## Linear-Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations with Jumps

Tang Maoning,, Meng Qingxin,

 基金资助: 国家自然科学基金.  11871121浙江省自然科学基金杰出青年基金项目.  LR15A010001

 Fund supported: the NSFC.  11871121the Natural Science Foundation of Zhejiang Province for Distinguished Young Scholar.  LR15A010001

Abstract

This paper is concerned with a linear quadratic optimal control problem for meanfield backward stochastic differential equations driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control is obtained. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a decoupling technique, the stochastic Hamilton system is decoupled by introducing two Riccati equations and a MF-BSDE with jumps. Then an explicit representation for the optimal control is obtained.

Keywords： Mean-field ; Optimal control ; Backward stochastic Differential equation ; Adjoint process

Tang Maoning, Meng Qingxin. Linear-Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations with Jumps. Acta Mathematica Scientia[J], 2019, 39(3): 620-637 doi:

## 1 引言

### 2.1 记号

$T$是一个固定的严格正实数. $(\Omega, {\cal F}, \{{\cal F}_t\}_{0\leq t\leq T}, P)$是一个完备的概率空间,其上定义一个一维标准布朗运动$\{W(t), 0\leq t\leq T\}$.${\cal P}$$[0, T]\times \Omega上的{\cal F}_t -可料\sigma -代数, {\cal B}(\Lambda)为任何拓扑空间\Lambda的Borel \sigma -代数.设(E, {\cal B} (E), \nu)是满足\nu(E)<\infty 的可测空间, \eta: \Omega\times D_\eta \longrightarrow E是具有特征测度\nu$${\cal F}_t$ -适应的平稳Poisson点过程,其中$D_\eta$$(0, \infty)的可数子集.那么,由\eta诱导的计数测度为 \tilde{\mu}({\rm d}e, {\rm d}t):=\mu({\rm d}e, {\rm d}t)-\nu({\rm d}e){\rm d}t是独立于布朗运动\{W(t), 0\leq t\leq T\}的补偿Poisson随机鞅测度.假设\{{\cal F}_t\}_{0\leq t\leq T}是由\{W(t), 0\leq t\leq T\}$$\{\iint_{A\times (0, t] }\tilde{\mu}({\rm d}e, {\rm d}s), 0\leq t\leq T, A\in {\cal B} (E) \}$共同生成的$P$ -完备自然流.接下来,给出文章中需要使用的基本记号.

$\bullet$$s: s\in[t, T). \bullet$$H$:范数为$\|\cdot\|_H$的Hilbert空间.

$\bullet$$\langle\alpha, \beta\rangle:$$\mathbb{R} ^n$空间上的内积. $\forall \alpha, \beta\in\mathbb{R} ^n.$

$\bullet$$|\alpha|=\sqrt{\langle\alpha, \alpha\rangle}:$$\mathbb{R} ^n$空间上的范数, $\forall \alpha\in\mathbb{R} ^n.$

$\bullet$$\langle A, B\rangle={\rm tr}(AB^\top):$$\mathbb{R} ^{n\times m}$空间上的内积, $\forall A, B\in \mathbb{R} ^{n\times m}.$这里$B^\top$表示$B$的转置.

$\bullet$$|A|=\sqrt{{\rm tr}(AA^\top)}:矩阵A的范数. \bullet$$ S^n:$$n\times n对称矩阵的全体. \bullet$$ S^n_+:$$S^n中非负定矩阵的全体. \bullet$$S_{{\cal F}}^2(t, T;H):$全体$H$ -值的右连左极${{\cal F}}_s$ -适应过程$f=\{f(s, \omega), \ (s, \omega)\in[t, T]\times\Omega\}$组成的空间,且满足

$\bullet$$M_{{\cal F}}^2(t, T;H):全体H -值的{{\cal F}}_s -适应过程f=\{f(s, \omega), \(s, \omega)\in[0, T]\times\Omega\}组成的空间,且满足 \bullet$${M}^{\nu, 2}( E; H):$定义在可测空间$(E, {\cal B}(E); \nu)$上的全体$H$ -值可测函数$r=\{r(e), e \in E\}$构成的空间,且满足

$\bullet$${M}_{{\cal F}}^{\nu, 2}{([t, T]\times E; H)}:全体{M}^{\nu, 2}( E; H) -值的{{\cal F}}_s -可料过程r=\{r(s, \omega, e), \(s, \omega, e)\in[t, T]\times\Omega\times E\}构成的空间,且满足 \bullet$$L^2(\Omega, {{\cal F}}, P;H):$定义在概率空间$(\Omega, {{\cal F}}, P)$上的全体$H$ -值随机变量$\xi$构成的空间,且满足

### 2.2 问题的提出

$$$\label{eq:1.1}\left\{\begin {array}{rl} {\rm d}Y(s)=&\bigg\{A(s)Y(s)+\bar A(s){\Bbb E} [Y(s)] +B(s)u(s)+\bar B(s){\Bbb E} [u(s)] \\& +C(s)Z(s)+\bar C(s){\Bbb E} [Z(s)]+\displaystyle\int_{E} D(s, e)R(s, e)\nu ({\rm d}e) \\[4mm] & +\displaystyle\int_{E}\bar D(s, e) {\Bbb E} [R(s, e)]\nu ({\rm d}e)\bigg\}{\rm d}s+ Z(s){\rm d}W(s) \\[4mm]& +\displaystyle \int_{E} R(s, e)\tilde{\mu}({\rm d}e, {\rm d}s), s\in [t, T], \\Y(T)=&\xi\end {array}\right.$$$

$\begin{eqnarray}\label{eq:1.2} J(t, \xi; u(\cdot))&=&{\Bbb E}\bigg[\int_t^T\bigg(\langle Q(s)Y(s), Y(s)\rangle+ \langle \bar {Q}(s){\Bbb E}[Y(s)], {\Bbb E}[Y(s)]\rangle+\langle N_1(s)Z(s), Z(s)\rangle\\&&+\langle\bar N_1(s){\Bbb E}[Z(s)], {\Bbb E}[Z(s)]\rangle+ \int _{E}\langle N_2(s, e )R(s, e), R(s, e)\rangle\nu({\rm d}e)\\&&+ \int _{E}\langle \bar N_2(s, e ){\Bbb E}[R(s, e)], {\Bbb E}[R(s, e)]\rangle\nu({\rm d}e)+\langle N_3(s)u(s), u(s)\rangle\\&&+\langle\bar{N}_3(s){\Bbb E}[u(s)], {\Bbb E}[u(s)]\rangle\bigg){\rm d}s+\langle GY(t), Y(t)\rangle +\langle \bar{ G}{\Bbb E}[Y(t)], {\mathbbE}[Y(t)]\rangle \bigg], \end{eqnarray}$

$$$\label{eq:b7}J(t, \xi;{u}^*(\cdot))=\displaystyle\inf\limits_{u(\cdot)\in {{\cal A}[t, T]}}J(t, \xi; u(\cdot))$$$

$\begin{eqnarray}\label{eq:1.7} |J(t, \xi; u(\cdot))|\leq K \bigg\{ ||\Lambda(\cdot)||_{M^2[t, T]}^2 +||u(\cdot)||_{{\cal A}[t, T]}^2\bigg\} \leq K \bigg\{||u(\cdot)||_{{\cal A}[t, T]}^2+ {\Bbb E}[|\xi|^2]\bigg\} < \infty, \end{eqnarray}$

$\begin{eqnarray} ||{\Bbb E} [\Phi]||^2_H\leq {\Bbb E}[||\Phi||_H^2]. \end{eqnarray}$

## 3 最优控制的存在唯一性

$\begin{eqnarray}\label{eq:5.10}&&| J (t, \xi; u (\cdot)) - J (t, \xi; \bar u(\cdot) ) |^2 \nonumber\\&\leq& K \Big \{ ||\Lambda(\cdot)-\bar\Lambda (\cdot)||^2_{M^2[t, T]}+||u(\cdot)-\bar u(\cdot)||^2_{{\cal A}[t, T]}\Big \} \nonumber \\&& \times \Big\{ ||\Lambda(\cdot)||^2_{M^2[t, T]}+||u(\cdot)||^2_{{\cal A}[t, T]}+||\bar\Lambda (\cdot)||^2_{M^2[t, T]}+||\bar u(\cdot)||^2_{{\cal A}[t, T]}\Big \} .\end{eqnarray}$

$\begin{eqnarray}| J (t, \xi; u (\cdot)) - J (t, \xi; \bar u (\cdot)) |^2 \nonumber&\leq& K \Big \{ ||u(\cdot)-\bar u(\cdot)||^2_{{\cal A}[t, T]}\Big \} \\ && \times \Big\{||u(\cdot)||^2_{{\cal A}[t, T]}+||\bar u(\cdot)||^2_{{\cal A}[t, T]}+{\Bbb E}[|\xi|^2]\Big \} , \end{eqnarray}$

$$$J (t, \xi; u (\cdot)) - J (t, \xi; \bar u (\cdot)) \rightarrow 0 , \quad {\rm as} \quad u (\cdot) \rightarrow \bar u (\cdot) \quad {\rm in} \quad {{\cal A}[t, T]} .$$$

$\begin{eqnarray}\label{eq:2.5} J(t, \xi; u(\cdot)) &\geq& {\Bbb E}\bigg[\int_t^T \bigg (\langle N_3(s)(u(s)-{\Bbb E}[u(s)]), u(s)-{\Bbb E}[u(s)]\rangle\\ && +\langle (N_3(s)+{\bar N}_3(s)){\Bbb E}[u(s)], {\Bbb E}[u(s)]\rangle \bigg ){\rm d}s\bigg]\\& \geq &\delta {\Bbb E}\bigg[\int_t^T\langle u(s)-{\Bbb E}[u(s)], u(s)-{\Bbb E}[u(s)]\rangle {\rm d}s\bigg]+ \delta{\Bbb E}\bigg[\int_t^T \langle {\Bbb E}[u(s)], {\Bbb E}[u(s)]\rangle {\rm d}s\bigg] \\ &=& \delta{\Bbb E} \bigg[\int_t^T |u(s)|^2{\rm d}s\bigg]=\delta ||u(\cdot)||^2_{{\cal A}[t, T]}, \end{eqnarray}$

$\begin{eqnarray}\label{eq:2.8}&& Y^{(t, \xi; u+v)}(s)=Y^{(t, \xi; u)}(s)+Y^{(t, 0; v)}(s), \\ &&Z^{(t, \xi; u+v)}(s)=Z^{(t, \xi; u)}(s)+Z^{(t, 0; v)}(s), \\ &&R^{(t, \xi; u+v)}(s)=R^{(t, \xi; u)}(s)+R^{(t, 0; v)}(s), t\leq s\leq T.\end{eqnarray}$

$J(t,\xi ;u( \cdot ) + v( \cdot )) - J(t,\xi ;u( \cdot )) = J(t,0;v( \cdot )) + {\Delta ^{u,v}}.$

$$$|J(t, 0;v(\cdot))| \leq K||v(\cdot) ||^2_{{\cal A}[t, T]}.$$$

$$$\displaystyle\lim\limits_ {\|v(\cdot) \|_{{\cal A}[t, T]} {\rightarrow0}}\frac{|J(t, \xi; u(\cdot) +v(\cdot) )-J(t, \xi; u(\cdot) )-\Delta^{u, v}|}{||v(\cdot) ||_{{\cal A}[t, T]}}=\displaystyle\lim\limits_ {\|v(\cdot) \|_{{\cal A}[t, T]} {\rightarrow0}}\frac{|J(t, 0; v(\cdot)|}{||v(\cdot) ||_{{\cal A}[t, T]}}=0.$$$

$\begin{eqnarray} \label{eq:2.10}\lim\limits_ {\varepsilon {\rightarrow0}}\frac{J(t, \xi; u(\cdot) +\varepsilon v(\cdot) )-J(t, \xi; u(\cdot) )}{\varepsilon}&=&\displaystyle\lim\limits_ {\varepsilon {\rightarrow0}}\frac{J(t, 0; \varepsilon v(\cdot))+\Delta^{u, \varepsilon v}}{\varepsilon }\\&=&\displaystyle\lim\limits_ {\varepsilon {\rightarrow0}}\frac{\varepsilon^2J(t, 0; v(\cdot))+\varepsilon\Delta^{u, v}}{\varepsilon}\\&=&\Delta^{u, v}\\&=&\langle J'(t, \xi; u(\cdot)), v(\cdot) \rangle.\end{eqnarray}$

既然可允许控制集${\cal A}[t, T]= M^2_{{\cal F}}(t, T;\mathbb{R} ^m)$是一个自反的Banach空间,根据引理3.1-3.3,问题2.1的最优控制的存在唯一性可以由文献[7]的命题2.12直接获得.(即定义在自反的Banach空间上的强制的,严格凸且下半连续的泛函存在一个唯一的最小值点).证明完毕.

$$$\label{eq:b16} \langle J'(t, \xi; u(\cdot) ), v(\cdot) \rangle = 0,$$$

$\begin{eqnarray}\label{eq:2.13} 0&=&2{\Bbb E}\bigg[\int_t^T\bigg(\langle Q(s)Y^{(t, \xi;u)}(s), Y^{(t, 0;v)}(s)\rangle +\langle \bar Q(s){\Bbb E} [Y^{(t, \xi;u)}(s)], {\Bbb E}[Y^{(t, 0;v)}(s)]\rangle\\ &&+\langle N_1(s)Z^{(t, \xi;u)}(s), Z^{(t, 0;v)}(s)\rangle +\langle \bar N_1(s){\Bbb E} [Z^{(t, \xi;u)}(s)], {\Bbb E}[Z^{(t, 0;v)}(s)]\rangle \\ &&+\int_{E}\langle N_2(s, e)R^{(t, \xi;u)}(s, e), R^{(t, 0;v)}(s, e)\rangle \nu({\rm d}e)\\&&+\int_{E}\langle \bar N_2(s){\Bbb E} [R^{(t, \xi;u)}(s, e)], {\Bbb E}[R^{(t, 0;\xi)}(s)]\rangle \nu({\rm d}e)\\&&+\langle N_3(s)u(s), v(s)\rangle+\langle \bar N_3(s){\Bbb E}[u(s)], {\Bbb E} [v(s)]\rangle \bigg){\rm d}s\bigg]\\&&+2{\Bbb E}\Big[\langle G Y^{(t, \xi;u)}(t), Y^{(t, 0;v)}(t)\rangle+\langle\bar G{\Bbb E}[Y^{(t, \xi;u)}(t)], {\Bbb E}[Y^{(t, 0;v)}(t)]\rangle\Big] , \ \forallu(\cdot), v(\cdot)\in{\cal A}[t, T].\\ &&\end{eqnarray}$

对于必要性,假设$u(\cdot)$是一个最优控制.由(3.12)式,对于任意可允许控制$v(\cdot)$$0< \varepsilon \leq 1, $$\langle J'(t, \xi; u(\cdot)), v(\cdot) \rangle= \displaystyle\lim\limits_ {\varepsilon {\rightarrow0^{+}}}\frac{J(t, \xi; u(\cdot) +\varepsilon v(\cdot) )-J(t, \xi; u(\cdot) )}{\varepsilon}\geq 0$$ $$-\langle J'(t, \xi; u(\cdot)), v(\cdot)\rangle=\langle J'(t, \xi; u(\cdot)), -v(\cdot) \rangle= \displaystyle\lim\limits_ {\varepsilon {\rightarrow0^{+}}}\frac{J(t, \xi; u(\cdot) +\varepsilon (-v(\cdot)) )-J(t, \xi; u(\cdot) )}{\varepsilon}\geq 0.$$ 这意味着 \langle J'(t, \xi; u(\cdot)), v(\cdot) \rangle = 0. 对于充分性,设u(\cdot)是一个给定的可允许控制使得对于任意可允许控制v(\cdot),$$ \langle J'(t, \xi; u(\cdot) ), $$v(\cdot) \rangle = 0.因为性能指标函数J$${\cal A}[t, T]$上是凸的,故

$J(t, \xi; v(\cdot))-J(t, \xi; u(\cdot)) \geq \langle J'(t, \xi; u(\cdot) ), v(\cdot) \rangle = 0,$

### 4.1 随机哈密顿系统

$\begin{eqnarray} \label{eq:3.1000}N_3(s)u(s)+\bar N_3(s){\Bbb E}[u(s)]+B(s)^\top k({s-})+\bar B(s)^\top {\Bbb E} [k({s-})]=0, \quad {\rm a.s.}, \end{eqnarray}$

$\begin {equation}\label{eq:3.2}\left\{\begin{array}{rl}{\rm d}k(s)=&-\Big[A(s)^\top k(s)+\bar A(s)^\top{\Bbb E} [k(s)]+Q(s)Y(s)+\bar Q(s){\Bbb E}[Y(s)]\Big]{\rm d}s\\&-\Big[C^\top(s)k(s)+\bar C(s)^\top{\Bbb E} [k(s)]+N_1(s)Z(s)+\bar N_1(s){\Bbb E}[Z(s)]\Big]{\rm d}W(s)\\&-\displaystyle\int_ E \Big[D(s, e)^\top k(s-)+\bar D(s, e)^\top{\Bbb E} [k(s-)]\\&+N_2(s, e)R(s, e)+\bar N_2(s, e){\Bbb E}[R(s, e)]\Big]\tilde{\mu}({\rm d}e, {\rm d}s), \\ k(t)=&-GY(t)-\bar G{\Bbb E}[Y(t)], s\in [t, T].\end{array} \right. \end {equation}$

设$u(\cdot)\in {\cal A}[t, T]$是一可允许控制过程,其相应的状态过程为$(Y(\cdot), Z(\cdot), R(\cdot, \cdot))$.因此对于任意可允许控制过程$v(\cdot) \in {{\cal A}[t, T]},$由引理3.3,有

$\begin{eqnarray}\label{eq:3.3}& &\langle J'(t, \xi; u(\cdot)), v(\cdot) \rangle \\ &=&2{\mathbbE}\bigg[\int_t^T\bigg(\langle QY^{(t, \xi;u)}, Y^{(t, 0;v)}\rangle +\langle \bar Q{\Bbb E} [Y^{(t, \xi;u)}], {\Bbb E}[Y^{(t, 0;v)}]\rangle +\langle N_1Z^{(t, \xi;u)}, Z^{(t, 0;v)}\rangle\\&&+\langle \bar N_1{\Bbb E} [Z^{(t, \xi;u)}], {\Bbb E}[Z^{(t, 0;v)}]\rangle +\int_{E}\langle N_2R^{(t, \xi;u)}, R^{(t, 0;v)}\rangle \nu({\rm d}e)\\&&+\int_{E}\langle \bar N_2{\Bbb E} [R^{(t, \xi;u)}], {\Bbb E}[R^{(t, 0;\xi)}]\rangle \nu({\rm d}e)+\langle N_3u, v\rangle+\langle \bar N_3{\Bbb E}[u], {\Bbb E} [v]\rangle \bigg){\rm d}s\bigg]\\&&+2{\Bbb E}\Big[\langle G Y^{(t, \xi;u)}(t), Y^{(t, 0;v)}(t)\rangle+\langle\bar G{\Bbb E}[Y^{(t, \xi;u)}(t)], {\Bbb E}[Y^{(t, 0;v)}(t)]\rangle\Big].\end{eqnarray}$

由定理3.4可知,问题2.1存在唯一最优的四元组$(u(\cdot), Y(\cdot), Z(\cdot), $$R(\cdot, \cdot)).假设k(\cdot)是与最优{四元组}相对应的对偶方程(4.2)的唯一解.根据定理4.1的必要性,最优控制u(\cdot)有对偶刻画(4.1).因此, (u(\cdot), Y(\cdot), Z(\cdot), R(\cdot, \cdot), k(\cdot))组成随机哈密尔顿系统(4.8)的一个适应解.下面证明随机哈密尔顿系统(4.8)解的唯一性.如果随机哈密尔顿系统(4.8)有另一个适应解(\bar u(\cdot), \bar Y(\cdot), \bar Z(\cdot), \bar R(\cdot, \cdot), \bar k(\cdot)).根据定理4.1的充分性, (\bar u(\cdot), \bar Y(\cdot), \bar Z(\cdot), \bar R(\cdot, \cdot))必定是问题2.1的最优四元组.因此,通过最优控制的唯一性,必有 u(\cdot)= \baru(\cdot).此外,根据MF-SDE和MF-BSDE解的唯一性,必有(\bar Y(\cdot), \bar Z(\cdot), \bar R(\cdot, \cdot), \bar k(\cdot))=(Y(\cdot), Z(\cdot), R(\cdot, \cdot), k(\cdot)).因此,随机哈密顿系统(4.8)存在唯一适应解.证明完毕. 总之,随机哈密尔顿系统(4.8)完全刻画了问题2.1的最优控制.因此,求解问题2.1等价于求解随机哈密尔顿系统(4.8),并且唯一的最优控制可以通过(4.1)式给出.对(4.1)式的两边取期望可得 $$\label{eq:3.8} N_3(s){\Bbb E}[u(s)]+\bar N_3(s){\Bbb E}[u(s)]+B(s)^\top{\Bbb E}[k({s-})]+\bar B(s)^\top {\Bbb E} [k({s-})]=0, \quad {\rm a.e.},$$ 这意味着 $$\label{eq:3.9} {\Bbb E}[u(s)] =-(N_3(s)+\bar N_3(s))^{-1}(B (s)+\bar B (s))^\top {\Bbb E} [k({s-})], \quad {\rm a.e.}.$$ 再由(4.1)式可以得到 $$\label{eq:3.10}N_3(s)u(s)=-\bar N_3(s){\Bbb E}[u(s)]-B(s)^\top k({s-})-\bar B(s)^\top {\Bbb E} [k({s-})], \quad {\rm a.e.}.$$ 再将(4.9)式代入(4.10)式,有 \begin{eqnarray} \label{eq:3.6} u(s)&=&-N_3^{-1}(s)\Big [B (s)^\top k({s-})+\bar B (s)^\top {\Bbb E} [k({s-})]\\&&-\bar N_3(s)(N_3(s)+\bar N_3(s))^{-1}(B (s)+\bar B (s))^\top {\Bbb E} [k({s-})]\Big], \quad {\rm {\rm a.e.}}. \end{eqnarray} ### 4.2 黎卡提方程的推导 由定理4.2可知假设2.1和2.2下,随机哈密尔顿系统(4.8)存在唯一解(u(\cdot), Y(\cdot), Z(\cdot),$$ R(\cdot, \cdot), k(\cdot)), $$(u(\cdot); Y(\cdot), Z(\cdot), R(\cdot, \cdot))是相应的最优四元组.注意到此时u(\cdot)可由(Y(\cdot), Z(\cdot),$$ R(\cdot, \cdot))$$k(\cdot)对偶表示出来.现在想仅用 k(\cdot)$$u(\cdot)$进行刻画.为此,如文献[11],利用对FBSDEs的通常的解耦技术,这将推导出两个Riccati方程.接下给出详细的推导过程.设$(u(\cdot), Y(\cdot), Z(\cdot), R(\cdot, \cdot), k(\cdot))$是随机哈密顿系统(4.8)的唯一解.对随机哈密顿系统(4.8)两边求期望,可得$({\Bbb E}[u(\cdot)], {\Bbb E}[Y(\cdot)], {\Bbb E}[Z(\cdot)], {\Bbb E}[R(\cdot, \cdot)], {\Bbb E}[k(\cdot)])$满足如下的正倒向常微分方程

$$$\label{eq:4.1}\left\{\begin {array}{rl}{\rm d}E[k(s)]=&-\Big[(A^\top(s)+\bar A(s)^\top){\Bbb E} [k(s)]+(Q(s)+\bar Q(s)){\Bbb E}[Y(s)]\Big], \\{\rm d}{\Bbb E}[Y(s)]=&\Big[(A(s)+\bar A(s)){\Bbb E} [Y(s)] +(B(s)+\bar B(s)){\Bbb E} [u(s)]+(C(s)+\bar C(s)){\Bbb E} [Z(s)] \\ &+\displaystyle\int_{E} (D(t, e) +\bar D(s, e)) {\Bbb E} [R(s, e)]\nu ({\rm d}e)\Big]{\rm d}s, \\{\Bbb E}[Y(T)]=& {\Bbb E}[\xi], \; {\Bbb E}[k(t)]=-(G+\bar G){\Bbb E}[Y(t)], \\ 0=&(N_3(s)+\bar N_3(s)){\Bbb E}[u(s)]+(B(s)^\top+\bar B(s)^\top) {\Bbb E} [k({s-})].\end {array}\right.$$$

$$$\label{eq:3.13}\left\{\begin {array}{rl} {\rm d}Y-{\Bbb E}[Y]=&\bigg\{A(Y-{\Bbb E} [Y]) +B(u-{\Bbb E} [u])+C(Z-{\Bbb E} [Z]) \\ & +\displaystyle\int_{E} D(R-{\Bbb E} [R])\nu ({\rm d}e)\bigg\}{\rm d}s+ Z{\rm d}W+\displaystyle \int_{E} R\tilde{\mu}({\rm d}e, {\rm d}s), \\{\rm d}k-{\Bbb E}[k]=&-\Big[A^\top(k-{\Bbb E} [k])+Q(Y-{\Bbb E}[Y])\Big]{\rm d}s+\Big[C^\top(k-{\Bbb E} [k])\\&+(C^\top+\bar C^\top){\Bbb E} [k]+N_1(Z-{\Bbb E}[Z])+(N_1+\bar N_1){\Bbb E}[Z]\Big]{\rm d}W\\&-\displaystyle\int_ E \Big[D^\top(k-{\Bbb E} [k])+(D^\top+\bar D^\top){\Bbb E} [k]+N_2(R-{\Bbb E}[R])\\&+(N_2+\bar N_2){\Bbb E}[R]\Big]\tilde{\mu}({\rm d}e, {\rm d}s), \\k(t)-{\Bbb E}[k(t)]=&-G(Y(t)-{\Bbb E}[Y(t)]), \; Y(T)-{\Bbb E}[Y(T)]=\xi-{\Bbb E}[\xi], \\ 0=&N_3(u-{\Bbb E}[u])+B^\top(k- {\Bbb E} [k]). \end {array}\right.$$$

$$$Y(s)=P(s)(k(s)-{\Bbb E}[k(s)]) +\Pi(s){\mathbb E}[k(s)]+\varphi(s),$$$

$$${\rm d}\varphi(s)=\alpha(s){\rm d}s+\beta(s){\rm d}W(s) +\Phi(s, e){\rm d}\tilde \mu({\rm d}e, {\rm d}s), \varphi(T)=\xi.$$$

$\begin{eqnarray} \label{eq:4.7}& &\bigg\{A(Y-{\Bbb E} [Y]) +B(u-{\Bbb E} [u])+C(Z-{\Bbb E} [Z]) +\displaystyle\int_{E} D(R-{\Bbb E} [R])\nu ({\rm d}e)\bigg\}{\rm d}s\\ &&+ Z{\rm d}W+\displaystyle \int_{E} R\tilde{\mu}({{\rm d}e}, {\rm d}s)\\&=&{\rm d}\big(Y-{\Bbb E}[Y])\\&=&{\rm d}P(k-{\Bbb E}[k])+{\rm d}\eta\\&=& \Big[\dot{P}(k-{\Bbb E}[k])-PA^\top(k-{\Bbb E} [k])-PQ(Y-{\Bbb E}[Y])\Big]{\rm d}s\\&&-P\Big[C^\top(k-{\Bbb E} [k])+(C^\top+\bar C^\top){\Bbb E} [k]+N_1(Z-{\Bbb E}[Z])+(N_1+\bar N_1){\Bbb E}[Z]\Big]{\rm d}W\\&&-\displaystyle\int_ E P \Big[D^\top(k-{\Bbb E} [k])+(D^\top+\bar D^\top){\Bbb E} [k]\\&&+N_2(R-{\Bbb E}[R])+(N_2+\bar N_2){\Bbb E}[R]\Big]\tilde{\mu}({\rm d}e, {\rm d}s)+\gamma{\rm d}s+\beta {\rm d}W +\Phi{\rm d}\tilde \mu({\rm d}e, {\rm d}s). \end{eqnarray}$

$$$\label{eq:4.8} Z =-P\Big[C^\top k +\bar C ^\top{\Bbb E} [k ]+N_1 Z +\bar N_1 {\Bbb E}[Z ]\Big]+\beta ,$$$

$$$\label{eq:4.9} R=-P \Big[D^\top k +\bar D^\top{\Bbb E} [k ]+N_2R+\bar N_2{\Bbb E}[R]\Big] +\Phi$$$

$\begin{array}{l}AP(k - \Bbb E[k]) + A\eta - BN_3^{ - 1}{B^{\top}}(k - \Bbb E[k]) + C(Z - \Bbb E[Z]) + \int_E D (R - \Bbb E[R])\nu ({\rm{d}}e)\\ = \dot P(k - \Bbb E[k]) - P{A^{\top}}(k - \Bbb E[k]) - PQP(k - \Bbb E[k]) - PQ\eta + \gamma ,\end{array}$

$\begin{array}{l}(\dot P - P{A^{\top}} - AP - PQP + BN_3^{ - 1}{B^{\top}})(k - \Bbb E[k])\\ - C(Z - \Bbb E[Z]) - \int_E D (R - \Bbb E[R])\nu ({\rm{d}}e) - (A + PQ)\eta (t) + \gamma = 0.\end{array}$

$0=-(PN_1 +P\bar N_1 +I){\Bbb E}[Z ]-P(C^\top +\bar C ^\top){\Bbb E} [k ]+{\Bbb E}[\beta ],$

$0=-(PN_2+\bar PN_2+I){\Bbb E}[R-P(D^\top+\bar D^\top){\Bbb E} [k ]+{\Bbb E}[\Phi],$

$0=-(PN_1(t)+I) (Z-{\Bbb E}[Z])-PC^\top (k-{\Bbb E} [k])+(\beta-{\Bbb E}[\beta]),$

$0=-(PN_2+I) ( R-E[R])-PD^\top(k-{\Bbb E} [k])+ \Phi-{\Bbb E}[\Phi].$

$\begin {equation}\label{eq:4.37}\left\{\begin{array}{lll} &&\dot{P}-PA^\top-AP -PQP+BN_3^{-1}B^\top+ C(PN_1 +I)^{-1}PC^\top\\&&+\int_ED(PN_2 +I)^{-1}PD^\top\nu({\rm d}e)=0, \\&& P(T)=0, \end{array} \right. \end {equation}$

$\begin {equation}\label{eq:4.38}\left\{\begin{array}{lll} &&\dot{\Pi} -\Pi(A^\top +\bar A ^\top)-(A +\bar A )\Pi -\Pi(Q +\bar Q )\Pi+(B +\bar B )(N_3 +\bar N_3 )^{-1}(B +\bar B )^\top \\&&+(C +\bar C )(PN_1+P\bar N_1+I)^{-1} P(C^\top +\bar C ^\top) \\ &&+\displaystyle\int_{E} (D+\bar D ) (PN_2+\bar PN_2 +I)^{-1}P(D^\top +\bar D ^\top)=0, \\ &&\Pi(T)=0, \end{array} \right. \end {equation}$

$\varphi(t)$应当满足如下MF-BSDE:

$\begin {equation}\label{eq:4.39}\left\{\begin{array}{rl}{\rm d}\varphi=&\bigg\{(PQ+A)\varphi+C(PN_1+I)^{-1}\beta\\&+\displaystyle\int_{E} D(PN_2+I)^{-1} \Phi\nu ({\rm d}e)+(\bar A+\Pi(Q+\bar Q)-PQ){\Bbb E}[\varphi]\\&+\big[(C+\bar C)(PN_1+P\bar N_1+I)^{-1}-C(PN_1+I)^{-1}\big]{\Bbb E}[\beta]\\&+\displaystyle\int_{E} \big[ (D +\bar D) (PN_2+\bar PN_2+I)^{-1}-D(PN_2+I)^{-1}\big]{\Bbb E}[\Phi]\nu ({\rm d}e)\bigg\}{\rm d}s\\&+\beta {\rm d}W +\Phi {\rm d}\tilde \mu({\rm d}e, {\rm d}s), \\\varphi(T)=& \xi.\end{array} \right. \end {equation}$

### 4.3 最优控制的表示

$Y = :P(k - \Bbb E[k]) + \Pi \Bbb E[k] + \varphi ,$

$\begin{array}{l}Z = : - {(P{N_1} + I)^{ - 1}}[P{C^{\top}})(k - \Bbb E[k]) - (\beta - \Bbb E[\beta ])]\\ - {(P{N_1} + P{{\bar N}_1} + I)^{ - 1}}[P({C^{\top}} + {{\bar C}^{\top}})\Bbb E[k] - \Bbb E[\beta ]],\end{array}$

$\begin{array}{l}R&=:&-(PN_2+I)^{-1} \bigg [PD^\top(k-{\Bbb E} [k])- (\Phi-{\Bbb E}[\Phi])\Big] \\ &&-(PN_2+ P\bar N_2+I)^{-1}\Big[P(D^\top+\bar D^\top){\Bbb E} [k]-{\Bbb E}[\Phi]\Big]. \end{array}$

$Y(s)=P(s)(k(s)-{\Bbb E}[k(s)])+\Pi(s){{\mathbb E}}[k(s)]+\varphi(s)$应用Itô公式, $( Y(\cdot), Z(\cdot), R(\cdot, \cdot))$满足以下带跳的MF-BSDE

$$$\label{eq:4.45}\left\{\begin {array}{ll} {\rm d}Y(s)=&\bigg\{A(s)Y(s)+\bar A(s){\Bbb E} [Y(s)] +B(s)u(s)+\bar B(s){\Bbb E} [u(s)]\\ &+C(s)Z(s)+\bar C(s){\Bbb E} [Z(s)]+\displaystyle\int_{E} D(s, e)R(s, e)\nu ({\rm d}e)\\& +\displaystyle\int_{E}\bar D(s, e) {\Bbb E} [R(s, e)]\nu ({\rm d}e)\bigg\}{\rm d}t+ Z(s){\rm d}W(s) \\ &+\displaystyle \int_{E} R(s, e)\tilde{\mu}({\rm d}e, {\rm d}s), \; s\in [t, T], \\Y(T)=&\xi.\end {array}\right.$$$

$\begin {equation}\label{eq:4.46}\left\{\begin{array}{rl}{\rm d}k(s)=&-\Big[A(s)^\top k(s)+\bar A(s)^\top{\Bbb E} [k(s)]+Q(s)Y(s)+\bar Q(s){\Bbb E}[Y(s)]\Big]{\rm d}s\\&-\Big[C^\top(s)k(s)+\bar C(s)^\top{\Bbb E} [k(s)]+N_1(s)Z(s)+\bar N_1(s){\Bbb E}[Z(s)]\Big]{\rm d}W(s)\\&-\displaystyle\int_ E \Big[D(s, e)^\top k(s-)+\bar D(s, e)^\top{\Bbb E} [k(s-)]\\&+N_2(s, e)R(s, e)+\bar N_2(s, e){\Bbb E}[R(s, e)]\Big]\tilde{\mu}({\rm d}e, {\rm d}s), \\ k(t)=&-GY(t)-\bar G{\Bbb E}[Y(t)].\end{array} \right. \end {equation}$

$\begin{eqnarray} \label{eq:4.47}N_3(s)u(s)+\bar N_3(s){\Bbb E}[u(s)]+B(s)^\top k({s-})+\bar B(s)^\top {\Bbb E} [k({s-})]=0, \quad {\rm {\rm a.s.}}. \end{eqnarray}$

$$$\left\{\begin {array}{ll}\label{eq:4.48} {\rm d}Y(s)=\bigg\{A(s)Y(s)+\bar A(s){\Bbb E} [Y(s)] +B(s)u(s)+\bar B(s){\Bbb E} [u(s)]\\\; \; +C(s)Z(s)+\bar C(s){\Bbb E} [Z(s)] -\displaystyle\int_{E} D(s, e)R(s, e)\nu ({\rm d}e)\\\; \; +\displaystyle\int_{E}\bar D(s, e) {\Bbb E} [R(s, e)]\nu ({\rm d}e)\bigg\}{\rm d}s + Z(s){\rm d}W(s) +\displaystyle \int_{E} R(t, e)\tilde{\mu}({\rm d}e, {\rm d}s), \nonumber\\{\rm d}k(s)=-\Big[A(s)^\top k(s)+\bar A(s)^\top{\Bbb E} [k(s)]+Q(s)Y(s)+\bar Q(s){\Bbb E}[Y(s)]\Big]{\rm d}s\nonumber\\\quad\quad\quad\quad-\Big[C(s)^\top k(s)+\bar C(s)^\top{\Bbb E} [k(s)]+N_1(s)Z(s)+\bar N_1(s){\Bbb E}[Z(s)]\Big]{\rm d}W(s)\nonumber\\\quad\quad\quad\quad-\displaystyle\int_ E \Big[D(s, e)^\top k(s)+\bar D(s, e)^\top{\Bbb E} [k(s)]\\\; \; +N_2(s, e)R(s, e)+\bar N_2(s, e){\Bbb E}[R(s, e)]\Big]\tilde{\mu}({\rm d}e, {\rm d}s), \\ Y(T)=\xi, k(t)=-GY(t)-\bar G{\Bbb E}[Y(t)], \nonumber \\ N_3(s)u(s)+\bar N_3(s){\Bbb E}[u(s)]+B(s)^\top k({s-})+\bar B(s)^\top {\Bbb E} [k({s-})]=0, \quad s\in [t, T]. \end {array}\right.$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Andersson D , Djehiche B .

A maximum principle for SDEs of mean-field type

Applied Mathematics and Optimization, 2011, 63: 341- 356

Baghery F , Baghery F , Øksendal B .

A maximum principle for stochastic control with partial information

Stochastic Analysis and Applications, 2007, 25 (3): 705- 717

Buckdahn R , Djehiche B , Li J .

A general stochastic maximum principle for SDEs of mean-field type

Applied Mathematics and Optimization, 2011, 64: 197- 216

Buckdahn R , Djehiche B , Li J , Peng S .

Mean-field backward stochastic differential equations:a limit approach

The Annals of Probability, 2009, 37 (4): 1524- 1565

Buckdahn R , Li J , Peng S .

Mean-field backward stochastic differential equations and related partial differential equations

Stochastic Processes and Their Applications, 2009, 119 (10): 3133- 3154

Du H , Huang J , Qin Y .

A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications

IEEE Transactions on Automatic Control, 2013, 38: 3212- 3217

Ekeland I , Témam R . Convex Analysis and Variational Problems. Amsterdam: North-Holland, 1976

Elliott R , Li X , Ni Y H .

Discrete time mean-field stochastic linear-quadratic optimal control problems

Automatica, 2013, 49 (11): 3222- 3233

Hafayed M .

A mean-field maximum principle for optimal control of forward-backward stochastic differential equations with Poisson jump processes

International Journal of Dynamics and Control, 2013, 1 (4): 300- 315

Huang J , Li X , Yong J .

A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon

Mathematical Control and Related Fields, 2015, 5 (1): 97- 139

Li X, Sun J, Xiong J. Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Applied Mathematics and Optimization.[2017-12-07]. http://doi.org/10.1007/s00245-017-9464-7

Li J .

Stochastic maximum principle in the mean-field controls

Automatica, 2012, 48 (2): 366- 373

Ma H , Liu B .

Linear quadratic optimal control problem for partially observed forward backward stochastic differential equations of mean-field type

Asian Journal of Control, 2017, 19 (1): 1- 12

Ma L , Zhang W .

Output feedback H control for discrete time mean-field stochastic systems

Asian Journal of Control, 2015, 17 (6): 2241- 2251

Ma L , Zhang T , Zhang W .

H control for continuous time mean-field stochastic systems

Asian Journal of Control, 2016, 18 (5): 1630- 1640

Meng Q , Shen Y .

Optimal control of mean-field jump-diffusion systems with delay:A stochastic maximum principle approach

Journal of Computational and Applied Mathematics, 2015, 279: 13- 30

Meyer-Brandis T , Øksendal B , Zhou X Y .

A mean-field stochastic maximum principle via Malliavin calculus

Stochastics, 2012, 84: 643- 666

Ni Y H , Li X , Zhang J F .

Finite-horizon indefinite mean-field stochastic linear-quadratic optimal control

IFAC-PapersOnLine, 2015, 48 (28): 211- 216

Ni Y H , Zhang J F , Li X .

Indefinite mean-field stochastic linear-quadratic optimal control

IEEE Transactions on Automatic Control, 2015, 60 (7): 1786- 1800

Zhang H, Qi Q. A Complete solution to optimal control and stabilization for mean-field systems: Part I, Discrete-time case. 2016, arXiv: 1608.06363

Qi Q, Zhang H. A Complete solution to optimal control and stabilization for mean-field systems: Part Ⅱ, Continuous-time case. 2016, arXiv: 1608.06475

Shen Y , Meng Q , Shi P .

Maximum principle for mean-field jump diffusion stochastic delay differential equations and its application to finance

Automatica, 2014, 50 (6): 1565- 1579

Shen Y , Siu T K .

The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem

Nonlinear Analysis:Theory, Methods and Applications, 2013, 86: 58- 73

Tang M, Meng Q. Linear-quadratic optimal control problems for mean-field stochastic differential equations with jumps. 2016, arXiv: 1610.03193

Wang G, Wu Z, Zhang C. Maximum principles for partially observed mean-field stochastic systems with application to financial engineering. 2014, DOI: 10.1109/ChiCC.2014.6895853

Wang G, Wu Z, Zhang C. A partially observed optimal control problem for mean-field type forwardbackward stochastic system. 2016, DOI: 10.1109/ChiCC.2016.7553351

Wang G, Xiao H, Xing G. A class of optimal control problems for mean-field forward-backward stochastic systems with partial information. 2015, arXiv: 1509.03729

Wang G , Zhang C , Zhang W .

Stochastic maximum principle for mean-field type optimal control under partial information

IEEE Transactions on Automatic Control, 2014, 59 (2): 522- 528

Yong J .

Linear-quadratic optimal control problems for mean-field stochastic differential equations

SIAM Journal on Control and Optimization, 2013, 51 (4): 2809- 2838

/

 〈 〉