  We give a new coherent description of the first-order Fermi acceleration of
particles in shock waves from the point of view of stochastic process of the
individual particles, under the test particle approximation. The time
development of the particle distribution function can be dealt with by this
description, especially for relativistic shocks. We formulate the acceleration
process of a particle as a two-dimensional Markov process in a logarithmic
momentum-time space, and relate the solution of the Markov process with the
particle distribution function at the shock front, for both steady and
time-dependent case. For the case where the probability density function of the
energy gain and cycle-time at each shock crossing of the particles obeys a
scaling law in momentum, which is usually assumed in the literature, it is
confirmed in more general form that the energy distribution of particles has
the power-law feature in steady state. The equation to determine the exact
power-law index which is applicable for any shock speed is derived and it is
shown that the power-law index, in general, depends on the shape of the
probability density function of the energy gain at each shock crossing; in
particular for relativistic shocks, the dispersion of the energy gain can
influence the power-law index. It is also shown that the time-dependent
solution has a self-similarity for the same case.
