  The representation theorem is obtained for functionals of non-Markov
processes and their first exit times from bounded domains. These functionals
are represented via solutions of backward parabolic Ito equations. As an
example of applications, analogs of forward Kolmogorov equations are derived
for conditional probability density functions of Ito processes being killed on
the boundary. In addition, a maximum principle and a contraction property are
established for SPDEs in bounded domains.
