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Impulse Control of Conditional McKean–Vlasov Jump Diffusions

Journal of Optimization Theory and Applications(2024)SCI 3区

KTH Royal Institute of Technology | University of Oslo

Cited 0|Views12
Abstract
In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ -algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.
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Jump diffusion,Impulse control,Common noise,49J53,49K99
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要点】:本文研究了条件 McKean–Vlasov 跳跃扩散的脉冲控制问题,证明了相关问题解的存在性,并提出了一个验证定理,应用于最优股息流问题。

方法】:通过研究条件 McKean–Vlasov 跳跃扩散的随机微分方程(SDEs)及其相关的 Fokker–Planck 随机偏微分方程(SPDEs),建立了脉冲控制问题的验证定理。

实验】:文中通过一个具体例子——在交易成本下的最优股息流问题,应用所提出的验证定理,找到了满足定理的函数和脉冲控制,实验未明确提及使用特定数据集。