One idea is to approach the problem through stochastic differential equations with reflections off the boundary of domains. For example, this paper by Tanaka (1979) considers a stochastic differential equation with reflections off the boundary of a convex set. They prove existence and uniqueness for coefficients defined on the domain only, and as far as I can see, in the proof they don't extend the coefficients to all of $\mathbb{R}^d$. If unique solutions exist with reflecting boundary conditions for all time, then presumably the original equation has a solution up to hitting the boundary. A more recent reference is Lions and Sznitman (1984).

This is a bit inelegant in that an SDE stopped at the boundary should be simpler than an SDE reflected off a boundary. But a least you don't need to extend the domain of the coefficients.

One idea is to approach the problem through stochastic differential equations with reflections off the boundary of domains. For example, this paper by Tanaka (1979) considers a stochastic differential equation with reflections off the boundary of a convex set. They prove existence and uniqueness for coefficients defined on the domain only, and as far as I can see, in the proof they don't extend the coefficients to all of $\mathbb{R}^d$. If unique solutions exist with reflecting boundary conditions for all time, then presumably the original equation has a solution up to hitting the boundary. A more recent reference is Lions and Sznitman (1984).

This is a bit inelegant in that an SDE stopped at the boundary should be simpler than an SDE reflected off a boundary. But a least you don't need to extend the domain of the coefficients.