Investment value transversality condition (${\lim }_{T\to \infty }\mathbb{E}[e^{-\rho T}V(w_T,s_T)]=0$)

A discounted-value transversality condition rules out residual value at infinity in an infinite-horizon Hamilton-Jacobi-Bellman equation verification. Under appropriate integrability, admissibility, and localization assumptions, $\mathbb{E}[e^{-\rho T}V(w_T,s_T)]\to 0$ identifies the economic value among formal differential-equation solutions.