Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations
Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations
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DOI：
10.1051/cocv/2021027
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发表时间：
2019-11
期刊：
影响因子：
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通讯作者：
Hanxiao Wang;J. Yong
中科院分区：
文献类型：
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作者：
Hanxiao Wang;J. Yong
考虑一类随机微分方程的最优控制问题，其代价泛函由一个倒向随机沃尔泰拉积分方程（BSVIE）确定.这种成本泛函具有递归的特性，可以覆盖一般的折现情形（包括指数和非指数）。众所周知，这样的问题一般是时间不一致的。因此，而不是找到一个全局最优控制，我们寻找一个时间一致的局部接近最优的平衡策略。利用多人微分对策的思想，构造了一类与时间区间划分相联系的近似均衡策略。通过将时间区间划分的网格尺寸设为零，导出了平衡Hamilton-Jacobi-Bellman（简称HJB）方程，并通过该方程得到了平衡值函数和平衡策略.在一定条件下，证明了一个验证定理，并建立了平衡HJB的适定性。作为平衡态HJB方程的Feynman-Kac公式，自然地引入了一类新的BSVIE（包含Z（r，r）的对角值Z（r，r）），并简要地给出了这类方程的适定性.
An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situations with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation is derived, through which the equilibrium value function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman–Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal value Z(r, r) of Z(⋅ , ⋅)) is naturally introduced and the well-posedness of such kind of equations is briefly presented.