﻿ 一个倒向随机微分时滞方程的最优控制问题 An Optimal Control Problem of Backward Stochastic Differential Delay Equation

Vol. 10  No. 01 ( 2021 ), Article ID: 39855 , 6 pages
10.12677/AAM.2021.101016

An Optimal Control Problem of Backward Stochastic Differential Delay Equation

Shuang Wu

College of Science, China University of Petroleum, Qingdao Shandong

Received: Dec. 16th, 2020; accepted: Jan. 5th, 2021; published: Jan. 20th, 2021

ABSTRACT

This paper studies an optimal problem of backward stochastic differential delay equation. By means of a three-coupled system of adjoint equations, we give a sufficient condition for optimal control. As an application, a financial example is presented to illustrate the theoretical result.

Keywords:Backward Stochastic Differential Delay Equation, Optimal Control, Adjoint Equation

1. 引言

2. 问题描述

${L}^{2}\left(r,s;R\right)=\left\{\phi \left(t\right)|\phi \left(t\right)是\text{ }R\text{ }值确定性的函数且{\int }_{r}^{s}{\phi }^{2}\left(t\right)\text{d}t<+\infty \right\};$

${L}^{2}\left({F}_{t};R\right)=\left\{\zeta |\zeta \text{ }是\text{ }R\text{ }值\text{ }{F}_{t}\text{ }可测随机变量且\text{ }E{\zeta }^{2}<+\infty \right\};$

${L}_{F}^{2}\left(r,s;R\right)=\left\{\psi \left(\cdot \right)|\psi \left(\cdot \right)是\text{ }R\text{ }值\text{ }{F}_{t}\text{ }适应的随机过程且E{\int }_{r}^{s}{\psi }^{2}\left(t\right)\text{d}t<+\infty \right\}.$

$\left\{\begin{array}{l}-\text{d}{y}^{v}\left(t\right)=f\left(t,{y}^{v}\left(t\right),{y}_{\delta }^{v}\left(t\right),{\stackrel{¯}{y}}^{v}\left(t\right),{z}^{v}\left(t\right),v\left(t\right)\right)-{z}^{v}\left(t\right)\text{d}W\left(t\right),\text{}t\in \left[0,T\right)\\ {y}^{v}\left(T\right)=\xi ,\text{}y\left(t\right)=\phi \left(t\right),\text{}t\in \left[-\delta ,0\right).\end{array}$ (1)

$J\left(v\left(\cdot \right)\right)=E{\int }_{0}^{T}l\left(t,{y}^{v}\left(t\right),{y}_{\delta }^{v}\left(t\right),{\stackrel{¯}{y}}^{v}\left(t\right),{z}^{v}\left(t\right),v\left(t\right)\right)\text{d}t+h\left({y}^{v}\left(0\right)\right),$

(H2.1) f是连续可微的；偏导 ${f}_{y},{f}_{{y}_{\delta }},{f}_{\stackrel{¯}{y}},{f}_{z},{f}_{v}$ 是一致有界的；

(H2.2) l和h是连续可微的，且存在 $C>0$，使得：

$|{l}_{y}|+|{l}_{{y}_{\delta }}|+|{l}_{\stackrel{¯}{y}}|+|{l}_{z}|+|{l}_{v}|+|{h}_{y}|\le C\left(1+|y|+|{y}_{\delta }|+|\stackrel{¯}{y}|+|z|+|v|\right)$

3. 充分条件

$H\left(t,y,{y}_{\delta },\stackrel{¯}{y},z,v,p,\stackrel{¯}{p}\right)=l\left(t,y,{y}_{\delta },\stackrel{¯}{y},z,v\right)-p\cdot f\left(t,y,{y}_{\delta },\stackrel{¯}{y},z,v\right)+\stackrel{¯}{p}\cdot \left(y-\lambda \stackrel{¯}{y}-{\text{e}}^{-\lambda \delta }{y}_{\delta }\right),$

$\left\{\begin{array}{l}\text{d}p\left(t\right)=-{H}_{y}\left(t\right)\text{d}t-{H}_{z}\left(t\right)\text{d}W\left(t\right),\\ p\left(0\right)={h}_{y}\left({y}^{u}\left(0\right)\right),\end{array}$ (2)

$\left\{\begin{array}{l}\text{d}\stackrel{¯}{p}\left(t\right)=-{H}_{\stackrel{¯}{y}}\left(t\right)\text{d}t-\stackrel{¯}{q}\left(t\right)\text{d}W\left(t\right),\\ \stackrel{¯}{p}\left(T\right)=0,\end{array}$ (3)

$\left\{\begin{array}{l}\text{d}\stackrel{˜}{p}\left(t\right)=-{H}_{{y}_{\delta }}\left(t\right)\text{d}t-\stackrel{˜}{q}\left(t\right)\text{d}W\left(t\right),\\ \stackrel{˜}{p}\left(T\right)=0.\end{array}$ (4)

$\begin{array}{c}J\left(v\left(\cdot \right)\right)-J\left(u\left(\cdot \right)\right)=E{\int }_{0}^{T}\left\{H\left(t,{\theta }^{v}\left(t\right),v\left(t\right),p\left(t\right),\stackrel{¯}{p}\left(t\right)\right)-H\left(t,{\theta }^{u}\left(t\right),u\left(t\right),p\left(t\right),\stackrel{¯}{p}\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+p\left(t\right)\left(f\left(t,{\theta }^{v}\left(t\right),v\left(t\right)\right)-f\left(t,{\theta }^{u}\left(t\right),u\left(t\right)\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\stackrel{¯}{p}\left(t\right)\left({y}^{v}\left(t\right)-\lambda {\stackrel{¯}{y}}^{v}\left(t\right)-{\text{e}}^{-\lambda \delta }{y}_{\delta }^{v}\left(t\right)-\left({y}^{u}\left(t\right)-\lambda {\stackrel{¯}{y}}^{u}\left(t\right)-{\text{e}}^{-\lambda \delta }{y}_{\delta }^{u}\left(t\right)\right)\right)\right\}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+h\left({y}^{v}\left(0\right)\right)-h\left({y}^{u}\left(0\right)\right)\end{array}$ (5)

$p\left(t\right)\left({y}^{v}\left(t\right)-{y}^{u}\left(t\right)\right)+\stackrel{¯}{p}\left(t\right)\left({\stackrel{¯}{y}}^{v}\left(t\right)-{\stackrel{¯}{y}}^{u}\left(t\right)\right)+\stackrel{˜}{p}\left(t\right)\left({y}_{\delta }^{v}\left(t\right)-{y}_{\delta }^{u}\left(t\right)\right)$ 利用伊藤公式，可得：

$\begin{array}{c}0=E{\int }_{0}^{T}\left\{-{H}_{y}\left(t\right)\left({y}^{v}\left(t\right)-{y}^{u}\left(t\right)\right)-p\left(t\right)\left(f\left(t,{\theta }^{v}\left(t\right),v\left(t\right)\right)-f\left(t,{\theta }^{u}\left(t\right),u\left(t\right)\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{H}_{\stackrel{¯}{y}}\left(t\right)\left({\stackrel{¯}{y}}^{v}\left(t\right)-{\stackrel{¯}{y}}^{u}\left(t\right)\right)-{H}_{{y}_{\delta }}\left(t\right)\left({y}_{\delta }^{v}\left(t\right)-{y}_{\delta }^{u}\left(t\right)\right)-{H}_{z}\left(t\right)\left({z}^{v}\left(t\right)-{z}^{u}\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\stackrel{¯}{p}\left(t\right)\left({y}^{v}\left(t\right)-\lambda {\stackrel{¯}{y}}^{v}\left(t\right)-{\text{e}}^{-\lambda \delta }{y}_{\delta }^{v}\left(t\right)-\left({y}^{u}\left(t\right)-\lambda {\stackrel{¯}{y}}^{u}\left(t\right)-{\text{e}}^{-\lambda \delta }{y}_{\delta }^{u}\left(t\right)\right)\right)\right\}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{h}_{y}\left({y}^{u}\left(0\right)\right)\left({y}^{v}\left(0\right)-{y}^{u}\left(0\right)\right)\end{array}$ (6)

$\begin{array}{c}J\left(v\left(\cdot \right)\right)-J\left(u\left(\cdot \right)\right)=E{\int }_{0}^{T}\left\{H\left(t,{\theta }^{v}\left(t\right),v\left(t\right),p\left(t\right),\stackrel{¯}{p}\left(t\right)\right)-H\left(t,{\theta }^{u}\left(t\right),u\left(t\right),p\left(t\right),\stackrel{¯}{p}\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{H}_{y}\left(t\right)\left({y}^{v}\left(t\right)-{y}^{u}\left(t\right)\right)-{H}_{\stackrel{¯}{y}}\left(t\right)\left({\stackrel{¯}{y}}^{v}\left(t\right)-{\stackrel{¯}{y}}^{u}\left(t\right)\right)-{H}_{{y}_{\delta }}\left(t\right)\left({y}_{\delta }^{v}\left(t\right)-{y}_{\delta }^{u}\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{H}_{z}\left(t\right)\left({z}^{v}\left(t\right)-{z}^{u}\left(t\right)\right)\right\}\text{d}t+h\left({y}^{v}\left(0\right)\right)-h\left({y}^{u}\left(0\right)\right)-{h}_{y}\left({y}^{u}\left(0\right)\right)\left({y}^{v}\left(0\right)-{y}^{u}\left(0\right)\right).\end{array}$ (7)

$J\left(v\left(\cdot \right)\right)-J\left(u\left(\cdot \right)\right)\le E{\int }_{0}^{T}{H}_{v}\left(t,{\theta }^{u}\left(t\right),u\left(t\right),p\left(t\right),\stackrel{¯}{p}\left(t\right)\right)\left(v\left(t\right)-u\left(t\right)\right)\text{d}t$

${H}_{v}\left(t,{\theta }^{u}\left(t\right),u\left(t\right),p\left(t\right),\stackrel{¯}{p}\left(t\right)\right)\left(v\left(t\right)-u\left(t\right)\right)\le 0$

4. 应用

$\text{d}B\left(t\right)=r\left(t\right)B\left(t\right)\text{d}t,\text{}B\left(0\right)={b}_{0}$

$\text{d}S\left(t\right)=S\left(t\right)\left[\mu \left(t\right)\text{d}t+\sigma \left(t\right)\text{d}W\left(t\right)\right],\text{}S\left(0\right)={s}_{0}$

(H4.1) $r\left(t\right),\mu \left(t\right),\sigma \left(t\right)$ 是确定性的函数；它们和 ${\sigma }^{-1}\left(t\right)$ 都是一致有界的。

$\left\{\begin{array}{l}\text{d}y\left(t\right)=\left[\stackrel{˜}{r}\left(t\right)y\left(t\right)-\beta \left(t\right){y}_{\delta }\left(t\right)-\alpha \stackrel{¯}{y}\left(t\right)+\pi \left(t\right)b\left(t\right)-c\left(t\right)\right]\text{d}t+\pi \left(t\right)\sigma \left(t\right)\text{d}W\left(t\right),\\ y\left(T\right)=\xi ,\text{}y\left(t\right)=0,\text{}t\in \left[-\delta ,0\right).\end{array}$ (7)

$\left\{\begin{array}{l}\text{d}y\left(t\right)=\left[\stackrel{˜}{r}\left(t\right)y\left(t\right)-\beta \left(t\right){y}_{\delta }\left(t\right)-\alpha \stackrel{¯}{y}\left(t\right)+b\left(t\right){\sigma }^{-1}\left(t\right)z\left(t\right)-c\left(t\right)\right]\text{d}t+z\left(t\right)\text{d}W\left(t\right),\\ y\left(T\right)=\xi ,\text{}y\left(t\right)=0,\text{}t\in \left[-\delta ,0\right).\end{array}$ (8)

$J\left(c\left(\cdot \right)\right)=E{\int }_{0}^{T}L{\text{e}}^{-\gamma t}\frac{c{\left(t\right)}^{1-R}}{1-R}\text{d}t-K\left(y\left(0\right)\right),$

$\begin{array}{c}H\left(t,y,{y}_{\delta },\stackrel{¯}{y},z,c,p,\stackrel{¯}{p}\right)=L{\text{e}}^{-\gamma t}\frac{{c}^{1-R}}{1-R}-p\left(\stackrel{˜}{r}\left(t\right)y-\beta \left(t\right){y}_{\delta }-\alpha \stackrel{¯}{y}+{\sigma }^{-1}\left(t\right)b\left(t\right)z-c\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\stackrel{¯}{p}\left(y-\lambda \stackrel{¯}{y}-{\text{e}}^{-\lambda \delta }{y}_{\delta }\right),\end{array}$

$\left\{\begin{array}{l}\text{d}p\left(t\right)=-\left[\stackrel{˜}{r}\left(t\right)p\left(t\right)+\stackrel{¯}{p}\left(t\right)\right]\text{d}t-{\sigma }^{-1}\left(t\right)b\left(t\right)p\left(t\right)\text{d}W\left(t\right),\\ p\left(0\right)=K,\end{array}$ (9)

$\left\{\begin{array}{l}\text{d}\stackrel{¯}{p}\left(t\right)=\left[\alpha p\left(t\right)+\lambda \stackrel{¯}{p}\left(t\right)\right]\text{d}t-\stackrel{¯}{q}\left(t\right)\text{d}W\left(t\right),\\ \stackrel{¯}{p}\left(T\right)=0,\end{array}$ (10)

$\left\{\begin{array}{l}\text{d}\stackrel{˜}{p}\left(t\right)=\left[\beta \left(t\right)p\left(t\right)+{\text{e}}^{-\lambda \delta }\stackrel{¯}{p}\left(t\right)\right]\text{d}t-\stackrel{˜}{q}\left(t\right)\text{d}W\left(t\right),\\ \stackrel{˜}{p}\left(T\right)=0,\end{array}$ (11)

$\beta \left(t\right)p\left(t\right)+{\text{e}}^{-\lambda \delta }\stackrel{¯}{p}\left(t\right)=0,$ (12)

$\text{d}\beta \left(t\right)p\left(t\right)=\left\{-\beta \left(t\right)\left[\stackrel{˜}{r}\left(t\right)p\left(t\right)+\stackrel{¯}{p}\left(t\right)\right]+\stackrel{˙}{\beta }\left(t\right)p\left(t\right)\right\}\text{d}t-\beta \left(t\right){\sigma }^{-1}\left(t\right)b\left(t\right)p\left(t\right)\text{d}W\left(t\right),$ (13)

$\left\{\begin{array}{l}\stackrel{˙}{\beta }\left(t\right)-\left(\stackrel{˜}{r}\left(t\right)+\lambda \right)\beta \left(t\right)+{\text{e}}^{\lambda \delta }{\beta }^{2}\left(t\right)+{\text{e}}^{-\lambda \delta }\alpha =0,\\ \beta \left(T\right)=0\end{array}$ (14)

$\left\{\begin{array}{l}\stackrel{˙}{\beta }\left(t\right)-\left(r\left(t\right)+\alpha +\lambda \right)\beta \left(t\right)+\left({\text{e}}^{\lambda \delta }-1\right){\beta }^{2}\left(t\right)+{\text{e}}^{-\lambda \delta }\alpha =0,\\ \beta \left(T\right)=0\end{array}$ (15)

$\left\{\begin{array}{l}{\stackrel{˙}{\beta }}_{1}\left(t\right)-\left(A\left(t\right)-2\sqrt{BC}\right){\beta }_{1}\left(t\right)+\sqrt{B}{\beta }_{1}^{2}\left(t\right)-\sqrt{C}\left(A\left(t\right)-2\sqrt{BC}\right)=0,\\ {\beta }_{1}\left(T\right)=-\sqrt{C}.\end{array}$ (16)

$A\left(t\right)-2\sqrt{BC}\ge 0,\forall t\in \left[0,T\right]$ 时，由文献 [6] 中的命题4.2可知，(16)有唯一的解 ${\beta }_{1}\left(t\right)<0$。从而(15)有解 $\beta \left(t\right)={\beta }_{1}\left(t\right)+\sqrt{C}/\sqrt{B}$

$\left\{\begin{array}{l}\text{d}\Psi \left(t\right)=-\left[r\left(t\right)+\alpha +\left(1-{\text{e}}^{\lambda \delta }\right)\beta \left(t\right)\right]\Psi \left(t\right)\text{d}t-{\sigma }^{-1}\left(t\right)b\left(t\right)\Psi \left(t\right)\text{d}W\left(t\right),\\ \Psi \left(0\right)=1.\end{array}$ (17)

${c}^{*}\left(t\right)={\left(L{\text{e}}^{-\gamma t}\right)}^{\frac{1}{R}}{\left(p\left(t\right)\right)}^{-\frac{1}{R}}={\left(\frac{L}{K}\right)}^{\frac{1}{R}}{\left(\Psi \left(t\right)\right)}^{-\frac{1}{R}}{\text{e}}^{-\frac{\gamma }{R}t}$ (18)

An Optimal Control Problem of Backward Stochastic Differential Delay Equation[J]. 应用数学进展, 2021, 10(01): 137-142. https://doi.org/10.12677/AAM.2021.101016

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