AI 中文总结
该研究针对异质路径提出坐标变换与对称化方法，得到带确定性增广过程的新期望特征公式，可用于解决矩问题与随机控制问题，克服相关计算瓶颈。
AI 中文摘要
我们研究分量正则性与概率结构可能存在差异的异质路径Y=(A,X)的特征变换。引入可逆坐标变换Ψ，使得变换后的特征Ψ∘Sig消除了对不规则分量X的混合积分，并可表示为X的特征坐标与对规则分量A的迭代积分的组合。此外，利用该表示进一步得到部分对称化特征的表达式。主要应用针对带确定性增广的过程给出新的期望特征公式，分析中用这些公式研究矩问题，数值上可精确计算期望特征，克服应用中的典型计算瓶颈。我们在由分数布朗运动驱动的基于特征的随机控制问题中展示了这些优势。
英文摘要
We study signature transformations of heterogeneous paths $Y=(A,X)$ whose components may differ in regularity and probabilistic structure. We introduce an invertible change of coordinates $Ψ$ such that the transformed signature $Ψ\circ\mathrm{Sig}$ eliminates mixed integration against the irregular component $X$ and admits a representation in terms of signature coordinates of $X$ and iterated integration against the regular component $A$. In addition, we exploit this representation to further represent partially symmetrized signatures. Our main application concerns new expected signature formulas for processes with deterministic augmentation. On the analytical side, these formulas are leveraged to study moment problems. On the numerical side, they enable accurate computation of expected signatures, thereby overcoming typical computational bottlenecks in applications. We illustrate these advantages in a signature-based stochastic control problem driven by fractional Brownian motion.
