Spacecraft guidance, navigation, and control systems operating during mission-critical phases such as atmospheric entry, powered descent, and planetary landing require reliable assessment of finite-time transient behavior under stochastic uncertainty. Existing stochastic stability frameworks primarily characterize asymptotic or infinitesimal behavior and therefore provide limited insight into finite-amplitude transient amplification over operationally relevant time horizons. This paper develops a finite-amplitude logarithmic measure for nonlinear Itô stochastic differential equations, extending classical matrix measures from infinitesimal to finite-amplitude perturbation evolution. The proposed framework establishes finite-time mean and variance bounds for logarithmic perturbation growth, derives Chernoff-type probabilistic bounds on transient amplification, and shows that mean transient stability does not necessarily guarantee finite-time pathwise safety. The theory is further extended to projected stochastic dynamics, leading to a transient-risk index that quantitatively balances deterministic contraction and diffusion-induced variability for transient-risk-aware system design. The proposed framework is validated through Monte Carlo simulations and flight-like lunar descent telemetry. The results demonstrate substantially improved prediction of nonlinear transient growth compared with classical Jacobian-based approaches and successfully distinguish trajectories exhibiting similar nominal behavior but significantly different transient-risk characteristics. The proposed framework provides a unified methodology for finite-time transient stability analysis, probabilistic transient-risk assessment, and transient-risk-aware guidance, navigation, and control of nonlinear stochastic aerospace systems.

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