  In recent years, instanton calculus has successfully been employed to
estimate tail probabilities of rare events in various stochastic dynamical
systems. Without further corrections, however, these estimates can only capture
the exponential scaling. In this paper, we derive a general, closed form
expression for the leading prefactor contribution of the fluctuations around
the instanton trajectory for the computation of probability density functions
of general observables. The key technique is applying the Gel'fand-Yaglom
recursive evaluation method to the suitably discretized Gaussian path integral
of the fluctuations, in order to obtain matrix evolution equations that yield
the fluctuation determinant. We demonstrate agreement between these predictions
and direct sampling for examples motivated from turbulence theory.
