发表机构
University of Calgary（卡尔加里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对带随机系数的退化随机最优控制问题对应的随机哈密顿-雅可比-贝尔曼方程，证明了其粘性解的存在性与唯一性，放宽了此前对扩散项的结构限制。
AI 中文摘要
我们研究由带随机系数且可能退化的随机最优控制问题产生的随机哈密顿-雅可比-贝尔曼方程。粘性解通过适配光滑随机测试场重新表述，该测试场几乎必然触及停止的时空邻域。状态系数是可预测的，不对其对基础布朗路径的依赖性做连续性假设。在漂移项和扩散项系数满足局部利普希茨线性增长、运行成本和终端成本系数满足多项式增长，以及控制集为非空可分度量空间的条件下，我们证明值函数是该表述中适配连续场的唯一粘性解，且该场具有平方可积的随机多项式增长界。比较证明通过受控轨迹和空间扰动的条件值构造适配场，利用弱半鞅分解及其正有限变差部分的正则化。特别地，该唯一性允许扩散项为随机、退化且依赖于时间、状态和控制，而此前Qiu [SIAM J. Control Optim. 56 (2018), pp. 3708--3730]和Qiu与Zhang [Ann. Appl. Probab. 33 (2023), pp. 889--930]的唯一性结果是在扩散项需满足额外结构限制（如确定性结合超抛物性或与状态和控制无关）的条件下得到的。
英文摘要
We study stochastic Hamilton--Jacobi--Bellman equations arising from stochastic optimal control problems with random coefficients and possible degenerateness. The viscosity solution is reformulated by using adapted smooth random test fields that touch on stopped space--time neighborhoods almost surely. The state coefficients are predictable, with no continuity assumption on their dependence on the underlying Brownian path. Under linear growth of locally Lipschitz drift and diffusion coefficients, polynomial growth of the running and terminal cost coefficients, and a nonempty separable metric space-valued control set, we show that the value function is the unique viscosity solution in this formulation among adapted continuous fields with square-integrable random polynomial growth bounds. The comparison proof constructs adapted fields from controlled trajectories and spatially shaken conditional values, using a weak semimartingale decomposition and regularization of its positive finite-variation part. In particular, the uniqueness allows for diffusion coefficients being random, degenerate, and depending on time, state, and control, whereas earlier uniqueness results of Qiu [SIAM J. Control Optim. 56 (2018), pp. 3708--3730] and Qiu and Zhang [Ann. Appl. Probab. 33 (2023), pp. 889--930] were obtained under additional structural restrictions on the diffusion coefficient, such as determinism combined with either superparabolicity or independence of the state and controls.
Comments64 pages
