具有无限时间 horizon 和 regime switching 的随机控制问题：出现在最优清算中的 semimartingale 策略
Stochastic Control Problems with Infinite Horizon and Regime Switching Arising in Optimal Liquidation with Semimartingale Strategies
AI总结：
本文研究了具有无限时间 horizon 和 regime switching 的随机控制问题，通过建立 BSDEs 系统并证明其解的存在性，得到最优清算策略的唯一性。
摘要：
我们研究了一个具有 semimartingale 策略、随机系数和 regime switching 的无限时间 horizon 最优化控制问题。价值函数和最优策略可以表示为三个具有无限时间 horizon 的反向随机微分方程（BSDEs）系统。其中之一是一个具有无界系数和无限时间 horizon 的线性 BSDE 系统，这在文献中似乎是新的。我们通过 BMO 分析和多维 BSDE 的比较定理建立了这些 BSDEs 解的存在性。接下来，我们证明了最优控制问题是良好的，即价值函数是有限的，当存在最优策略时，它唯一。这通过将成本函数重新表述为二次函数和候选价值函数的和来实现。重新表述关键依赖于已建立的 BSDE 系统的良好性结果。最后，在额外假设下，我们获得了唯一的最优策略。
英文摘要：
We study an optimal control problem on infinite time horizon with semimartingale strategies, random coefficients and regime switching. The value function and the optimal strategy can be characterized in terms of three systems of backward stochastic differential equations (BSDEs) with infinite horizon. One of them is a system of linear BSDEs with unbounded coefficients and infinite horizon, which seems to be new in literature. We establish the existence of the solutions to these BSDEs by BMO analysis and comparison theorem for multi-dimensional BSDEs. Next, we establish that the optimal control problem is well posed, in the sense that the value function is finite and the optimal strategy-when it exists-is unique. This is achieved by reformulating the cost functional as the sum of a quadratic functional and the candidate value function. The reformulation crucially relies on the well-established well-posedness results for systems of BSDEs. Finally, under additional assumptions, we obtain the unique optimal strategy.
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