  In this paper we show the strong existence and the pathwise uniqueness of an
infinite-dimensional Stochastic Differential Equation (SDE) corresponding to
the bulk limit of Dyson's Brownian Motion (DBM), for all $\beta\geq 1$. Our
construction applies to an explicit and general class of initial conditions,
including the lattice configuration $\{x_i\}=\mathbb{Z}$ and the sine process.
We further show the convergence of the finite to infinite-dimensional SDE. This
convergence concludes the determinantal formula of Katori and Tanemura (2010)
for the solution of this SDE at $\beta=2$.
