  We present Monte Carlo-Euler methods for a weak approximation problem related
to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic
differential equations in infinite dimensional spaces, and prove strong and
weak error convergence estimates. The weak error estimates are based on
stochastic flows and discrete dual backward problems, and they can be used to
identify different error contributions arising from time and maturity
discretization as well as the classical statistical error due to finite
sampling. Explicit formulas for efficient computation of sharp error
approximation are included. Due to the structure of the HJM models considered
here, the computational effort devoted to the error estimates is low compared
to the work to compute Monte Carlo solutions to the HJM model. Numerical
examples with known exact solution are included in order to show the behavior
of the estimates.
