Stochastic Trajectory Influence Functions for LQR: Joint Sensitivity Through Dynamics and Noise Covariance
We present a three-level influence hierarchy for data valuation in stochastic LQR. At the \emph{model level}, the trajectory influence surrogate $\IFm_k := H^{-1}g_k$ approximates the leave-one-trajectory parameter shift. At the \emph{control level} with fixed covariance, the usual fixed-noise score is obtained by composing $\IFm_k$ with the Riccati gradient of $\tr(P(θ)Σ)$. At the \emph{stochastic control level}, the plug-in cost depends additionally on the residual covariance estimate $\hat W$, so removing a trajectory perturbs the cost through both the dynamics and the covariance channels. We derive an exact leave-one-trajectory decomposition of the covariance shift into a \emph{direct-removal} term and a \emph{parameter-shift} term, show that the additional first-order contribution is a simple residual cross-moment, and obtain a stochastic influence score built directly on $\IFm_k$. The resulting method preserves the amortized structure of prior work: after one Hessian factorization and one adjoint Lyapunov solve, each trajectory requires only a dot product plus an $O(n_x^2)$ direct-removal correction. The shared Hessian solves can also be performed iteratively by conjugate gradients when explicit factorization is undesirable. The new covariance remainder is explicit and does not involve Lyapunov-operator amplification; the only amplified term is the familiar Riccati remainder inherited from fixed-covariance influence analysis. Numerical results on two linear systems show that accounting for the estimated covariance substantially improves agreement with exact leave-one-trajectory retraining, especially under heterogeneous noise.