Lyapunov Continuation for Uncertain Differential Equations Beyond Linear Growth

This paper studies finite-horizon existence and distribution formulas for Liu-driven uncertain differential equations under assumptions weaker than the usual global Lipschitz and linear growth conditions. The main existence result replaces linear growth by a Lyapunov continuation condition. Under local Lipschitz continuity and a coercive [Formula: see text] function [Formula: see text] satisfying [Formula: see text] maximal pathwise solutions are constructed by localization and are shown not to explode on [Formula: see text]. The proof uses the fact that, on a full-measure set, Liu sample paths are Lipschitz and the equation reduces to a deterministic Carathéodory equation. In the scalar case, the paper proves that the Yao–Chen formula remains valid under these continuation hypotheses, provided the relevant [Formula: see text]-paths stay in an interval where the diffusion coefficient is positive and the Liu process satisfies the required slope-event estimates. Thus the role of linear growth is to guarantee global continuation, not to drive the distribution formula itself. An example with finite-time blow-up shows why some continuation condition is necessary.