  Bosonic mean-field theories can approximate the dynamics of systems of $n$
bosons provided that $n \gg 1$. We show that there can also be an exact
correspondence at finite $n$ when the bosonic system is generalized to include
interactions with the environment and the mean-field theory is replaced by a
stochastic differential equation. When the $n \to \infty$ limit is taken, the
stochastic terms in this differential equation vanish, and a mean-field theory
is recovered. Besides providing insight into the differences between the
behavior of finite quantum systems and their classical limits given by $n \to
\infty$, the developed mathematics can provide a basis for quantum algorithms
that solve some stochastic nonlinear differential equations. We discuss
conditions on the efficiency of these quantum algorithms, with a focus on the
possibility for the complexity to be polynomial in the log of the stochastic
system size. A particular system with the form of a stochastic discrete
nonlinear Schr\"{o}dinger equation is analyzed in more detail.
