  In this paper, we consider an arbitrary irreducible unitary representation
$(\pi_{\lambda},V_{\lambda})$ of a compact connected, simply connected
semisimple Lie group $G$ with highest weight $\lambda$, and apply the idea of
Daubechies--Klauder (1985) and Yamashita (2011) on rigorous coherent-state path
integrals to this representation, where the orbit of the highest weight vector
is interpreted as the manifold of coherent states. Our main theorem is
two-fold: the first main theorem is in terms of Brownian motions and stochastic
integrals, and proven using the Feynman--Kac--It\^o formula on a vector bundle
of a Riemannian manifold, due to G\"uneysu (2010). In the second main theorem,
we consider a sequence $(\mu_{n})$ of finite measures on the space of smooth
paths, and a `path integral' is defined to be a limit of the integrals with
respect to $(\mu_{n})$. The formulation and the proof of the second main
theorem employ \emph{rough path theory} originated by Lyons (1998).
