On approximation of a conditional transition function of right-continuous Markov processes (Q2740304)

Let \(z(t)=(x(t),y(t))\), \(t\geq 0,\) be a right-continuous Markov process with continuous time. Let \({\mathcal F}_{[0,t]}=\sigma\{y(s)\), \(0\leq s\leq t\}\) be a minimal \(\sigma\)-algebra generated by \(y(s),0\leq s\leq t\). An approximation of the conditional transition function \(\widehat\pi_{x}(t,A)=P\{x(t)\in A\mid x(0)=x,{\mathcal F}_{[0,t]}\}\) of the process \(x(t)\) is proposed when the trajectory of the \(y(t)\) component of the process is given.