[kkylin]

It depends a bit on exactly what you want to calculate, but in general things like the probability density function of the solution of a stochastic differential equation (SDE) at time t satisfies a partial differential equation (PDE) that is first order in time and second order in space [0]. (This PDE is known to physicists as the Fokker-Planck equation and to mathematicians as the Kolmogorov forward equation.) Except in special examples, the PDE will not have exact analytical solutions, and a numerical solution is needed. Such a numerical solution will be very expensive in high dimensions, however, so in high-dimensional problems it is cheaper to solve the SDE and do Monte Carlo sampling, rather than try to solve the PDE.

Edit: sometimes people are interested in other types of questions, for example the solution when certain random events occur. Analogous comments apply. Also, while stochastic calculus is very useful for working with SDEs, if your interest is other types of Markov (or even non-Markov) processes you may need other tools.

Edit again: as another commenter mentioned, in special cases the SDE itself may also have exact solutions, but in general not.

[0] This statement is specific to stochastic differential equations, i.e., a differential equation with (gaussian) white noise forcing. For other types of stochastic processes, e.g., Markov jump processes, the evolution equation for distributions have a different form (but some general principles apply to both, e.g., forms of the Chapman-Kolmogorov equation, etc).