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Representation of functionals of Ito processes and their first exit times

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, analogues of forward Kolmogorov equations are derived for conditional probability density functions of Ito processes killed on the boundary. In addition, a maximum principle and a contraction property are established for stochastic partial differential equations in bounded domains.