  We first provide a stochastic formula for the Carath\'eodory distance in
terms of general Markovian couplings and prove a comparison result between the
Carath\'eodory distance and the complete K\"ahler metric with a negative lower
curvature bound using the Kendall-Cranston coupling. This probabilistic
approach gives a version of the Schwarz lemma on complete non-compact K\"ahler
manifolds with a further decomposition Ricci curvature into the orthogonal
Ricci curvature and the holomorphic sectional curvature, which cannot be
obtained by using Yau--Royden's Schwarz lemma. We also prove coupling estimates
on quaternionic K\"ahler manifolds. As a byproduct, we obtain an improved
gradient estimate of positive harmonic functions on K\"ahler manifolds and
quaternionic K\"ahler manifolds under lower curvature bounds.
