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\title{Stochastic simulation of adjoint problems for jump processes}
\author{Uwe Jaekel, Ivan Dimov, Dietmar Wendt}

\maketitle

\begin{abstract}
  We consider a class of adjoint problems to problems whose dynamics
  is determined by a Master equation for the probability distribution
  of the possible states. The forward problem can be simulated by the
  minimal process method (Gillespie algorithm). The general backward
  problem, however, cannot be interpreted as a Master equation, as it
  is not probability conserving, and correspondingly the  adjoint
  distribution can neither be regarded as a probability distribution,
  nor is its integral a constant. We illustrate the problem for a
  transport equation whose continuum limit has been examined by Dimov
  et al. and show that the adjoint solution can be obtained as the
  solution of a related Master equation, multiplied by the same time
  dependent scaling factor as in the continuum limit. We generalize
  the result, such that we can solve the adjoint equation with a
  slight variation of the minimal process method for a large class of
  problems.
\end{abstract}


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