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For birth and death processes with finite state space, we consider stochastic processes induced by conditioning on hitting the right boundary point before hitting the left boundary point. We call the induced stochastic processes the conditional processes. We show that the conditional processes are again birth and death processes when the right boundary point is absorbing. On the other hand, it is shown that the conditional processes do not have Markov property and they are not birth and death processes when the right boundary point is reflecting.

For one-dimensional diffusion processes on

An important class of ODGDPs which is used as stochastic models in various fields is that of birth and death processes. For example, Moran [

In this paper we prove that Assertions

In Section

Let

The generator

We show that the birth and death process

For a function

We set

There exist a function

For each

Throughout this paper we denote by

In order to make the boundary conditions at

Let

First we show that

Assume that

Theorem

By means of (

We turn to the case that

The second and the third terms of the right-hand side of (

Let

This theorem is proved by applying the following simple proposition for sample path's behavior after hitting the boundary

Let

We prove this proposition in the following section.

We use the same notations as those in Section

First we prepare the following lemma. The proof of this lemma is easy and we omit it.

Let

We assume that

It follows from Theorem

We introduce the Green function corresponding to

First we prove Proposition

We divide the proof into four cases.

Let

Combining this with (

Since (

Let

Let

In this section, we consider a simple birth and death process. Let

We first consider

We next consider

We finally consider

for

Here we consider two stochastic models in population genetics and their conditional processes. In this section, we use notations different from those of the previous sections to emphasize the difference between the original models and the induced models of conditional processes. We denote the conditional process by

We consider the following diffusion model for a randomly mating population consisting of

First we consider the case that

Next we consider the case that

Moran [

First we consider the case that

Next we consider the case that

The authors thank Thomas Nagylaki for suggesting them to consider conditional processes for Moran model in population genetics. They also thank the anonymous reviewer for comments on the previous version of this paper.