| 胡世培.由L\'{e}vy过程驱动的仿射方程关联的无限时区的最优二次控制[J].数学年刊A辑,2013,34(2):179~204 |
| 由L\'{e}vy过程驱动的仿射方程关联的无限时区的最优二次控制 |
| Infinite Horizontal Optimal Quadratic Control for an AffineEquation Driven by L\'{e}vy Processes |
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| DOI： |
| 中文关键词: 线性二次, 无限时区, 倒向随机 Riccati 微分方程, L\'{e}vy 过程 |
| 英文关键词:Linear quadratic, Infinite horizon, Backward stochastic Riccati & differential equation, L\'{e}vy processes |
| 基金项目: |
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| 中文摘要: |
| 讨论线性二次最优控制问题, 其随机系统是由 L\'{e}vy 过程驱动的具有随机系数而且还具有仿射项的线性随机微分方程.
伴随方程具有无界系数, 其可解性不是显然的. 利用 $\mathscr{B}\mathscr{M}\mathscr{O}$ 鞅理论, 证明伴随方程在有限
时区解的存在唯一性. 在稳定性条件下, 无限时区的倒向随机 Riccati 微分方程和伴随倒向随机方程的解的存在性是通过对应有限
时区的方程的解来逼近的. 利用这些解能够合成最优控制. |
| 英文摘要: |
| The author studies a linear quadratic
optimal control problem for stochastic systems driven by L\'{e}vy processes where the
linear state equation has stochastic coefficients and moreover, an
affine term. The adjoint equation has unbounded coefficients, and
its solution is not known. Employing
$\mathscr{B}\mathscr{M}\mathscr{O}$-martingale theory, the author proves the
existence and uniqueness of the solutions to the adjoint equation in a
finite horizon. In the case of an infinite horizon, assuming some
stabilizability, the author proves via suitable finite horizontal
approximation the existence of the solutions to the backward
stochastic Riccati differential equation and the adjoint backward
stochastic equation. Using these solutions, the author performs the synthesis
of the optimal control. |
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