪问题和不确定性订单变现问题进行建模与求解。

本文针对最优交易执策略以及延伸出的控制算法，综合运用随机微分系统中的 Ito引理、动态规划原理、HJB 方程和神经网络逼近技术等，分别得到了价格过程服从跳–扩散模型的最优变现策略解析表达式，讨论交易执行期间的不同风险测度，分析了具有控制约束下的最优变现策略弱解的存在与唯一性，设计出非线性随机系统的自适应控制器，并将其应用到机构投资的实际交易执行问题中。

The financial market is regarded as the ”barometer” and ”meteorological station” of the national economy. It is usually a dynamic and complex systems consisting of investors, financial assets, financial intermediaries, and government regulators. Traditional financial theory assumes that the financial market is completely efficient, and arbitrarily large asset positions can be traded
at current market prices. However, this is not the case. Large trading often has a worse execution price than small trades. In addition, large amount trading will aggravate the volatility of financial asset price, cause the instability of financial market and increase the trading execution risk of investors. In order to reduce the adverse effects of large trading, investors need to design optimal execution strategies according to their own risk preferences, and divide large trading orders into multiple sub orders. Traditional research on optimal trading execution strategy often uses diffusion model to describe the dynamic characteristics of financial asset price. However, due to the complexity of financial market, the arrival of new information will affect the price of financial asset and make jumps in the price evolution process. The diffusion model can not meet this kind of characteristics. In addition, the essence of the optimal trading execution strategy is the optimal control of complex system. Most of the current research is based on the linear system, which is due to the computational complexity of the traditional dynamic programming algorithm in solving the optimal control of nonlinear system. In this paper, the jump and nonlinear characteristics of the price model are used as the starting point to conduct research on the optimal trading execution strategy and control algorithm. The main work is as follows:

In view of the fact that the traditional diffusion model is unable to describe the jump phenomenon of financial asset price, and that the distribution of return rate shows the characteristics of peak and thick tail, this paper first expands the financial asset price model to discrete jump diffusion process, and describes the jump part of the model with composite Poisson process. Based on the Almgren–Chriss liquidity model, our goal is to maximize the expected cash flow of liquidated stocks, under the constraints of the final value of the inventory, then the dynamic programming method is used to obtain the recurrence formula of the optimal liquidation strategy formula. Furthermore, the execution risk of the remaining inventory is added to the optimization model, and the recurrence formula of the optimal realization strategy is obtained under the mean–value at risk criterion. Finally, the performance of the optimal strategy is evaluated through trading simulation.

Because the discrete-time model has many problems, such as complex calculation process, high calculation intensity and difficulty in dealing with the risk measurement during the execution of trading, this paper further assumes that the trading in the financial market can be carried out continuously under the continuous time model, and the price process of financial assets is composed of additive Gaussian noise and Poisson noise. On the basis of the Almgren–Chriss liquidity model, the optimal liquidation strategy under four different risk criteria is discussed, including: expected total wealth, Constant Absolutely Risk Aversion (CARA) utility function, mean–quadratic variation, and mean–value at risk. Next, it is transformed into the Hamilton-Jacobi-Bellman Partial Integro–differential Equation (HJB–PIDE) of the stochastic optimal control problem by dynamic programming method. Using the heuristic method to obtain the analytic expression of the optimal liquidation strategy under the expected total wealth, mean–quadratic variation, and mean–value at risk value criterion, and for the CARA utility function, we proof that the strategy is a deterministic function of time. Based on the function approximation idea, an approximate expression of the liquidation strategy is obtained. The results of the derivation and the sensitivity of the parameters are explained through numerical simulation.

In view of the deficiency of negative price of financial assets caused by additive noise, this paper assumes that the price of financial assets follows the geometric jump diffusion process, considering the trading rate constraint and trading fee, and on the basis of Almgren–Chriss liquidity model, maximizes the expected cash flow of financial assets. At this time, the classical solution of the optimal liquidation problem does not exist. We discuss the properties of HJB–PIDE from the point of view of weak solution, use the viscosity solution theory of optimal control to establish the relationship between the optimal value function and HJB–PIDE, and use the dynamic programming principle (DPP) proves that the optimal value function is the viscosity solution of HJB–PIDE. Furthermore, according to the comparison principle, the uniqueness of viscosity solution of HJB–PIDE and the conditions for the optimal value function to satisfy the convexity are obtained.

Due to the nonlinear and uncertainty characteristics in the financial asset price, such as mean–reverting, elasticity of variance and long memory of volatility etc., simply using linear systems to describe its characteristics obviously cannot meet the actual situation. Therefore, we propose a type of nonlinear stochastic system intelligent control algorithm based on these characteristics. In the infinite time optimal control problem, we first design an online policy iterative algorithm
based on Adaptive Dynamic Programming (ADP), and give a proof of the convergence of the algorithm in stochastic systems. Secondly, a single critic neural network is used to approximate the optimal value function of the stochastic system, and the activation function needs to satisfy the second-order mixed partial derivatives of the input variables continuously. Compared with the traditional actor–critic dual networks structure, the algorithm reduces the calculation complex and weight updating errors. Finally, Lyapunov’s direct method is used to prove that the dynamic characteristics of neural network weight estimation errors are uniformly ultimately bounded. In the finite–time optimal control problem, the HJB equation of the optimal control problem has time–varying characteristics due to terminal constraints on the index functional, which makes the
finite–time optimal control problem difficult to handle. In view of the characteristics of the time–varying HJB equation, we propose an offline learning algorithm based on the least square error. A single-critic neural network with time–varying weights is used to approximate the unknown continuous value function, and the nonlinear time-varying HJB equation is transformed into nonlinear ordinary differential equations about the weight vector by a least square approximation on a compact set. The equations are solved by the time inverse integration method. We proof that with the increase of the linearly independent activation function, the approximate time–varying HJB equation, the approximate value function, and the approximate control law will converge to the true value uniformly. For these two algorithms, we first studied the effectiveness of the algorithm through simulation, and then applied the algorithm to the optimal trade execution problem in the financial market. We model the stock inventory tracking problem and the uncertainty order filling problem respectively. The approximate solution of the optimal execution strategy can be obtained by the intelligent control method.

In this paper, we use Ito lemma, dynamic programming principle and HJB equation in stochastic differential systems and neural network approximation technique are used to solve the control problem extended by the optimal trading execution strategy. The analytic expression of the optimal liquidation strategy of the jump–diffusion model is obtained, the different risk measures during the execution of the trading are discussed, the existence and uniqueness of the weak solution of the optimal liquidation strategy with control constraints are analyzed, the adaptive controller of the nonlinear stochastic system is designed, and it is applied to the practical trading execution of the institutional investment.