Most stochastic problems do not admit analytical solutions. Numerical methods can only solve problems involving finite sets of random variables. For example, these methods cannot deliver the distribution of the extreme $\sup_{t\in[0,\tau]}|X(t)|$ of a real-valued, continuous-time stochastic process $X(t)$ since these processes are uncountable families of random variables indexed by time. Numerical methods can only deliver estimates of extremes $\sup_{t\in[0,\tau]}|X_d(t)|$ of finite dimensional (FD) surrogates $X_d(t)$ of $X(t)$, i.e., deterministic functions of time and $d$ random variables. These numerical solutions are useful only if the distribution of $\sup_{t\in[0,\tau]}|X_d(t)|$ converges to that of $\sup_{t\in[0,\tau]}|X(t)|$ as $d$, referred to as stochastic dimension, increases to infinity. We develop conditions under which the distributions of functionals of $X_d(t)$ converge to those of functionals of target process $X(t)$, where $X(t)$ can be a real/vector-valued Gaussian/non-Gaussian process denoting the input to or the output of dynamical systems. Under these conditions, the distributions of extremes of FD processes can be used as surrogates for those of target processes provided that the stochastic dimension $d$ is sufficiently large. These theoretical results are illustrated by numerical examples which show consistency with theoretical developments.