I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process:

$$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg] $$ where $(Y_t)_t$ is a Yule process of intensity $\lambda$ and $k$ an arbitrary positive integer.

Are there any results regarding the $Z_t$'s distribution?

  • $\begingroup$ Are you sure you want to condition on $Y_t=k$ rather than $Y_0=k$? $\endgroup$ – Yuval Peres Jun 21 at 7:31
  • $\begingroup$ Yes. Although I should have mentioned that the Yule process starts in $Y_0=1$. I updated the post. $\endgroup$ – Gabriel Jun 21 at 9:27
  • $\begingroup$ Where is the random variable here? Are you interested in the conditional expectation of the integral, or in its conditional distribution? $\endgroup$ – Kostya_I Jun 21 at 10:06
  • $\begingroup$ The random variables are the Yule process $(Y_t)_t$ and the conditional expectation $(Z_t)_t$. I am interested in the distribution of the conditional expectation above. I think intuitively one might compare it to the integral of a Brownian bridge (although BB is not non-negative and increasing). $\endgroup$ – Gabriel Jun 21 at 11:35
  • $\begingroup$ The conditional expectation is a deterministic function of $k$. $\endgroup$ – Yuval Peres Jun 21 at 12:40

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