  Kinetic descriptions of runaway electrons (RE) are usually based on
Fokker-Planck models that determine the probability distribution function (PDF)
of RE in 2-dimensional momentum space. Despite of the simplification involved,
the Fokker-Planck equation can rarely be solved analytically and direct
numerical approaches (e.g., continuum and particle-based Monte Carlo (MC)) can
be time consuming, especially in the computation of asymptotic-type observables
including the runaway probability, the slowing-down and runaway mean times, and
the energy limit probability. Here we present a novel backward MC approach to
these problems based on backward stochastic differential equations (BSDEs) that
describe the dynamics of the runaway probability by means of the Feynman-Kac
theory. The key ingredient of the backward MC algorithm is to place all the
particles in a runaway state and simulate them backward from the terminal time
to the initial time. As such, our approach can provide much faster convergence
than direct MC methods (by significantly reducing the number of particles
required to achieve a prescribed accuracy) while at the same time maintaining
the advantages of particle-based methods (compared to continuum approaches).
The proposed algorithm is unconditionally stable, can be parallelized as easy
as the direct MC method, and its extension to dimensions higher than two is
straightforward, thus paving the way for conducting large-scale RE simulation.
