 Solution

Both coefficient functions are linear, hence globally Lipschitz. Their norms are bounded by a constant times $|(x,y)|$, so they satisfy the linear growth condition for an SDE. The existence and pathwise uniqueness theorem for a stochastic differential equation gives a unique global strong solution of a stochastic differential equation for every deterministic starting point $(x_0,y_0)\in {\mathbb{R}}^2$. Linear growth prevents finite-time explosion.

The theorem gives pathwise uniqueness, and strong existence with pathwise uniqueness gives uniqueness in law. Thus every initial value is permitted, and both forms of uniqueness hold. The coefficients need not be bounded for this theorem; their global Lipschitz and linear-growth properties are sufficient.