不对称信息随机微分博弈的充要最大值原理及其在高频金融市场中的应用

Sufficient and Necessary Maximum Principles for Stochastic Differential Games with Asymmetric Information and Their Applications in High-Frequency Financial Markets

摘要: 本文研究了一类多人不对称信息下的非零和随机博弈问题，其中系统方程是由布朗运动驱动的随机微分方程来描述，博弈者获取信息存在延迟且不对称。利用凸变分推导了充分和必要最大值原理，给出了纳什均衡点的充要条件。进一步将理论结果应用于高频金融市场中算法交易商、一般交易商和做市商的不对称随机微分博弈。

Abstract: In this paper, we investigate a class of non-zero-sum stochastic differential games with asymmetric information for multiple players, where the system equations are described by stochastic differential equations driven by Brownian motion. The players experience delayed and asymmetric information acquisition. By employing convex variation, we derive sufficient and necessary maximum principles to provide optimal conditions for the Nash equilibrium. The theoretical results are further applied to asymmetric stochastic differential games for algorithm traders, general traders, and market traders in high-frequency financial markets.

1. 引言

近年来，高频交易在国际金融市场中的作用日益凸显，全球各大交易所纷纷引入高速交易平台，以满足机构投资者及对冲基金等对快速执行交易策略的需求。然而，伴随而来的也有不少争议，如是否存在市场操纵、是否加剧了市场的不稳定等，因此高频金融市场的研究是学者们广泛关注的热点问题，例如：Aït-Sahalia和Sağlam在文献[1]中描述了高频金融市场中做市商如何利用速度和信息优势与低频交易者互动，研究最优策略和市场均衡；文献[2]利用高频交易数据和深度强化学习算法构建量化模型，以提高高频交易效果和企业盈利能力；文献[3]发现高频交易者通过永久性价格变化和暂时性定价错误方向交易，利用流动性需求订单促进价格效率，并且其交易方向与公共信息相关，从而能够据此预测短期内的价格变化；Saito和Takahashi在文献[4]中考虑高频金融市场中算法交易商、一般交易商和做市商的相互合作影响的交易行为，他们的目标各不相同且合作共赢，将其建模为一个完全信息下非零和的随机微分博弈问题。

文献[4]考虑的算法交易商和做市商获取的是完全信息，且用示性函数χ_t来表示一般交易商获取信息存在延迟，也就是说只考虑了[0, δ]时间区间的延迟，此后这三类交易者获取的信息是同步的情况，这不符合真实高频金融市场中一般交易商的特性。真实市场交易中，三者不可避免地获取信息都存在延迟，其中算法交易者和做市商是利用复杂算法和高速计算自动执行交易策略，算法交易者是非做市的程序化交易者，他们会在较大的价格波动中寻找交易机会，而做市商则是通过在买卖价差中获利的程序化交易者，他们同时挂出买入和卖出订单，并在紧邻的范围内执行这些订单；一般交易商包括私人和机构投资者的长期头寸，以及非自动化的对冲基金，这些基金采用趋势跟踪策略，手动买卖金融资产。他们的反应速度不如算法交易者和做市商快，并且在市场出现意外事件时，他们会被迫减少其风险资产。因此，本文发展了文献[4]的问题模型，考虑了真实市场交易中三类交易者获取信息都是存在延迟的，并且他们的延迟是不同步的，即控制过程适应于完全信息的不同子滤子流。

关于完全信息下的博弈问题，最常用的两种求解方法是用动态规划原理或最大值原理来刻画纳什均衡点。例如，Savku和Weber在文献[5]中基于行为金融框架，在连续时间马尔可夫机制转换环境下，应用随机博弈与动态规划，通过马尔可夫链建模经济状态与投资者心理，建立HJB-Isaacs方程并求得最优策略；Feng在文献[6]中研究了竞争布朗粒子的规律性之后，建立了上下值函数的动态规划原理，并证明了HJB方程具有唯一粘性解。另一方面，文献[7]利用随机最大值原理给出了一类具有马尔可夫跳跃和泊松跳跃的无穷时间区域随机微分对策满足的充要条件；Nie和Yan在文献[8]中基于控制域非凸且控制变量在扩散项的条件，研究了一类平均场型的部分信息非零和随机微分博弈问题，推导了最大值原理和凸域验证定理，并将其应用于线性二次标量交互模型分析；文献[9]研究了部分信息下含时滞和平均场的混合正则–奇异非零和随机微分博弈，建立了奇异方程解的存在唯一性及最大原理，求得纳什均衡与鞍点，并应用于模型不确定性的最优投资与分红策略。

对于控制过程基于部分信息的博弈问题，由于系统的非马尔可夫特性，使得贝尔曼最优性准则不成立，因此动态规划原理失效(见文献[10])。近年来，很多学者推导部分信息下最大值原理来解决这种类型的随机微分博弈问题，例如，文献[11]研究了一类在部分信息下向后随机微分延迟方程的新型微分博弈问题，通过偶性关系，提出了非零和博弈的最大原；Wu和Liu在文献[12]中研究了部分信息下的平均场型零和随机微分博弈，应用对偶方法和倒向平均场型随机微分方程得出充分和必要最大值原理，并将结果应用于投资组合中的博弈问题；Zhang在文献[13]中研究了具有离散分布时滞性的非零和随机微分博弈的最大值原理，通过对偶方法和倒向随机微分方程，建立了必要最大值原理，并应用于动态广告中的博弈问题；文献[14]研究了Stackelberg随机微分再保险–投资博弈，考虑到延迟对平衡策略的影响，推导最大值原理并得到了均衡策略。

本文考虑高频金融市场中算法交易商、一般交易商和做市商的非零和随机微分博弈，在模型中假设三类博弈者获取信息都存在延迟且不对称。文献[10]中，An和Øksendal推导了不对称信息下零和博弈的最大值原理框架，本文进一步发展和补充了文献[10]的理论结果，针对多人不对称信息下的非零和随机微分博弈，推导了必要和充分最大值原理来刻画纳什均衡点，并将理论结果应用于高频金融市场中的随机微分博弈。

2. 模型

设$\left(\Omega ,ℱ,{\left\{{\widetilde{ℱ}}_t\right\}}_{t\ge 0},P\right)$ 是完备的滤子概率空间，$W\left(t\right)$ 是定义在此空间上的$m$ 维标准布朗运动，系统状态$X\left(t\right)$ 满足如下的随机微分方程：

$\{\begin{array}{l}\text{d}X\left(t\right)=b\left(t,X\left(t\right),u_1\left(t\right),u_2\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t+\sigma \left(t,X\left(t\right),u_1\left(t\right),u_2\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}W\left(t\right)\text{, }t\in \left[0,T\right],\\ X\left(0\right)=x\in {ℝ}^d,\end{array}$ (1)

其中，$b:\left[0,T\right]\times {ℝ}^d\times K_1\times K_2\times \cdots \times K_n\to {ℝ}^d$ ，$\sigma :\left[0,T\right]\times {ℝ}^d\times K_1\times K_2\times \cdots \times K_n\to {ℝ}^{d\times m}$ 是给定的函数，在这里$K_i$ 是${ℝ}^{l_i}$ 的一个闭子集，$i=1,2,\cdots ,n$ 。

本文研究一类n个参与者组成的非零和随机微分博弈问题，控制$u_i:\left[0,T\right]\times \Omega \to K^{l_i}$ 代表的是参与者i的策略，$i=1,2,\cdots ,n$ 。假设参与者获取信息是存在延迟的，且延迟的时间${\delta }_i$ 是不同步的，因此令${\left\{{ℱ}_t\right\}}_{t\ge 0}$ 是由布朗运动$W\left(t\right)$ 生成的滤子流，则有：

${ℱ}_{t-{\delta }_i}\subseteq {ℱ}_t,t\ge 0,i=1,2,\cdots ,n,$

其中，${ℱ}_{t-{\delta }_i}$ 代表$t-{\delta }_i$ 时刻系统包含的所有信息，要求$u_i\left(t\right)$ 关于滤子流${ℱ}_{t-{\delta }_i}$ 是适应的，以表示参与者获取信息存在延迟且延迟时间${\delta }_i$ 不同步。

接下来定义容许控制集，首先我们定义容许控制$u_i(\cdot ),i=1,2,\cdots ,n$ ：

定义2.1 一个控制$u_i(\cdot ),i=1,2,\cdots ,n$ 。被称为是一个容许控制，如果

1) 在控制$u_i\left(t\right):\left[0,T\right]\times \Omega \to K^{l_i}$ 下，系统方程(1)存在唯一的解；

2) $u_i\left(t\right)$ 是${\left\{{ℱ}_{t-{\delta }_i}\right\}}_{t-{\delta }_i\ge 0}$ 渐近可测的随机过程；

3) $E\left[{\displaystyle {\int }_0^T{\left|u_i\left(t\right)\right|}^2\text{d}t}\right]<\infty$ 。

所有容许控制组成的集合就是容许控制集，此处记${\mathcal{A}}_i$ 为参与者i策略的容许控制集，$i=1,2,\cdots ,n$ 。

因为本文考虑的是非零和随机微分博弈问题，所以这n个参与者是合作式的，都有其各自的目标，定义参与者i的代价泛函为：

$J_i\left(u_1,u_2,\cdots ,u_n\right)=E\left[{\displaystyle {\int }_0^T{f}_i\left(t,X\left(t\right),u\left(t\right)\right)\text{d}t+g_i\left(X\left(T\right)\right)}\right],i=1,2,\cdots ,n.$ (2)

其中，$f_i:\left[0,T\right]\times {ℝ}^d\times K_1\times K_2\times \cdots \times K_n\to ℝ$ ，$g_i:{ℝ}^d\to ℝ$ 是给定的函数。

现在我们的目标是去寻找一组控制$\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_n\right)\in {\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n$ ，使得：

$\begin{array}{l}{J}_1\left(u_1,{\widehat{u}}_2,{\widehat{u}}_3,\cdots ,{\widehat{u}}_n\right)\le J_1\left({\widehat{u}}_1,{\widehat{u}}_2,{\widehat{u}}_3,\cdots ,{\widehat{u}}_n\right),\forall u_1\left(t\right)\in A_1,\\ J_2\left({\widehat{u}}_1,u_2,{\widehat{u}}_3,\cdots ,{\widehat{u}}_n\right)\le J_2\left({\widehat{u}}_1,{\widehat{u}}_2,{\widehat{u}}_3,\cdots ,{\widehat{u}}_n\right),\forall u_2\left(t\right)\in A_2,\\ \cdots \\ J_n\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_{n-1},u_n\right)\le J_n\left({\widehat{u}}_1,{\widehat{u}}_2,{\widehat{u}}_3,\cdots ,{\widehat{u}}_n\right),\forall u_n\left(t\right)\in A_n.\end{array}$ (3)

这一组控制$\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_n\right)\in {\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n$ 被称为是博弈问题(3)的纳什均衡点。

3. 不对称信息下非零和博弈问题的最大值原理

在求解博弈问题时，动态规划和最大值原理(见文献[15])是两种经典的方法。但在博弈问题(3)中，由于控制过程$u_i\left(t\right),t\in \left[0,T\right]$ 是关于${ℱ}_{t-{\delta }_i}$ 适应的，而不是${ℱ}_t$ 适应的，故此问题不符合贝尔曼最优性准则，从而动态规划不适用，因此本文要推导最大值原理。事实上，在文献[10]中，已经给出了二人零和博弈问题的最大值原理，以及二人非零和博弈的充分最大值原理，本文将进一步发展和补充文献[12]的工作，推导多人非零和博弈问题(3)的充分最大值原理和必要最大值原理来刻画纳什均衡点。

对于博弈问题(3)，定义哈密尔顿函数

$H_i:\left[0,T\right]\times {ℝ}^d\times K^1\times \cdots \times K^n\times {ℝ}^d\times {ℝ}^{d\times m}\times ℝ\to ℝ,i=1,2,\cdots ,n,$

为

$H_i\left(t,x,u,p_i,q_i\right)=f_i\left(t,x,u\right)+b^{\text{T}}\left(t,x,u\right)p_i+tr\left({\sigma }^{\text{T}}\left(t,x,u\right)q_i\right),$ (4)

其中，$\left(p_i(\cdot ),q_i(\cdot )\right)$ 满足下列伴随方程：

$\{\begin{array}{l}\text{d}{p}_i\left(t\right)=-{\nabla }_x{H}_i\left(t,X\left(t\right),u\left(t\right),p_i\left(t\right),q_i\left(t\right)\right)\text{d}t+q_i\left(t\right)\text{d}W\left(t\right),\\ p_i\left(T\right)=\nabla g_i\left(X\left(T\right)\right),\end{array}$ (5)

其中，T代表矩阵的转置，$tr\left(A\right)$ 代表矩阵A的迹，${\nabla }_x\phi (\cdot )={\left(\frac{\partial \phi }{\partial x_1},\frac{\partial \phi }{\partial x_2},\cdots ,\frac{\partial \phi }{\partial x_d}\right)}^{\text{T}}$ 代表函数$\phi :{ℝ}^d\to ℝ$ 对x的梯度。

本文的研究基于以下假设：

假设3.1 (A1) 假设控制${\beta }_i\left(s\right)=\left(0,\cdots ,{\beta }_i^j\left(s\right),\cdots ,0\right)\in {\mathcal{A}}_i,i=1,\cdots ,n;j=1,\cdots ,l_i$ ，其中：

${\beta }_i^j\left(s\right)={\alpha }_i^j{\chi }_{\left[t,t+h\right]}\left(s\right),$ (6)

其中，$t,h$ 满足$0\le t<t+h\le T$ ，且${\alpha }_i^j$ 是有界${ℱ}_{t-{\delta }_i}$ 可测的随机变量，此外${\chi }_{\left[t,t+h\right]}\left(s\right)$ 代表示性函数。

(A2) 对于给定的$u_i(\cdot )\in {\mathcal{A}}_i$ 和有界的${\beta }_i(\cdot )\in {\mathcal{A}}_i$ ，存在$\delta >0$ ，使得：

$u_i+y_i{\beta }_i\in {\mathcal{A}}_i,$ (7)

对于所有的$y_i\in \left(-\delta ,\delta \right)$ 成立。

记$X^{u_i+y{\beta }_i}\left(t\right)=X^{\left(u_1,\cdots ,u_i+y{\beta }_i,\cdots ,u_n\right)}\left(t\right)$ ，对于给定的$u_i(\cdot )\in {\mathcal{A}}_i$ 和有界的${\beta }_i(\cdot )\in {\mathcal{A}}_i$ ，定义变分过程$Y^{u_i}\left(t\right)$ 为：

$Y^{u_i}\left(t\right)={\frac{\text{d}}{\text{d}y}{X}^{u_i+y\beta }|}_{y=0}={\left(Y_1^{u_i}\left(t\right),\cdots ,Y_d^{u_i}\left(t\right)\right)}^{\text{T}},i=1,2,\cdots ,n,$ (8)

根据系统方程(1)可得：

$\text{d}{Y}_j^{u_i}\left(t\right)={\lambda }_j^{u_i}\left(t\right)\text{d}t+{\displaystyle \sum _{k=1}^m{\xi }_{jk}^{u_i}}\left(t\right)\text{d}{W}_k\left(t\right),j=1,2,\cdots ,d,$ (9)

这里

${\lambda }_j^{u_i}\left(t\right)={\nabla }_x{b}_j{\left(t,X\left(t\right),u_1\left(t\right),u_2\left(t\right),\cdots ,u_n\left(t\right)\right)}^{\text{T}}{Y}^{u_i}\left(t\right)+{\nabla }_{u_i}{b}_j{\left(t,X\left(t\right),u_1\left(t\right),u_2\left(t\right),\cdots ,u_n\left(t\right)\right)}^{\text{T}}{\beta }_i\left(t\right)$ (10)

和

${\xi }_{jk}^{u_i}\left(t\right)={\nabla }_x{\sigma }_{jk}{\left(t,X\left(t\right),u_1\left(t\right),u_2\left(t\right),\cdots ,u_n\left(t\right)\right)}^{\text{T}}{Y}^{u_i}\left(t\right)+{\nabla }_{u_i}{\sigma }_{jk}{\left(t,X\left(t\right),u_1\left(t\right),u_2\left(t\right),\cdots ,u_n\left(t\right)\right)}^{\text{T}}{\beta }_i\left(t\right).$ (11)

假设3.2 对于$i=1,2,\cdots ,n$ ，可积性条件

$E\left[{\displaystyle {\int }_0^T{Y}^{{\widehat{u}}_i{}^{\text{T}}}\left(t\right){\widehat{q}}_i\left(t\right){\widehat{q}}_i{\left(t\right)}^{\text{T}}{Y}^{{\widehat{u}}_i}\left(t\right)\text{d}t}\right]<\infty$ (12)

和

$E\left[{\displaystyle {\int }_0^T{\widehat{p}}_i^{\text{T}}\left(t\right){\xi }^{{\widehat{u}}_i}\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right){\xi }^{{\widehat{u}}_i{}^{\text{T}}}\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right){\widehat{p}}_i\left(t\right)\text{d}t}\right]<\infty$ (13)

成立。

现在给出博弈问题(3)的必要最大值原理：

定理3.1 (必要最大值原理) 在假设3.1和假设3.2成立的条件下，如果$\left({\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)$ 是博弈问题(3)的一个纳什均衡点，也就是说，$\left(0,0,\cdots ,0\right)$ 是函数

$h_i\left(y_1,\cdots ,y_n\right)=J_i\left({\widehat{u}}_1+y_1{\beta }_1,{\widehat{u}}_2+y_2{\beta }_2,\cdots ,{\widehat{u}}_n+y_n{\beta }_n\right),i=1,2,\cdots ,n$ (14)

的一个稳定点，即：

$\frac{\partial h_i}{\partial y_j}\left(0,0,\cdots ,0\right)=0$

对于$j=1,2,\cdots ,n$ 成立，那么$\left({\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)$ 满足：

$E\left[{\nabla }_{u_i}{H}_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}\right]=0,i=1,2,\cdots ,n.$ (15)

证明：因为$h_i\left(y_1,\cdots ,y_n\right)$ 在$\left(0,0,\cdots ,0\right)$ 处有一个稳定点，从而：

$\begin{array}{l}{\frac{\partial }{\partial y_i}h\left(y_1,y_2,\cdots ,y_n\right)|}_{y_i=0}=E[{\displaystyle {\int }_0^T{\nabla }_{x}f{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}^{\text{T}}{\widehat{Y}}^{{\widehat{u}}_i}\left(t\right)\text{d}t}\\ +{\displaystyle {\int }_0^T{\nabla }_{u_i}f{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}^{\text{T}}{\beta }_i\left(t\right)\text{d}t+\nabla g\left(\widehat{X}\left(T\right)\right)Y^{{\widehat{u}}_i}\left(T\right)}]=0,\end{array}$ (16)

根据方程(5)的终值条件，可得：

$E\left[\nabla g\left(\widehat{X}\left(T\right)\right)Y^{{\widehat{u}}_i}\left(T\right)\right]=E\left[{\widehat{p}}_i^{\text{T}}\left(T\right)Y^{{\widehat{u}}_i}\left(T\right)\right],$ (17)

然后，对${\widehat{p}}_i^{\text{T}}\left(t\right)Y^{{\widehat{u}}_i}\left(t\right)$利用伊藤公式，并根据哈密尔顿函数的定义，可得：

$\begin{array}{l}E\left[{\widehat{p}}_i^{\text{T}}\left(T\right)Y^{{\widehat{u}}_i}\left(T\right)\right]=E\left[{\displaystyle {\int }_0^{T}d\left({\widehat{p}}_i^{\text{T}}\left(t\right)Y^{{\widehat{u}}_i}\left(t\right)\right)}\right]\\ =E[{\displaystyle \sum _{j=1}^d{\displaystyle {\int }_0^T\{{\widehat{p}}_i^j\left(t\right){({\nabla }_x{b}^j\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}^{\text{T}}{Y}^{{\widehat{u}}_i}\left(t\right)}}\\ +{\nabla }_{u_i}{b}^j{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}^{\text{T}}{\beta }_i\left(t\right))\\ +Y_j^{{\widehat{u}}_i}\left(t\right)\left(-{\nabla }_x{H}_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)\right)\\ +{\displaystyle \sum _{k=1}^m{\widehat{q}}_i^{jk}({\nabla }_x}{\sigma }_{jk}{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}^{\text{T}}{Y}^{{\widehat{u}}_i}\left(t\right)\\ +{\nabla }_{u_i}{\sigma }_{kj}{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}^{\text{T}}{\beta }_i\left(t\right))\}\text{d}t],\end{array}$ (18)

由于

$\begin{array}{l}{\nabla }_x{H}_i\left(t,x,u_1,u_2,\cdots ,u_n,p_i,q_i\right)\\ ={\nabla }_{x}f\left(t,x,u_1,u_2,\cdots ,u_n\right)+{\displaystyle \sum _{j=1}^d{\nabla }_x{b}_j\left(t,x,u_1,u_2,\cdots ,u_n\right)p_i^j}+{\displaystyle \sum _{j=1}^d{\displaystyle \sum _{k=1}^m{\nabla }_x{\sigma }_{k,j}\left(t,x,u_1,u_2,\cdots ,u_n\right)q_i^{k,j}}},\end{array}$ (19)

式(16)可化为：

$\begin{array}{l}E[{\displaystyle {\int }_0^T{\displaystyle \sum _{j=1}^{l_i}\{\frac{\partial f}{\partial u_i^j}\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}}\\ -{\displaystyle \sum _{k=1}^d({\widehat{p}}_i^k\left(t\right)\frac{\partial b_k}{\partial u_i^j}\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right){\beta }_i^j\left(t\right)}\\ +{\displaystyle \sum _{z=1}^m\left[{\widehat{q}}_i^{z,k}\left(t\right)\frac{\partial {\sigma }_{z,k}}{\partial u_i^j}\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)\right]}){\beta }_i^j\left(t\right)\}\text{d}t]=0.\end{array}$ (20)

又因为

$\begin{array}{l}{\nabla }_{u_i}{H}_i\left(t,x,u_1,u_2,\cdots ,u_n,p_i,q_i\right)\\ ={\nabla }_{u_i}f\left(t,x,u_1,u_2,\cdots ,u_n\right)+{\displaystyle \sum _{j=1}^d{\nabla }_{u_i}{b}_j\left(t,x,u_1,u_2,\cdots ,u_n\right)p_i^j}+{\displaystyle \sum _{k=1}^d{\displaystyle \sum _{j=1}^m{\nabla }_{u_i}{\sigma }_{k,j}\left(t,x,u_1,u_2,\cdots ,u_n\right)q_i^{k,j}}},\end{array}$ (21)

(21)结合式(20)可得：

$E\left[{\displaystyle {\int }_0^T{\nabla }_{u_i}{H}_i{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)}^{\text{T}}{\beta }_i\left(t\right)\text{d}t}\right]=0.$ (22)

根据${\beta }_i\left(t\right)$ 的定义，有：

$E\left[{\displaystyle {\int }_t^{t+h}\frac{\partial }{\partial u_i^j}{H}_i\left(t,\widehat{X}\left(s\right),{\widehat{u}}_1\left(s\right),{\widehat{u}}_2\left(s\right),\cdots ,{\widehat{u}}_n\left(s\right),{\widehat{p}}_i\left(s\right),{\widehat{q}}_i\left(s\right)\right){\alpha }^j\text{d}s}\right]=0.$ (23)

(23)式两端同时对h求导，并令$h=0$ ，可得：

$E\left[\frac{\partial }{\partial u_i^j}{H}_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right){\alpha }^j\right]=0.$ (24)

进一步利用条件数学期望的性质，(24)可化为：

$E\left[E\left[\frac{\partial }{\partial u_i^j}{H}_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}\right]{\alpha }^j\right]=0.$ (25)

因为${\alpha }^j$ 是任意有界${ℱ}_{t-{\delta }_i}$ 可测的随机变量，因此最大值条件(15)成立，定理得证。

定理3.1给出了纳什均衡点的必要性条件，下面考虑纳什均衡点的充分性条件。首先补充如下假设：

假设3.3 假设下列可积性条件成立：

$E\left[{\displaystyle {\int }_0^T{\left(\widehat{X}\left(t\right)-X^{\left(u_i\right)}\left(t\right)\right)}^{\text{T}}{\widehat{q}}_i\left(t\right){\widehat{q}}_i{\left(t\right)}^{\text{T}}\left(\widehat{X}\left(t\right)-X^{\left(u_i\right)}\left(t\right)\right)}\right]<\infty ,$ (26)

$E\left[{\displaystyle {\int }_0^T{\widehat{p}}_i{\left(t\right)}^{\text{T}}\sigma \left(t,X\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right){\sigma }^{\text{T}}\left(t,X\left(t\right),{\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right){\widehat{p}}_i\left(t\right)}\right]<\infty .$ (27)

定理3.2 (充分最大值原理) 设$\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_n\right)\in {\mathcal{A}}_1\times {\mathcal{A}}_2\times \cdots \times {\mathcal{A}}_n$ 满足：

$\begin{array}{l}E\left[H_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}\right]\\ \le E\left[H_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}\right],\forall u_i\left(t\right)\in {\mathcal{A}}_i,P-a.s.\end{array}$ (28)

其中，$\widehat{X}\left(t\right)$ 是$\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_n\right)$ 对应的系统状态，且$\left({\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)$ 是$\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_n\right)$ 对应的伴随方程(5)的一组解，$i=1,2,\cdots ,n$ 。此外，对任意$t\in \left[0,T\right]$ ，哈密尔顿函数$H_i\left(t,x,u_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,u_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)$ 关于 $x,u_1,u_2,\cdots ,u_n$ 是凹的且$g_i\left(x\right)$ 关于x是凹的。则$\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_n\right)$ 是博弈问题(3)的纳什均衡点。

证明：首先考虑：

$\begin{array}{l}{J}_i\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)-J_i\left({\widehat{u}}_1,\cdots ,u_i,\cdots ,{\widehat{u}}_n\right)\\ =E[{\displaystyle {\int }_0^T\{f_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}\\ -{\displaystyle {\int }_0^T{f}_i\left(t,X^{\left(u_i\right)}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}\}\text{d}t+g_i\left(\widehat{X}\left(T\right)\right)-g_i\left(X^{\left(u_i\right)}\left(T\right)\right)],\end{array}$ (29)

由于g关于x的凹性，从而：

$E\left[g_i\left(\widehat{X}\left(T\right)\right)-g_i\left(X^{\left(u_i\right)}\left(T\right)\right)\right]\ge E\left[{\left(\widehat{X}\left(T\right)-X^{\left(u_i\right)}\left(T\right)\right)}^{\text{T}}\nabla g_i\left(\widehat{X}\left(T\right)\right)\right]$ (30)

伴随方程(5)的终值条件，可得：

$E\left[{\left(\widehat{X}\left(T\right)-X^{\left(u_i\right)}\left(T\right)\right)}^{\text{T}}\nabla g_i\left(\widehat{X}\left(T\right)\right)\right]=E\left[{\left(X^{\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)}\left(T\right)-X^{\left({\widehat{u}}_1,\cdots ,u_i,\cdots ,{\widehat{u}}_n\right)}\left(T\right)\right)}^{\text{T}}{\widehat{p}}_i\left(T\right)\right]$ (31)

对${\left(X^{\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)}\left(t\right)-X^{\left({\widehat{u}}_1,\cdots ,u_i,\cdots ,{\widehat{u}}_n\right)}\left(t\right)\right)}^{\text{T}}{\widehat{p}}_i\left(t\right)$用伊藤公式，可得：

$\begin{array}{l}E\left[{\left(X^{\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)}\left(T\right)-X^{\left({\widehat{u}}_1,\cdots ,u_i,\cdots ,{\widehat{u}}_n\right)}\left(T\right)\right)}^{\text{T}}{\widehat{p}}_i\left(T\right)\right]\\ =E[{\displaystyle {\int }_0^T{\left(\widehat{X}\left(t\right)-X^{\left(u_i\right)}\left(t\right)\right)}^{\text{T}}\left(-{\nabla }_x{H}_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)\right)\text{d}t}]\\ +{\displaystyle {\int }_0^T{\widehat{p}}_i^T\left(t\right)\{b\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}\\ -b\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)\}\text{d}t\\ +{\displaystyle {\int }_0^{T}tr[\sigma \left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}\\ -\sigma {\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i^{*}\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}^{\text{T}}{\widehat{q}}_i\left(t\right)]\text{d}t].\end{array}$ (32)

另一方面，根据哈密尔顿函数$H_i$ 的定义，有：

$\begin{array}{l}E[{\displaystyle {\int }_0^T\{f_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}-{\displaystyle {\int }_0^T{f}_i\left(t,X^{\left(u_i\right)}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}\}\text{d}t]\\ =E[{\displaystyle {\int }_0^T\{H_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}-H_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)]\\ -E[{\displaystyle {\int }_0^T{\widehat{p}}_i\left(t\right)\{b\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}\\ -b\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)\}\text{d}t]\\ -E[{\displaystyle {\int }_0^{T}tr\{\sigma \left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)}\\ {-\sigma \left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right)\right)\}}^{\text{T}}{\widehat{q}}_i\left(t\right)\text{d}t],\end{array}$ (33)

对比(29)和(32)得到：

$\begin{array}{l}{J}_i\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)-J_i\left({\widehat{u}}_1,\cdots ,u_i,\cdots ,{\widehat{u}}_n\right)\\ =E[{\displaystyle {\int }_0^T\{H_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)}\\ -H_1\left(t,X^{\left(u_i\right)}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)\}\text{d}t,\end{array}$ (34)

根据哈密尔顿函数$H_i$ 关于$x$ 和$u_i$ 的凹性，有：

$\begin{array}{l}{H}_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)\\ -H_i\left(t,X^{\left(u_i\right)}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)\\ \ge {\nabla }_x{H}_i{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)}^{\text{T}}\left(\widehat{X}\left(t\right)-X^{\left(u_i\right)}\left(t\right)\right)\\ +{\nabla }_{u_i}{H}_i{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)}^{\text{T}}\left({\widehat{u}}_i\left(t\right)-u_i\left(t\right)\right).\end{array}$ (35)

当$u_i={\widehat{u}}_i\left(t\right)$ 时，$E\left[H_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}\right]$ 达到最大，并且$u_i\left(t\right)$ 和${\widehat{u}}_i\left(t\right)$ 是${ℱ}_{t-{\delta }_i}$ 可测的，因此：

$\begin{array}{l}E\left[{\nabla }_{u_i}{H}_1{\left(t,X^{u_i}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)}^{\text{T}}\left({\widehat{u}}_i\left(t\right)-u_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}\right]\\ =\left({\widehat{u}}_i\left(t\right)-u_i\left(t\right)\right){\nabla }_{u_i}E\left[H_i\left(t,X^{u_i}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}\right]\ge 0,\end{array}$ (36)

式(35)结合式(34)，有：

$\begin{array}{l}E[H_i\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)\\ -H_i\left(t,X^{\left(u_i\right)}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,u_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)|{ℱ}_{t-{\delta }_i}]\\ \ge {\nabla }_x{H}_i{\left(t,\widehat{X}\left(t\right),{\widehat{u}}_1\left(t\right),\cdots ,{\widehat{u}}_i\left(t\right),\cdots ,{\widehat{u}}_n\left(t\right),{\widehat{p}}_i\left(t\right),{\widehat{q}}_i\left(t\right)\right)}^{\text{T}}\left(\widehat{X}\left(t\right)-X^{\left(u_i\right)}\left(t\right)\right).\end{array}$ (37)

因此，式(33)可以写为：

$J_i\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)-J_i\left({\widehat{u}}_1,\cdots ,u_i,\cdots ,{\widehat{u}}_n\right)\ge 0,$ (38)

因为式(37)对所有的$u_i\in {\mathcal{A}}_i$ ，$i=1,2,\cdots ,n$ ，都是成立的，我们有 $J_i\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)=\underset{u_i\in {\mathcal{A}}_i}{\sup }{J}_i\left({\widehat{u}}_1,\cdots ,u_i,\cdots ,{\widehat{u}}_n\right)$ ，即$\left({\widehat{u}}_1,\cdots ,{\widehat{u}}_i,\cdots ,{\widehat{u}}_n\right)$ 是博弈问题(3)的纳什均衡点。

4. 不对称信息下非零和博弈问题在高频金融市场中的应用

本节研究一类高频金融市场，利用随机微分博弈对其中三类参与者的交易行为进行建模，这三类参与者分别是算法交易商、一般交易商和做市商。其中算法交易商利用复杂算法和高速计算自动执行交易策略，一般交易商手动买卖金融资产，交易频率和速度低于高频算法交易，做市商持续提供买卖价差盈利，且这三者合作寻求最大利润，是非对抗性的，因此这是一个非零和的随机微分博弈问题。这类问题在文献[4]中已经进行了深入的研究，给出了纳什均衡点的解析表达式。本文在文献[4]的基础上，考虑了三类参与者获取信息存在不同步的时间延迟，更加符合真实的市场交易情况。

令$X_1\left(t\right),X_2\left(t\right),X_3\left(t\right)$ 分别表示t时刻算法交易商、普通交易商和做市商持有的头寸，且$X_0\left(t\right)$ 是风险资产的中间价格过程，则$X\left(t\right)=\left(X_0\left(t\right),X_1\left(t\right),X_2\left(t\right),X_3\left(t\right)\right)$ 满足如下的方程：

$\text{d}X\left(t\right)=\left(\begin{array}{c}\text{d}{X}_0\left(t\right)\\ \text{d}{X}_1\left(t\right)\\ \text{d}{X}_2\left(t\right)\\ \text{d}{X}_3\left(t\right)\end{array}\right)=\left(\begin{array}{c}\mu +\left({\gamma }_1{u}_1\left(t\right)+{\gamma }_2{}_t{u}_1\left(t\right)+\delta u_3\left(t\right)\right)\\ u_1\left(t\right)\\ u_2\left(t\right)\\ -\left(u_1\left(t\right)+u_2\left(t\right)\right)+k{u}_3\left(t\right)\end{array}\right)\text{d}t+\left(\begin{array}{c}{\sigma }_t\\ 0\\ 0\\ 0\end{array}\right)\text{d}W\left(t\right),t\in \left[0,T\right],$ (39)

$X\left(0\right)={\left(\begin{array}{cccc}{x}_0& 0& x_2& 0\end{array}\right)}^{\text{T}},$

其中，控制$u_i\left(t\right):\left[0,T\right]\times \Omega \to ℝ,i=1,2,\cdots ,n$ 分别是三种博弈者的策略，算法交易商、普通交易商分别控制他们的交易速度，做市商控制交易价格与中间价格的距离；$\mu ,{\gamma }_1,{\gamma }_2,\delta ,k$ 是常数，这里$\mu$ 表示了发生在市场之外的全球市场冲击，且在交易期间不断影响风险资产的中间价格；代表了交易速度对于中间价格的影响；当采取最佳出价订单时，订单被修改，然后下一个最佳出价和要价${\gamma }_1{u}_1\left(t\right)+{\gamma }_2{u}_1\left(t\right)$ 订单是被最后一个中间价格的某一个点为中心。通过$\delta u_3\left(t\right)$ 来表示中间价格的转变；$k{u}_3\left(t\right)$ 间价格的是做市场错过的算法交易商、普通交易商与第三方的成交量，${\sigma }_t$ 是一个确定的函数，具体表示为${\sigma }_t={\sigma }_0+vt$ 。

接下来，定义博弈者在$T$ 时刻的预期利润为他们的代价泛函$J_i\left(u_1,u_2,u_3\right),i=1,2,3$ ，具体形式如下：

• 算法交易商

$\begin{array}{l}{J}_1\left(u_1;u_2,u_3\right)=E[-{\displaystyle {\int }_0^T{u}_1\left(t\right)\left(X_0\left(t\right)+u_3\left(t\right)+\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)\right)}\text{d}t+X_1\left(T\right)X_0\left(T\right)\\ -\frac{{\eta }_1}{2}{\displaystyle {\int }_0^T{\left(X_t^1\right)}^2{\sigma }_t^2\text{d}t}-\frac{1}{2}{c}_1{X}_1{\left(T\right)}^2]\end{array}$ (40)

• 一般交易商

$\begin{array}{l}{J}_2\left(u_2;u_1,u_2\right)=E[-{\displaystyle {\int }_0^T{u}_2\left(t\right)\left(X_0\left(t\right)+u_3\left(t\right)+\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)\right)}\text{d}t+\left(X_2\left(T\right)X_0\left(T\right)-X_2\left(0\right)X_0\left(0\right)\right)\\ -\frac{{\eta }_2}{2}{\displaystyle {\int }_0^T{\left(X_t^2\right)}^2{\sigma }_t^2\text{d}t}-\frac{1}{2}{c}_2{X}_2{\left(T\right)}^2]\end{array}$ (41)

• 做市商

$\begin{array}{l}{J}_3\left(u_3;u_1,u_2\right)=E[-{\displaystyle {\int }_0^T\left(-\left(u_1\left(t\right)+u_2\left(t\right)\right)+k{u}_3\left(t\right)\right)\left(X_0\left(t\right)+u_3\left(t\right)+\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)\right)}\text{d}t\\ +X_3\left(T\right)X_3\left(T\right)-\frac{1}{2}{c}_3{X}_3{\left(T\right)}^2]\end{array}$ (42)

其中，$\left(X_0\left(t\right)+u_3\left(t\right)+\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)\right)$ 是最后实际的交易价格，第三项$\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)$ 表示的是算法交易商和一般交易商单位时间的成交量使得价格下滑，$\frac{1}{2}{c}_i{X}_i{\left(T\right)}^2,i=1,2,3$ 是持有的风险资产在时刻T的风险成本，无论在周期结束时的位置符号如何，它都是存在的；用成交总额的二次变差也就是$-\frac{{\eta }_i}{2}\langle X^0{X}^i\rangle ,i=1,2$ 来表示算法交易和一般交易商的风险厌恶。

现在我们的目标是去寻找纳什均衡点$\left({\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_n\right)\in {\mathcal{A}}_1\times {\mathcal{A}}_2\times {\mathcal{A}}_3$ ，使得：

$\begin{array}{l}{J}_1\left(u_1,{\widehat{u}}_2,{\widehat{u}}_3\right)\le J_1\left({\widehat{u}}_1,{\widehat{u}}_2,{\widehat{u}}_3\right),\forall u_1\left(t\right)\in {\mathcal{A}}_1,\\ J_2\left({\widehat{u}}_1,u_2,{\widehat{u}}_3\right)\le J_2\left({\widehat{u}}_1,{\widehat{u}}_2,{\widehat{u}}_3\right),\forall u_2\left(t\right)\in {\mathcal{A}}_2,\\ J_n\left({\widehat{u}}_1,{\widehat{u}}_2,u_3\right)\le J_n\left({\widehat{u}}_1,{\widehat{u}}_2,{\widehat{u}}_3\right),\forall u_n\left(t\right)\in {\mathcal{A}}_3.\end{array}$ (43)

这就是高频金融市场中的一个不对称信息下非零和的博弈问题，接下来就用我们推导出的定理3.1来刻画博弈问题(43)的纳什均衡点。

具体来说，在这个高频金融市场中的博弈问题中，系统的系数矩阵为：

$b\left(t,X\left(t\right),u\left(t\right)\right)=\left(\begin{array}{c}\mu +\left({\gamma }_1{u}_1\left(t\right)+{\gamma }_2{u}_1\left(t\right)+\delta u_3\left(t\right)\right)\\ u_1\left(t\right)\\ u_2\left(t\right)\\ -\left(u_1\left(t\right)+u_2\left(t\right)\right)+k{u}_3\left(t\right)\end{array}\right)$ (44)

和

$\sigma \left(t,X\left(t\right),u\left(t\right)\right)=\left(\begin{array}{c}{\sigma }_t\\ 0\\ 0\\ 0\end{array}\right).$ (45)

代价泛函中的过程效应如下：

$f_1\left(t,X\left(t\right),u\left(t\right)\right)=-u_1\left(t\right)\left(X_0\left(t\right)+u_3\left(t\right)+\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)\right)-\frac{{\eta }_1}{2}{\left(X_t^1\right)}^2{\sigma }_t^2,$ (46)

$f_2\left(t,X\left(t\right),u\left(t\right)\right)=-u_2\left(t\right)\left(X_0\left(t\right)+u_3\left(t\right)+\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)\right)-\frac{{\eta }_2}{2}{\left(X_t^2\right)}^2{\sigma }_t^2$ (47)

和

$f_3\left(t,X\left(t\right),u\left(t\right)\right)=-\left(-\left(u_1\left(t\right)+u_2\left(t\right)\right)+k{u}_3\left(t\right)\right)\left(X_0\left(t\right)+u_3\left(t\right)+\lambda \left(u_1\left(t\right)+u_2\left(t\right)\right)\right).$ (48)

终端效应分别为：

$g_1\left(X\left(T\right)\right)=X_1\left(T\right)X_0\left(T\right)-\frac{1}{2}{c}_1{X}_1{\left(T\right)}^2,$ (49)

$g_2\left(X\left(T\right)\right)=\left(X_2\left(T\right)X_0\left(T\right)-X_2\left(0\right)X_0\left(0\right)\right)-\frac{1}{2}{c}_2{X}_2{\left(T\right)}^2$ (50)

和

$g_3\left(X\left(T\right)\right)=X_3\left(T\right)X_3\left(T\right)-\frac{1}{2}{c}_3{X}_3{\left(T\right)}^2.$ (51)

令$P_i=\left(p_{i,0},p_{i,1},p_{i,2},p_{i,3}\right),i=1,2,3,4$ ，哈密尔顿函数(4)具体表示为：

$\begin{array}{l}{H}_1\left(t,X,u,p_1,q_1\right)=-u_1\left(X_0+u_3+\lambda \left(u_1+u_2\right)\right)+\left(\mu +\left({\gamma }_1{u}_1+{\gamma }_2{u}_2+\delta u_3\right)\right)p_{1,0}\\ +u_1{p}_{1,1}+{\chi }_t{u}_2{p}_{1,2}-\left(u_1+u_2\right)p_{1,3}+k{u}_3{p}_{1,3}-\frac{{\eta }_1}{2}{\left(X^1\right)}^2{\sigma }_t^2,\end{array}$ (52)

$\begin{array}{l}{H}_2\left(t,X,u,p_2,q_2\right)=-{\chi }_t{u}_2\left(X_0+u_3+\lambda \left(u_1+u_2\right)\right)+\left(\mu +\left({\gamma }_1{u}_1+{\gamma }_2{u}_2+\delta u_3\right)\right)p_{2,0}\\ +u_1{p}_{2,1}+u_2{p}_{2,2}-\left(u_1+u_2\right)p_{2,3}+k{u}_3{p}_{2,3}-\frac{{\eta }_2}{2}{\left(X_t^2\right)}^2{\sigma }_t^2\end{array}$ (53)

和

$\begin{array}{l}{H}_3\left(t,X,u,p_3,q_3\right)=\left(-\left(u_1+u_2\right)+k{u}_3\right)\left(X_0+u_3+\lambda \left(u_1+u_2\right)\right)+\left(\mu +\left({\gamma }_1{u}_1+{\gamma }_2{}_t{u}_2+\delta u_3\right)\right)p_{3,0}\\ +u_1{p}_{3,1}+u_2{p}_{3,2}-\left(u_1+u_2\right)p_{3,3}+k{u}_3{p}_{3,3}.\end{array}$ (54)

伴随方程(5)表示为如下的倒向随机微分方程组：

$\{\begin{array}{l}d{P}_{1,0}\left(t\right)=-u_1\left(t\right)\text{d}t+q_{1,0}\left(t\right)\text{d}{W}_t,P_{1,0}\left(T\right)=X_1\left(T\right);\\ \text{d}{P}_{1,1}\left(t\right)=-{\eta }_1{X}_t^1{\sigma }_t^2\text{d}t+q_{1,1}\left(t\right)\text{d}{W}_t,P_{1,1}\left(T\right)=X_0\left(T\right)-c_1{X}_1\left(T\right);\\ \text{d}{P}_{1,2}\left(t\right)=q_{1,2}\left(t\right)\text{d}{W}_t,P_{1,2}\left(T\right)=0;\\ \text{d}{P}_{1,3}\left(t\right)=q_{1,3}\left(t\right)\text{d}{W}_t,P_{1,3}\left(T\right)=0;\\ \text{d}{P}_{2,0}\left(t\right)=-u_2\left(t\right)\text{d}t+q_{2,0}\left(t\right)\text{d}{W}_t,P_{2,0}\left(T\right)=X_2\left(T\right);\\ \text{d}{P}_{2,1}\left(t\right)=q_{2,1}\left(t\right)\text{d}{W}_t,P_{2,1}\left(T\right)=0;\\ \text{d}{P}_{2,2}\left(t\right)=-{\eta }_2{X}_t^2{\sigma }_t^2\text{d}t+q_{2,2}\left(t\right)\text{d}{W}_t,P_{2,2}\left(T\right)=X_0\left(T\right)-c_2{X}_2\left(T\right);\\ \text{d}{P}_{2,3}\left(t\right)=q_{2,3}\left(t\right)\text{d}{W}_t,P_{2,3}\left(T\right)=0;\\ \text{d}{P}_{3,0}\left(t\right)=-\left(-\left(u_1\left(t\right)+u_2\left(t\right)\right)+k{u}_3\left(t\right)\right)\text{d}t+q_{3,0}\left(t\right)\text{d}{W}_t,P_{3,0}\left(T\right)=X_3\left(T\right);\\ \text{d}{P}_{3,1}\left(t\right)=q_{3,1}\left(t\right)\text{d}{W}_t,P_{3,1}\left(T\right)=0;\\ \text{d}{P}_{3,2}\left(t\right)=q_{3,2}\left(t\right)\text{d}{W}_t,P_{3,2}\left(T\right)=0;\\ \text{d}{P}_{3,3}\left(t\right)=q_{3,3}\left(t\right)\text{d}{W}_t,P_{3,3}\left(T\right)=X_0\left(T\right)-c_3{X}_3\left(T\right).\end{array}$ (55)

进一步根据倒向随机微分方程理论，由于终值条件满足：

$p_{1,2}\left(T\right)=p_{1,3}\left(T\right)\equiv 0,p_{2,1}\left(T\right)=p_{2,3}\left(T\right)=0,p_{3,1}\left(T\right)=p_{3,2}\left(T\right)=0,$ (56)

则可解出：

$q_{1,2}\left(t\right)=q_{1,3}\left(t\right)\equiv 0,q_{2,1}\left(t\right)=q_{2,3}\left(t\right)=0,q_{3,1}\left(t\right)=q_{3,2}\left(t\right)=0,$ (57)

从而伴随方程(54)可以简化为：

$\{\begin{array}{l}\text{d}{P}_{1,0}\left(t\right)=-u_1\left(t\right)dt+q_{1,0}\left(t\right)d{W}_t,\begin{array}{cc}{P}_{1,0}\left(T\right)=X_1\left(T\right);& \end{array}\\ \text{d}{P}_{1,1}\left(t\right)=-{\eta }_1{X}_t^1{\sigma }_t^2\text{d}t+q_{1,1}\left(t\right)\text{d}{W}_t,P_{1,1}\left(T\right)=X_0\left(T\right)-c_1{X}_1\left(T\right);\\ \text{d}{P}_{1,2}\left(t\right)=0,P_{1,2}\left(T\right)=0;\\ \text{d}{P}_{1,3}\left(t\right)=0,P_{1,3}\left(T\right)=0;\\ \text{d}{P}_{2,0}\left(t\right)=-u_2\left(t\right)\text{d}t+q_{2,0}\left(t\right)\text{d}{W}_t,P_{2,0}\left(T\right)=X_2\left(T\right);\\ \text{d}{P}_{2,1}\left(t\right)=0,P_{2,1}\left(T\right)=0;\\ \text{d}{P}_{2,2}\left(t\right)=-{\eta }_2{X}_t^2{\sigma }_t^2\text{d}t+q_{2,2}\left(t\right)\text{d}{W}_t,P_{2,2}\left(T\right)=X_0\left(T\right)-c_2{X}_2\left(T\right);\\ \text{d}{P}_{2,3}\left(t\right)=0,P_{2,3}\left(T\right)=0;\\ \text{d}{P}_{3,0}\left(t\right)=-\left(-\left(u_1\left(t\right)+u_2\left(t\right)\right)+k{u}_3\left(t\right)\right)\text{d}t+q_{3,0}\left(t\right)\text{d}{W}_t,P_{3,0}\left(T\right)=X_3\left(T\right);\\ \text{d}{P}_{3,1}\left(t\right)=0,P_{3,1}\left(T\right)=0;\\ \text{d}{P}_{3,2}\left(t\right)=0,P_{3,2}\left(T\right)=0;\\ \text{d}{P}_{3,3}\left(t\right)=q_{3,3}\left(t\right)\text{d}{W}_t,P_{3,3}\left(T\right)=X_0\left(T\right)-c_3{X}_3\left(T\right).\end{array}$ (58)

最后根据定理3.1得到最大值条件：

$E\left[-2\lambda {\widehat{u}}_1-\lambda {\widehat{u}}_2-{\widehat{u}}_3-X_0+{\gamma }_1{P}_{1,0}+P_{1,1}|{ℱ}_{t-{\delta }_1}\right]=0,$ (59)

$E\left[-2\lambda {\widehat{u}}_2-\lambda {\widehat{u}}_1-{\widehat{u}}_3-X_0+{\gamma }_2{P}_{2,0}+P_{2,2}|{ℱ}_{t-{\delta }_2}\right]=0$ (60)

和

$E\left[2k{\widehat{u}}_3+\left(1-k\lambda \right)\left({\widehat{u}}_1+{\widehat{u}}_2\right)-k{X}_0+\delta P_{3,0}+k{P}_{3,3}|{ℱ}_{t-{\delta }_3}\right]=0.$ (61)

综合以上推导过程，我们总结定理如下：

定理4.1 在滤子概率空间$\left(\Omega ,ℱ,{\left\{{ℱ}_t\right\}}_{t\ge 0},P\right)$ 中，建立随机博弈系统(1)~(3)，可以得到此纳什均衡点$\widehat{u}$ 满足正倒向随机微分方程组：

$\{\begin{array}{l}\text{d}X\left(t\right)=b\left(t,X\left(t\right),\widehat{u}\left(t\right)\right)\text{d}t+\sigma \left(t,X\left(t\right),\widehat{u}\left(t\right)\right)\text{d}W\left(t\right),\\ \text{d}{p}_i\left(t\right)=-{\nabla }_x{H}_i\left(t,X\left(t\right),\widehat{u}\left(t\right),p_i\left(t\right),q_i\left(t\right)\right)\text{d}t+q_i\left(t\right)\text{d}W\left(t\right),\\ X\left(0\right)=x\in R^n{p}_i\left(T\right)=\nabla g_i\left(X\left(T\right)\right).\end{array}$ (62)

其中控制$\left({\widehat{u}}_1\left(t\right),{\widehat{u}}_2\left(t\right),{\widehat{u}}_3\left(t\right)\right)$ 满足如下式子：

$\{\begin{array}{l}{\widehat{u}}_1=1/2\lambda E[-\lambda {\widehat{u}}_2-{\widehat{u}}_3-X_0+{\gamma }_1{P}_{1,0}+P_{1,1}|{ℱ}_{t-{\delta }_1}],\\ {\widehat{u}}_2=1/2\lambda E[-\lambda {\widehat{u}}_1-{\widehat{u}}_3-X_0+{\gamma }_2{P}_{2,0}+P_{2,2}|{ℱ}_{t-{\delta }_2}],\\ {\widehat{u}}_3=1/\left(-2k\right)E[\left(1-k\lambda \right)\left({\widehat{u}}_1+{\widehat{u}}_2\right)-k{X}_0+\delta P_{3,0}+k{P}_{3,3}|{ℱ}_{t-{\delta }_3}].\end{array}$ (63)

5. 结论

本文推导了不对称信息下非零和随机微分博弈的充分和必要最大值原理来刻画纳什均衡点，并应用于高频金融市场中算法交易商、一般交易商和做市商的随机微分博弈。在理论上，我们补充和发展了文献[14]中的理论结果，得到了多人非零和随机微分博弈的最大值原理。在应用上，基于文献[3]的模型，进一步假设三类博弈者获取信息存在延迟且不同步，利用最大值原理将求解纳什均衡点的问题转化成求解FBSDE的问题，从而用FBSDE的解来刻画纳什均衡点。但是需要指出的是：由于FBSDE中存在条件数学期望，使得它不是一个线性的FBSDE，从而不能用经典的线性方法来直接求解，因此本文没有给出纳什均衡点的显示解，FBSDE以及纳什均衡点的求解方法将是我们接下来的研究方向。

NOTES

^*通讯作者。