发表机构
Université Gustave Eiffel; University College London（巴黎-塞纳河大学; 伦敦大学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过结合马利亚万微积分、唯一延拓和向后唯一性，在生成元结构相容性条件下，仅要求终端条件非常数，证明了多维正向过程标量倒向分量的密度存在性。
AI 中文摘要
我们给出充分条件，确保在每一个固定的正时刻，具有多维正向过程的马尔可夫型正倒向随机微分方程的标量倒向分量关于勒贝格测度具有密度。现有的倒向随机微分方程密度准则通常通过符号或单调性假设获得马利亚万非退化性。我们为具有任意维正向状态的标量倒向分量开发了一条不同的路径。在正则性假设和生成元的结构相容性条件下，终端条件仅需非常数。关键思想是从解耦场的临界集的确定性刚性推导出马利亚万非退化性。我们将马利亚万微积分与相关半线性抛物型方程的唯一延拓和向后唯一性相结合。唯一延拓阻止了解耦场的空间梯度在正测度集上消失，除非它在该时间切片上恒为零，而向后唯一性则将这种消失传播到终端时间。在此过程中，我们建立了从正测度集出发的唯一延拓性质，以及由微分半线性方程产生的线性抛物型系统在全空间上的向后唯一性结果。
英文摘要
We give sufficient conditions ensuring that, at every fixed positive time, the scalar backward component of a Markovian forward-backward stochastic differential equation with multidimensional forward process admits a density with respect to Lebesgue measure. Existing density criteria for BSDEs often obtain Malliavin non-degeneracy through sign or monotonicity assumptions. We develop a different route for a scalar backward component with an arbitrary-dimensional forward state. Under regularity assumptions and a structural compatibility condition on the generator, the terminal condition is only required to be non-constant. The key idea is to deduce Malliavin non-degeneracy from deterministic rigidity of the critical set of the decoupling field. We combine Malliavin calculus with unique continuation and backward uniqueness for the associated semilinear parabolic equation. Unique continuation precludes the spatial gradient of the decoupling field from vanishing on a set of positive measure unless it vanishes identically on that time slice, while backward uniqueness propagates such vanishing to the terminal time. Along the way, we establish a unique continuation property from sets of positive measure and a backward uniqueness result on the whole space for the linear parabolic systems arising from differentiated semilinear equations.
Comments46 pages
