[FabHK]

Certain simple stochastic differential equations can be solved explicitly analytically (like some integrals and simple ordinary differential equations can be solved explicitly), for example the classic Black Scholes equation. More complicated ones typically can't be solved in that way.

What one often wishes to have is the expectation of a function of a stochastic process at some point, and what can be shown is that this expectation obeys a certain (deterministic) partial differential equation. This then can be solved using numerical PDE solvers.

In higher dimensions, though, or if the process is highly path-dependent (not Markovian), one resorts to Monte Carlo simulation, which does indeed simulate "many possible unfolding of events".