Statistical simulation of one type of pairs of random variables with the use of fictitious jumps

Statistical simulation based on the sampling technique for a pair of random variables $(T,\mathscr U)$, where $T\in[0,+\infty)$ and $\mathscr U\in\mathscr R^d$ ($d\geq1$) is considered. The simultaneous distribution of the pair is specified in the form that is common for analogous problems in various fields. It has the form
$$ \mathbf P\{T\in dt,\mathscr U\in du\}=f(t,u)\exp\biggl(-\int_0^t\int_{\mathscr R^d}f(t',u')m(du')dt'\biggr)dt\,m(du), $$
where $f$ is a function and $m$ is a measure. The first variable $T$ is the well-known random waiting time. A simulation method for the pair $(T,\mathscr U)$ is constructed using a realization of an auxiliary Markov sequence of trial pairs. Applications of this method in particle transport theory and in kinetics of rarefied gases are discussed.