| 徐茂中,唐矛宁,孟庆欣.一类由Lévy过程驱动的正倒向随机微分方程及在LQ问题中的应用[J].数学年刊A辑,2025,46(4):403~434 |
| 一类由Lévy过程驱动的正倒向随机微分方程及在LQ问题中的应用 |
| A Class of Forward-Backward Stochastic Differential Equations Driven by L'évy Processes and Application to LQ Problems |
| 投稿时间：2025-04-11 修订日期：2025-10-31 |
| DOI：10.16205/j.cnki.cama.2025.0026 |
| 中文关键词: 时滞 正倒向随机微分方程 Lévy过程 参数延拓法 控制单调性条件 随机LQ问题 |
| 英文关键词:Delay Forward-backward stochastic differential equation Lévy processes Method of continuation Domination-monotonicity condition Stochastic linear-quadratic problem |
| 基金项目:国家自然科学基金(No. 12271158, No. 11871121)；浙江省自然科学基金(No. Z22A013952) |
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| 中文摘要: |
| 作者重点研究一类含时滞与超前项、由Lévy过程驱动的非线性完全耦合正倒向随机微分方程
. 结合含时滞与Lévy过程的线性二次 (LQ)
最优控制问题实例, 针对该类FBSDELDAs采用一组控制单调性条件,
运用参数延拓法, 得到其解的唯一存在性及解对生成元的连续依赖性结果.
这些结果对一系列LQ问题具有重要意义,
其中的随机哈密顿系统恰好是满足控制单调性条件的FBSDELDAs,
因此可利用相应随机哈密顿系统的解, 建立唯一最优控制的显式对偶表达式. |
| 英文摘要: |
| The authors’ focus lies in the thorough investigation of a specific category of nonlinear fully coupled forward-backward stochastic differential equations involving time delays and advancements with the incorporation of L'évy processes, which shall be abbreviated as FBSDELDAs. Drawing inspiration from diverse examples of linear-quadratic (LQ for short) optimal control problems featuring delays and L'évy processes, the authors proceed to employ a set of domination-monotonicity conditions tailored to this class of FBSDELDAs. Through the application of the continuation method, the authors achieve the pivotal results of unique solvability and the derivation of a pair of estimates for the solutions of these FBSDELDAs. These findings, in turn, carry significant implications for a range of LQ problems. Specifically, they are relevant when stochastic Hamiltonian systems perfectly align with the FBSDELDAs that fulfill the domination-monotonicity conditions. Consequently, the authors are able to establish explicit expressions for the unique optimal controls by utilizing the solutions of the corresponding stochastic Hamiltonian systems. |
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