Conditional Processes Induced by Birth and Death Processes

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.


Introduction
For one-dimensional diffusion processes on 0, 1 related to diffusion models in population genetics, Ewens 1 considered stochastic processes induced by conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The boundary points 0 and 1 are accessible and absorbing boundaries for the diffusion processes that he considered and the induced stochastic processes are again diffusion processes.Then the induced stochastic processes are referred to as the conditional diffusion processes by Ewens 1 see also 2 .Motivated by this work, Iizuka et al. 3 were concerned with one-dimensional generalized diffusion processes ODGDPs for brief on l 1 , l 2 whose speed measures are right-continuous and strictly increasing functions.They considered stochastic processes induced by conditioning on hitting the right boundary point l 2 before hitting the left boundary point l 1 .The induced stochastic processes are called the conditional processes.They showed as Theorem 2.1 that the conditional processes are again ODGDPs when the boundary point l 2 is accessible with the absorbing boundary condition Assertion 1 .If the original process x t is a one-dimensional diffusion process with the generator L a x 2 then the conditional process x * t induced by conditioning on hitting l 2 before hitting l 1 is again a one-dimensional diffusion process and its generator can be expressed as Here we put where c is a point with l 1 < c < l 2 see 4, 5 .On the other hand, Iizuka et al. 3 showed as Theorem 2.2 that the probability distributions of the conditional processes do not satisfy the Chapman-Kolmogorov equation when the boundary point l 2 is accessible with the reflecting boundary condition.Hence the conditional processes cannot be Markov processes when the boundary point l 2 is accessible with the reflecting boundary condition Assertion 2 .An important class of ODGDPs which is used as stochastic models in various fields is that of birth and death processes.For example, Moran 6 introduced a birth and death process as one of fundamental stochastic models in population genetics called Moran model we will consider this model in Section 5 .However, the speed measure of any birth and death process is not a strictly increasing function see 7 and we cannot apply the results of 3 to birth and death processes.
In this paper we prove that Assertions 1 and 2 hold for the case that the speed measure is a nondecreasing step function.The motivation of this paper is to investigate the properties of the conditional processes induced by conditioning on hitting the right boundary point before hitting the left boundary point when the original processes are birth and death processes.The proof of Theorem 2.2 in 3 is analytical nonprobabilistic and it is not easy to see that the conditional processes do not satisfy Markov property when the right boundary point is accessible with the reflecting boundary condition.The proof presented in this paper for Assertion 2 is based on the fact that the state space is discrete.The proof is probabilistic and we can see intuitively that the conditional processes do not satisfy Markov property when the right boundary point is reflecting.It is our extra purpose to see this by considering birth and death processes.
In Section 2 we state our results more precisely.Section 3 is devoted to their proofs.In Section 4 we introduce a very simple birth and death process and present concrete expressions of its conditional processes considering all the boundary conditions.Finally we discuss some stochastic models in population genetics and their conditional processes in Section 5.
International Journal of Mathematics and Mathematical Sciences 3

Main Results
Let e be an exponentially distributed random variable with the mean 1 and let {e 1 , e 2 , e 3 , . ..} be a sequence of independent copies of e.We put τ 0 0 and τ k k i 1 e i k 1, 2, . . . .For {e 1 , e 2 , e 3 , . ..}, an integer N N ≥ 2 , and points a i i 0, 1, . . ., N such that a 0 < a 1 < a 2 < • • • < a N , we consider a birth and death process D X t , P x with the state space Σ {a 0 , a 1 , a 2 , . . ., a N } satisfying the following conditions.For a i ∈ Σ and τ k < t < τ k 1 , conditional probabilities conditional on X τ k a i satisfy where 0 ≤ p 0 ≤ 1, p N 0, q 0 0, 0 ≤ q N ≤ 1, and p i > 0, q i > 0, r i ≥ 0 for i 1, 2, . . ., N − 1.Here P x denotes the probability measure concentrated at the event {X 0 x}, that is, P x X 0 x 1.The end boundary point a 0 resp., a N is called to be absorbing or reflecting according to p 0 0 resp., q N 0 or p 0 > 0 resp., q N > 0 .
The generator L of D is given by for u ∈ D L , where D L is the set of all functions u on Σ such that u a 0 0 if a 0 is absorbing, Here is a proof of 2.2 .By means of 2.1 , we find that

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Therefore we obtain the following: 2.5 see also 7 .
We show that the birth and death process D can be described as an ODGDP.We set where ρ 0 resp., ρ N is an increasing continuous function on l 1 , a 0 resp., a N , l 2 such that ρ 0 a 0 0 and ρ 0 l 1 −∞ resp., ρ N a N 0 and ρ N l 2 ∞ .Further we set Here s is a real-valued continuous increasing function on S l 1 , l 2 , and m is a right-continuous nondecreasing function on R.They are called the scale function and the speed measure, respectively.We set m {x} m x − m x− , m i m {a i } , and s i s a i , i 0, 1, . . ., N. We note that m 0 ∞ resp., m N ∞ if a 0 resp.a N is absorbing.
For a function f on S, we simply write f l 1 resp., f l 2 in place of f l 1 resp., f l 2 − provided f l 1 resp., f l 2 − exists.Further, f resp., f − stands for the right resp., left derivative of f with respect to s if it exists, that is, We set Σ * Σ ∩ S. Let D G be the space of all bounded continuous functions u on S satisfying the following conditions.
G.1 There exist a function f on Σ * and two constants A 1 , A 2 such that Throughout this paper we denote by c an arbitrarily fixed point of Σ * .The operator G is defined by the mapping from u ∈ D G to f that appeared in 2.9 .The operator G is called the one-dimensional generalized diffusion operator ODGDO for brief with s, m .It is known that there exists a strong Markov process D * with the generator G, which is called an ODGDP on S see 8, 9 .It is also known that D can be identified with D * see 7-9 .Indeed, it is easy to see that u ∈ D G satisfies the following:

2.10
In order to make the boundary conditions at a 0 and a N clear, we use D IJ and P IJ x in place of D and P x , respectively.Here I, J ∈ {A, R}, and I A resp., J A means that a 0 resp., a N is absorbing i.e., p 0 0 resp., q N 0 and I R resp., J R means that a 0 resp., a N is reflecting i.e., p 0 > 0 resp., q N > 0 .It is known that there is the transition probability density p IJ t, x, y of D IJ with respect to m, that is, Let Σ o {a 1 , . . ., a N } and let σ a be the first hitting time at a, that is, σ a inf{t > 0 : X t a}.In this paper we consider stochastic processes induced by the following conditional probability:

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We set It is known that We note that h is independent of boundary conditions I, J ∈ {A, R}.
First we show that Q IA induces a birth and death process for I ∈ {A, R}.
Theorem 2.1.Assume that a N is absorbing.Then Q IA x X t y is independent of I ∈ {A, R}, and it is represented as x induces a birth and death process D o on Σ o for which the end point a 1 is reflecting, the end point a N is absorbing, and the generator L o is given by Theorem 2.1 shows that the relation between 1.1 and 1.2 for diffusion processes corresponds to the relation between 2.2 and 2.16 for birth and death processes.We note that the generator is given by 2.17 and the boundary condition 2.18 when N 2.

2.19
International Journal of Mathematics and Mathematical Sciences 7 Combining these with 2.16 , we find that Q IA x satisfies the following.For

2.20
We turn to the case that a N is reflecting.When m x is strictly increasing, a representation of Q IR x is given by 2.11 of 3 .We note that this representation is available even if m x is not strictly increasing.Therefore we obtain the following representation for birth and death processes:

2.27
We note that N A •, •, • 0. The second and the third terms of the right-hand side of 2.21 come from sample path's behavior after hitting the boundary a N .This representation suggests that Q IR does not satisfy Markov property.Indeed we obtain the following theorem.We prove this proposition in the following section.

Proofs of Theorems
We use the same notations as those in Section 2.

Proof of Theorem 2.1
First we prepare the following lemma.The proof of this lemma is easy and we omit it.

3.3
Proof of Theorem 2.1.We assume that a N is absorbing.Let I ∈ {A, R} and t > 0. Then Then by means of 2.13 , and 2.24 ,

3.5
Let x, y ∈ Σ o \ {a N }.Then by using Markov property of D IA , 2.13 and 3.1 , we see that
for a function u on Σ o \ {a N } and i 2, 3, . . ., N − 1.By means of 2.7 , 2.8 , and 3.7 , we see that In the same way, we have Therefore we get 12 for i 2, 3, . . ., N − 1.Since a 0 is entrance, we see that by virtue of general theory on ODGDOs.Thus we find that D o is a birth and death process on Σ o , the generator is given by 2.16 , the end point a 1 is reflecting with 2.17 , and the end point a N is absorbing.The proof is completed.

Proof of Theorem 2.3
We introduce the Green function corresponding to D IJ .For I, J ∈ {A, R}, k 1, 2, and α > 0, let g IJ k •, α be a continuous function on S satisfying the following properties:

3.20
Here x, α .Note that W IJ α is a positive number independent of x ∈ S. We put

3.25
The right-hand side of this formula is negative if x ≥ y.This implies that 2.29 does not hold true for x ≥ y.
Case 3. I A and J R. By means of 3.3 ,

3.26
Since a 0 is absorbing, we get

3.27
Combining these equalities with 3.2 and 3.3 , we see that

3.28
The right-hand side of this formula is positive if x ≤ y.This implies that 2.29 does not hold true for x ≤ y. x .Here we note that 3.32 is valid for y a N .Indeed,

3.33
Combining this with 3.22 , we see that By virtue of 3.17 , 3.20 , and 3.21 , we see that

3.36
We take a point Then by virtue of 3.19 , 3.20 , and 3.21 , , α dm z .

3.37
Thus we obtain that

3.38
It is known that

3.40
This contradicts the fact that the last term is positive.Thus 2.29 does not hold true for t > 0, x ∈ Σ o \ {a N }, and y ∈ Σ o .

3.41
where I ∈ {A, R}.Therefore that is, 2.29 is valid for x y a 1 .Proposition 2.4 implies, however, that 2.29 does not hold for x a 1 and y a 2 .

3.43
Then l 1 0 and l 2 3. Further 2.7 and 2.8 are reduced to 4.6

Case That the End Point 0 Is Absorbing and the End Point 3 Is Reflecting
We next consider D AR with q 3 1, that is,

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For simplicity, we put K 2. Then l 1 0 and l 2 ∞.Further 2.7 and 2.8 are reduced to

4.8
By virtue of 11 , we obtain that P AR x X t y p AR t, x, y m y , x,y ∈ {1, 2, 3},

Case That the End Points 0 and 3 Are Reflecting
We finally consider D RR with p 0 1 and q 3 1, that is,

4.12
For simplicity, we put K 2. Then l 1 −∞ and l 2 ∞.Further 2.7 and 2.8 are reduced to

4.13
By virtue of 11 , we obtain that P RR x X t y p RR t, x, y m y , x,y ∈ {0, 1, 2, 3}, 4.14 by 1.2 and Theorem 2.1 of 3 .Note that the effect of conditioning is that it inflates the mutation rate u 2 to u 2 1/N.Ewens 1 considered the case that u 1 u 2 0 and the induced diffusion process is referred to as the conditional diffusion process by Ewens 1 see also 2 .
Next we consider the case that 0 < 2Nu 1 < 1.The point 1 is regular boundary in this case see 16 and we can pose various boundary conditions there.If we pose the absorbing boundary condition, then the induced process is again a diffusion process with the generator 3 by 1.2 and Theorem 2.1 of 3 .On the other hand, if we pose the reflecting boundary condition as it is usually done in population genetics see 17-19 , then the induced conditional process does not satisfy the Chapman-Kolmogorov equation and this process is not a diffusion process due to Theorem 2.2 of 3 .These results imply that we cannot use the diffusion model whose generator is given by 5.3 as the conditional process when we pose the reflecting boundary condition at the boundary point 1.

5.4
where p j q j r j 1 0 ≤ j ≤ N .Note that p 0 ν 2 , q 0 0, r 0 1 − ν 2 , p N 0, q N ν 1 , and r N 1 − ν 1 .Note also that p j > 0 unless ν 1 1 and ν 2 0, q j > 0 unless ν 1 0 and ν 2 1, and r j > 0 for 0 < j < N. The process X t does not jump at time τ k if the types of newborn individual and the dead are the same even though a "birth and death" event occurs at time τ k .One of the end points 0 is absorbing resp., reflecting when ν 2 0 resp., ν 2 > 0 and the other end point 1 is absorbing resp., reflecting when ν 1 0 resp., ν 1 > 0 .Let σ i be the first hitting time to i i 0, 1 .First we consider the case that ν 1 ν 2 0. This is the case without mutation and both boundary points are absorbing with p j q j j N − j N 2 , r j 1 − p j − q j 1 − 2j N − j N 2 .

5.5
By 2.7 and 2.8 we have

5.6
Then Theorem 2.1 implies that the conditional process X * t conditional on {σ 1 < σ 0 } is again a birth and death process on {1/N, 2/N, . . ., N − 1 /N, 1} with the transition law 5.9 if N 3 and this is essentially the same as the simple birth and death process discussed in Section 4.1.
Next we consider the case that 0 < ν 1 < 1.The boundary point 1 is reflecting.Then the induced conditional process does not satisfy Markov property and this conditional process is not a birth and death process by Theorem 2.3.
implies 3.32 with y a N .International Journal of Mathematics and Mathematical Sciences Since 3.32 holds true for x ∈ Σ o \ {a N } and y ∈ Σ o , we have 0

for 1 ≤ 1 2N
j < N. The end point 1/N is reflecting since p * − 1 /N 2 > 0. Note that m x of the original Moran model X t reduces to m x This theorem is proved by applying the following simple proposition for sample path's behavior after hitting the boundary a N .
t 1 y 2.28 does not hold for some x, y, z ∈ Σ o .This implies that Q IR x does not satisfy Markov property.does not hold for some x ∈ Σ o \ {a N } and y ∈ Σ o .
Lemma 3.1.Let I, J ∈ {A, R}, t > 0, and x, y ∈ Σ o \ {a N }.Then it holds true that 5 , and 3.6 show that Q IA x X t y is independent of I ∈ {A, R}.The formula 2.15 follows from 2.11 , 2.14 , and 3.6 .It follows from Theorem 2.2 and Propositions 3.1 and 3.4 of 12 that Q IA x induces an ODGDP D o on a 0 , a N , the boundary a 0 is entrance in the sense of Feller see 8, 13 , the boundary a N is absorbing, and the generator is the ODGDO G o with s o , m o , where We divide the proof into four cases.
by means of 3.3 .Since there are x, y ∈ Σ o \ {a N } such that x / y for N ≥ 3, 3.23 shows that 2.29 does not hold true for x / y for N ≥ 3.
Case 4. I J R. Suppose that 2.29 holds true for t > 0, x ∈ Σ o \ {a N }, and y ∈ Σ o .Then ∞ 0 e −αt H t, x, y dt 0, 3.29 for α > 0, x ∈ Σ o \ {a N }, and y ∈ Σ o , where for α > 0 and x, y ∈ Σ o \ {a N }. for α > 0 and x, y ∈ Σ o \ {a N }, where E IJ x stands for the expectation with respect to P IJ Moran 6introduced the following birth and death process as one of the fundamental stochastic models in population genetics called continuous-time Moran model see 4 for discrete time Moran model .We refer to this model as Moran model for brief.Let N be the number of individuals in a haploid population with two types A 1 and A 2 , where N is an integer greater than 2. Let τ 0 0 and τ k , k 1, 2, • • • be a sequence of random times introduced in Section 2. At time τ k an individual is chosen randomly and it reproduces a new individual k ≥ 1 .The type of the newborn individual is A 1 resp.,A 2 with probability 1 − ν 1 resp., 1 − ν 2 and it is A 2 resp.,A 1 with probability ν 1 resp., ν 2 if the parent is A 1 resp.,A 2 , where 0 ≤ ν 1 , ν 2 ≤ 1.Then at this time τ k an individual except newborn individual is chosen randomly to die.There is no change at time t / τ k k ≥ 1 .Denoting by X t the relative frequency of A 1 at time t, X t is a birth and death process on {0, 1/N, 2/N, . . ., N − 1 /N, 1} with the transition law