Schrödinger bridges for time series (SBTS) generate synthetic paths by projecting, in relative entropy, a Brownian reference onto the path laws that match the joint distribution of the data on the observation grid. The Brownian reference, however, fixes the quadratic variation of the generated paths, which is restrictive when stochastic volatility, correlated noise, or rank-deficient covariance structures must be reproduced. We introduce "Triangular-Reference Schrödinger Bridges for Time Series" (TR-SBTS), which keeps the entropy-projection backbone of SBTS but replaces the Brownian reference by a triangular, volatility-informed, intervalwise frozen reference on a state augmented with latent covariance descriptors. The construction remains a single entropy projection on the augmented state: the minimiser is the \(h\)-transform of the reference, and on each frozen interval the optimal drift has the logarithmic-gradient form \(b^\star(t,x)=A\,\nabla\log H(t,x)\), intrinsic to the active covariance directions when the frozen covariance \(A\) is degenerate. We prove stability of the frozen approximation and consistency of the associated regularised kernel estimators, describe a reference-aware Nadaraya--Watson implementation of the conditional next-increment law, and evaluate the construction on numerical experiments.

翻译：时间序列的薛定谔桥（SBTS）通过相对熵将布朗参考投影到与观测网格上数据联合分布相匹配的路径法则上，从而生成合成路径。然而，布朗参考固定了生成路径的二次变分，这在需要再现随机波动率、相关噪声或秩缺陷协方差结构时具有局限性。我们提出“时间序列的三角参考薛定谔桥”（TR-SBTS），该方法保留SBTS的熵投影框架，但将布朗参考替换为在具有隐协方差描述符的增广状态上定义的三角型、波动率感知的分段冻结参考。该构造仍为增广状态上的单一熵投影：其最小化器是参考的h变换，在每个冻结区间内，最优漂移具有对数梯度形式b⋆(t,x)=A∇logH(t,x)，当冻结协方差A退化时，该形式内蕴于活跃协方差方向。我们证明了冻结近似的稳定性及相应正则化核估计量的一致性，描述了条件下一增量法则的参考感知Nadaraya-Watson实现方法，并通过数值实验评估了该构造。