I have a stochastic variable X(t) which changes at a discrete set of random times $t_1, t_2, \dots$. I can simulate this stochastic process to obtain a series $(t_1,X(t_1)), (t_2, X(t_2)), \dots$
However, the transitions are so common that saving the entire simulation is too memory intensive. My first approach was to only store the state of the process $(t_i, X(t_i))$ if $t_i-t_{i-1}> \Delta t$. So $\Delta t$ is a sampling interval and $1/\Delta t$ is a sampling rate. However, downsampling in this way, only recording the state of the system about every $\Delta t$, biases the statistics of transition times. For example, it's difficult to estimate the distribution of the time between transitions, i.e., of $\delta t = t_i-t_{i-1}$ if I downsample in this way, since I ignored all $\delta t < \Delta t$.
Is there a way to downsample a stochastic time series that (even approximately) preserves the statistics of transition times? Please note I have to downsample on the fly, i.e., as the simulation runs.