In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary value problems in mathematical finance?

I came across this Q&A: "Hyperbolic and Elliptic PDEs in Quant Finance" on the Quantitative Finance StackExchange, but I cannot find a clear mathematical formulation of a "perpetual exchange option" (whatever that is)...