# The Limit Behavior of a Stochastic Logistic Model with Individual Time-Dependent Rates

Authors: Yilun Shang
Journal: Journal of Mathematics
 (2013)

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## Abstract

We investigate a variant of the stochastic logistic model that allows individual variation and time-dependent infection and recovery rates. The model is described as a heterogeneous density dependent Markov chain. We show that the process can be approximated by a deterministic process defined by an integral equation as the population size grows.


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## Body

## 1. Introduction

The stochastic logistic model, also called the endemic SIS model in the epidemiological context, was first discussed by Weiss and Dishon [1]. This model describes the evolution of an infection in a fixed population of size n by a continuous-time Markov chain for the number of infected individuals Yn. The state space is {0,1,…,n}, and the transition rates are given by
(1)Yn⟶Yn+1atraten-1λYn(n-Yn),Yn⟶Yn-1atrateμYn,
where λ and  μ are the infection rate of susceptibles and the recovery rate of infectives, respectively. This simple model has found applications in a variety of fields, including population biology, metapopulation ecology, chemistry, and physics. Properties such as mean epidemic size [2, 3], mean extinction time [1, 4, 5], and quasi-stationary distributions [6–8] have been extensively studied under different initial conditions.The stochastic logistic model has an interesting limit property that it can be approximated by deterministic differential equations. In particular, based on Theorem  3.1 of [9], the rescaled process n-1Yn converges in probability uniformly on finite time intervals to the solution of an ordinary differential equation. The issue of differential equation approximations for stochastic processes traces back to the pioneer work of Kurtz [9, 10], where deterministic limit of pure jump density-dependent Markov processes was addressed via Trotter type approximation theorems. Recently, McVinish and Pollett [11] show that a deterministic limit process can be established for a stochastic logistic model with individual variation, where the coefficients of the transition rates of the Markov chain can vary with the nodes. We refer the interested reader to a comprehensive survey [12] for numerous sufficient conditions for this type of convergence.In this paper, along the above line of study, we investigate a variant of the stochastic logistic model where both individual variation and time-dependent infection and recovery rates are allowed. Specifically, we consider a continuous-time Markov chainXn=(X1n,…,Xnn) on the state space  {0,1}n with transition rates given by
(2)Xn⟶Xn+einatrateλi,t(1-Xin)f(n-1∑j=1n‍gj(Xjn)),Xn⟶Xn-einatrateμi,tXin,
where i=1,2,…,n, λi,t, μi,t∈ℝ+, and ein is the n-dimensional vector whose ith entry is 1, and other entries are 0. Let Xin=0 represent that individual i is susceptible and Xin=1 represent that individual i is infected. Then Yn=∑i=1n‍Xin is equivalent to the stochastic logistic model by setting λi,t=λ, μi,t=μ, f(x)=x, and gj(x)=x for all i, j, and t.The above model (2) can be viewed as a generalization of that treated in [11] in twofolds. Firstly, the infection and recovery rates are time varying, incorporating realistic scenarios where the infection and recovery capacities may change over time [13]. As such, the transition rates of the Markov chain are explicitly time dependent, making the previously obtained sufficient conditions no longer applicable.Secondly, the introduction of functionsgj(x) accommodates needed flexibility for some applications. Indeed, since Xjn∈{0,1}, the linear form gj(x)=κjx used in [11] implies that no contribution can be made by susceptible individuals. This is only a rough approximation. For one thing, susceptible individuals aware of a disease in their proximity can take measures (such as wearing masks, avoiding public places, and frequent hand washing) to reduce their susceptibility, which in turn affect the epidemic dynamics dramatically [14, 15]. For another, the epidemic progression strongly depends on the contact patterns between susceptible and infected individuals, especially in the network context. Identifying individual role is an interesting and demanding task [16]. In the present framework, each node j applies individual contribution gj(0) (and gj(1)), indicating the underlying interaction structure/strength among individuals.The rest of the paper is organized as follows. We state the main result in Section2 and provide the proof in Section 3.


## 2. The Result

In what follows, we assume naturally thatgj(1)≥gj(0)≥0 for all  j. We will show that the stochastic logistic model (2) converges weakly to the solution of an integral equation as the population size n→∞.Let(S,ℬ) be a measurable space with S⊆ℝ+2 and ℬ the Borel σ-algebra on S. Denote by Cb(S) the set of bounded, continuous functions on S. Let Ω be the set of σ-finite measures on S. For n∈ℕ and h∈Cb(S), we define the measure-valued nonrandom process {σtn,t∈ℝ+} and the measure-valued Markov process {ρtn,t∈ℝ+} by
(3)σtn(h)≔∫S‍h(μ,λ)σtn(dμ,dλ),σtn(h)=n-1∑i=1n‍(gi(1)-gi(0))h(μi,t,λi,t),ρtn(h)≔∫S‍h(μ,λ)ρtn(dμ,dλ)ρtn(h)=n-1∑i=1n‍(gi(1)-gi(0))Xi,tnh(μi,t,λi,t).
For some set A, let D(ℝ+,A) be the set of right-continuous functions with left-hand limits mapping ℝ+ to A. Thus, for n∈ℕ, σn∈D(ℝ+,Ω) and the sample paths of ρn belong to D(ℝ+,Ω), since the sample paths of Xn belong to D(ℝ+,{0,1}n) (application of Theorem  16.16 in [17, page 316]).We assume that the following assumptions hold.(A1)
n-1∑i=1n‍gi(0)→g-<∞, and supng-1,n<∞, where g-a,n=n-a∑i=1n‍(gi(1)-gi(0))a for a∈ℕ.(A2)
σtn→dσt uniformly for t∈ℝ+, and ρ0n→dρ0 in Ω as n→∞, where σt and ρ0 are nonrandom measures, and →d means weak convergence (i.e., limn→∞σtn(h)=σt(h), and ρ0n(h) converges to ρ0(h) in distribution as n→∞ for all h∈Cb(S) [17, page 316]).(A3)
S is bounded in ℝ+2.(A4)
f:ℝ+→ℝ+ is Lipschitz continuous.(A5)
λi,t and μi,t as functions of t are piecewise constant for i=1,…,n.Theorem 1.
If assumptions (A1)–(A5) hold, then the measure-valued Markov processρtn converges weakly to a measure-valued nonrandom process ρt∈D(ℝ+,Ω), that is,
(4)ρn→diρ,
as n→∞, where ρ is the unique solution of
(5)0=ρt(h)-ρ0(h)-∫0t‍Gρs(h)ds,
with
(6)ρt(h)=∫S‍h(μ,λ)ρt(dμ,dλ),Gρt(h)=∫S‍λh(μ,λ)f(∫S‍ρt(dμ′,dλ′)+g-)σt(dμ,dλ)-∫S‍λh(μ,λ)f(∫S‍ρt(dμ′,dλ′)+g-)ρt(dμ,dλ)-∫S‍μh(μ,λ)ρt(dμ,dλ).LetL1(S) be the space of σt-integrable functions on S. We have the following corollary.Corollary 2.
Suppose thatρ is the unique solution to (5). Then there exists a unique function
(7)φ:ℝ+⟶L1(S)t↦φt
with 0≤φt≤1 such that
(8)ρt(A)=∫A‍φt(μ,λ)σt(dμ,dλ)
for all Borel set A∈S, and
(9)φt=φ0+∫0t(λf(∫S‍φs(μ′,λ′)σs(dμ′,dλ′)+g-)‍φt=φ0+∫S‍D×(1-φs)-μφs∫S‍)ds.


## 3. Proofs

In this section, Theorem1 will be proved through a series of lemmas by tightness and uniqueness arguments [18]. The idea of proofs is similar to that in [11], and we include the complete proofs here, not only for the convenience of the reader but also to convince the reader that the results do hold in our setting.Lemma 3.
The sequenceρn is tight in D(ℝ+,Ω).Proof.
Recall that [17, Theorem  16.27, page 324] ρn is tight in D(ℝ+,Ω) if and only if ρtn(h) is tight in D(ℝ+,ℝ) for every h∈Cb(S). According to Aldous's tightness criterion [18, Theorem  16.10, page 178], the tightness of ρtn(h) holds, if the following two conditions are satisfied.  (A)
For eachε>0, η>0, and r≥1, there exist a δ0>0 and an n0∈ℕ such that if  τn is a discrete ℱtn-stopping time satisfying τn≤r, then
(10)supn≥n0supδ≤δ0P(|ρτnn(h)-ρτn+δn(h)|≥η)≤ε.(B)
For eachr>0,
(11)lima→∞limsupn→∞P(supt≤rρtn(h)>a)=0.
To show (A), we express the generator ofXn by
(12)Gnl(x)=∑i=1n‍(l(x+ein)-l(x))λi,t(1-xi)Gnl(x)=200×f(n-1∑j=1n‍gj(xj))Gnl(x)=+∑i=1n‍(l(x-ein)-l(x))μi,txi,
for all real-valued functions l(x) with x=(x1,…,xn)∈{0,1}n. Hence, using (3), we obtain
(13)Gnρtn(h)=∫S‍λh(μ,λ)f(∑j=1n‍∫S‍ρtn(dμ′,dλ′)=∫S‍λh(μ,λ)iiiiif+n-1∑j=1n‍gj(0))σtn(dμ,dλ)=-∫S‍λh(μ,λ)f(∑j=1n‍∫S‍ρtn(dμ′,dλ′)=-∫S‍λh(μ,λ)iiiif+n-1∑j=1n‍gj(0))ρtn(dμ,dλ)=-∫S‍μh(μ,λ)ρtn(dμ,dλ).
Applying Dynkin's formula to ρtn(h) (see, e.g., [19, Proposition  1.7, page 162]), we have that
(14)Mtn(h)=ρtn(h)-ρ0n(h)-∫0t‍Gnρsn(h)ds
is a martingale with respect to the filtration ℱtn=σ{Xi,sn,0≤s≤t}, and M0n(h)=0. Moreover, Mtn(h) is square integrable. The condition (A) holds if for each ε>0, η>0, and r≥1, there exist a δ0>0 and an n0∈ℕ such that for any discrete ℱtn-stopping time satisfying τn≤r,
(15)supn≥n0supδ≤δ0P(|∫τnτn+δ‍Gnρsn(h)ds|≥η)≤ε,(16)supn≥n0supδ≤δ0P(|Mτn+δn(h)-Mτnn(h)|≥η)≤ε.
To see (15), we note that
(17)|Gnρtn(h)|≤∫S‍λh(μ,λ)f(∫S‍ρtn(dμ′,dλ′)+n-1∑j=1n‍gj(0))×σtn(dμ,dλ)+∫S‍λh(μ,λ)f(∫S‍ρtn(dμ′,dλ′)+n-1∑j=1n‍gj(0))×ρtn(dμ,dλ)+∫S‍μh(μ,λ)ρtn(dμ,dλ)≤∫S‍h(μ,λ)(2λf(g-1,n+n-1∑j=1n‍gj(0))+μ)×σtn(dμ,dλ)≤g-1,n×sup(μ,λ)∈S{h(μ,λ)(2λf(g-1,n+n-1∑j=1n‍gj(0))+μ)},
by using (3). It follows from Markov's inequality and (17) that
(18)P(|∫τnτn+δ‍Gnρsn(h)ds|≥η)≤η-1E(∫τnτn+δ‍|Gnρsn(h)|ds)≤η-1δg-1,n×sup(μ,λ)∈S{h(μ,λ)(2λf(g-1,n+n-1∑j=1n‍gj(0))+μ)}.
From assumptions (A1) and (A4), we obtain supnf(g-1,n+n-1∑j=1n‍gj(0))<∞. Therefore, we can find δ0 and n0 such that (15) is satisfied by using the assumptions (A1), (A3), (A4), and the fact that h∈Cb(S).
To prove (16), we need to introduce the quadratic variation process [Mn(h)] for Mn(h). Since Mtn(h) is a square integrable martingale, it follows from Proposition  6.1 in [19, page 79] that (Mn(h))2-[Mn(h)] is also a martingale. By using Markov's inequality and martingale properties, we obtain
(19)P(|Mτn+δn(h)-Mτnn(h)|≥η)≤η-2E((Mτn+δn(h)-Mτnn(h))2)=η-2E(E((Mτn+δn(h)-Mτnn(h))2∣ℱτnn))=η-2(E((Mτn+δn(h))2)+E((Mτnn(h))2)-2E(Mτnn(h)E(Mτn+δn(h)∣ℱτnn))((Mτn+δn(h))2))=η-2(E((Mτn+δn(h))2)-E((Mτnn(h))2))=η-2(E((Mτn+δn(h))2-[Mn(h)]τn+δ)+E([Mn(h)]τn+δ)-E((Mτnn(h))2-[Mn(h)]τn)-E([Mn(h)]τn))=η-2(E([Mn(h)]τn+δ-[Mn(h)]τn)).
Applying Dynkin's formula to(ρtn(h))2 similarly as in the derivation of (14), we have that
(20)M~tn(h)=(ρtn(h))2-(ρ0n(h))2-2∫0t‍ρsn(h)Gnρsn(h)ds-n-1∫0t‍G~nρsn(h)ds
is a martingale, where
(21)G~nρtn(h)=∫S‍λh2(μ,λ)f(∫S‍ρtn(dμ′,dλ′)+n-1∑j=1n‍gj(0))×σ~tn(dμ,dλ)-∫S‍λh2(μ,λ)f(∫S‍ρtn(dμ′,dλ′)+n-1∑j=1n‍gj(0))×ρ~tn(dμ,dλ)-∫S‍μh2(μ,λ)ρ~tn(dμ,dλ),
with
(22)∫S‍h(μ,λ)σ~tn(dμ,dλ)=n-1∑i=1n‍(gi(1)-gi(0))2h(μi,t,λi,t),∫S‍h(μ,λ)ρ~tn(dμ,dλ)=n-1∑i=1n‍(gi(1)-gi(0))2Xi,tnh(μi,t,λi,t).
By using (14) and (20), we obtain for any 0≤s<t,
(23)(ρtn(h)-ρsn(h))2=(ρtn(h))2-(ρsn(h))2-2ρsn(h)(ρtn(h)-ρsn(h))=M~tn(h)-M~sn(h)+n-1∫st‍G~nρrn(h)dr-2ρsn(h)(Mtn(h)-Msn(h))+2∫st‍ρrn(h)Gnρrn(h)dr-2ρsn(h)∫st‍Gnρrn(h)dr.
It follows from (3) and the assumption (A5) that ρtn(h) has piecewise constant sample paths, and ∫0t‍Gnρsn(h)ds is a continuous finite variation process. Therefore, (14), (23), and the fact  M~0n(h)=0 imply that
(24)[Mn(h)]t=[ρn(h)]t[Mn(h)]t=M~tn(h)-2∫0t‍ρsn(h)dMsn(h)[Mn(h)]t+n-1∫0t‍G~nρsn(h)ds.
From (19), (24), and martingale properties, we obtain
(25)P(|Mτn+δn(h)-Mτnn(h)|≥η)≤n-1η-2E(∫τnτn+δ‍G~nρsn(h)ds)≤n-1η-2×∫τnτn+δ‍∫S‍h2(μ,λ)(2λf(g-1,n+n-1∑j=1n‍gj(0))+μ)×σ~sn(dμ,dλ)ds≤n-1η-2δg-2,n×sup(μ,λ)∈S{h2(μ,λ)(2λf(g-1,n+n-1∑j=1n‍gj(0))+μ)}.
The assumption (A1) implies that limsupi→∞(gi(1)-gi(0))<∞, and hence supng-2,n<∞. Therefore, as in the derivation of (15), we can choose some n0 and δ0 such that (16) is satisfied.
Now, the only thing remaining to verify is the condition (B). Sinceh∈Cb(S) and the assumption (A3) holds, there exists a constant C>0 such that
(26)supt≤r|ρtn(h)|≤Cg-1,n
for all n∈ℕ and r>0. From Markov's inequality and the assumption (A1), we have
(27)limsupn→∞P(supt≤rρtn(h)>a)≤limsupn→∞a-1E(supt≤r|ρtn(h)|)≤a-1Csupng-1,n⟶0,
as a→∞. The proof of Lemma 3 is complete.Lemma 4.
For anyh∈Cb(S), Mn(h)→d0 in D(ℝ+,ℝ) as n→∞.Proof.
By the corollary in [18, page 28], it is sufficient to show that d∞0(Mn(h),0)→p0 as n→∞, where →p means convergence in probability. Here, by definition, the space D(ℝ+,ℝ) is the so-called Skorohod space D[0,∞), and d∞0 is the metric on it, which defines the Skorohod topology (see, e.g., [18, page 168]). Theorem  16.7 in [18, page 174] further implies that it is sufficient to show that dt0(Mn(h),0)→p0 for each t≥0. The metric dt0 generates the topology of Skorohod space D[0,t] (see [18, pages 166 and 125] for definitions).
Sincedt0(Mn(h),0)≤sups≤t|Msn(h)|, by Doob's martingale inequality, we have for any ε>0,
(28)P(dt0(Mn(h),0)≥ε)≤P(sups∈[0,t]|Msn(h)|≥ε)≤ε-2E((Mtn(h))2)=ε-2E[Mn(h)]t=ε-2n-1E(∫0t‍G~nρsn(h)ds),
where the first equality uses the fact that (Mtn(h))2-[Mn(h)]t is a martingale with expectation 0, and the second equality follows from (24). It then follows from the derivation in (25) that
(29)limn→∞P(dt0(Mn(h),0)≥ε)=0,
which concludes the proof of Lemma 4.From Lemma3, the sequence ρn is tight, and then, there exists a weakly convergent subsequence ρnk in D(ℝ+,Ω). We still denote it by ρn in the next lemma for convenience.Lemma 5.
Ifρn→dρ as n→∞, then for any h∈Cb(S), Mn(h)→dM(h) in D(ℝ+,ℝ) as n→∞, where
(30)Mt(h)=ρt(h)-ρ0(h)-∫0t‍Gρs(h)ds.Proof.
For eachn∈ℕ, define a random process Nn(h) by
(31)Ntn(h)=ρtn(h)-ρ0n(h)-∫0t‍Gρsn(h)ds.
From Theorem  3.1 in [18, page 28] and the arguments in the proof of Lemma 4, it is sufficient to show the following two conditions:  (A)
sups≤t|Msn(h)-Nsn(h)|→p0 as n→∞, for each t≥0,(B)
Nn(h)→dM(h) as n→∞.
To show (A), note that(32)sups≤t|Msn(h)-Nsn(h)|=∫0t|∫S‍λh(μ,λ)f(∫S‍ρsn(dμ′,dλ′)+n-1∑j=1n‍gj(0))‍×(σsn(dμ,dλ)-σs(dμ,dλ))∑j=1n‍|ds.
From the argument following (17), there is some constant C>0 satisfying
(33)f(∫S‍ρsn(dμ′,dλ′)+n-1∑j=1n‍gj(0))≤C.
Hence, by the dominated convergence theorem,
(34)sups≤t|Msn(h)-Nsn(h)|≤C∫0t‍|∫S‍λh(μ,λ)(σsn(dμ,dλ)-σs(dμ,dλ))|ds→pi0,
as n→∞, since λh∈Cb(S) and the assumption (A2) holds.
To show (B), it is sufficient to show that [18, Theorem  2.1, page 16]
(35)E(l(Nn(h)))⟶E(l(M(h))),
as n→∞, for any bounded, uniformly continuous function l:D(ℝ+,ℝ)→ℝ. By the definition of Nn(h) and the assumption ρn→dρ, we only need to show that l(Nn(h)) is a continuous function of ρn mapping D(ℝ+,Ω) to ℝ, invoking the dominated convergence theorem. Furthermore, it is sufficient to show that Nn(h) is a continuous function of ρn mapping D(ℝ+,Ω) to D(ℝ+,ℝ). In the following, we take one term in Nn(h) as an example to show the continuity with respect to ρn. Other terms can be shown analogously.
Define(36)ltn:=∫S‍λh(μ,λ)f(∫S‍ρtn(dμ′,dλ′)+g-)ρtn(dμ,dλ),lt:=∫S‍λh(μ,λ)f(∫S‍ρt(dμ′,dλ′)+g-)ρt(dμ,dλ).
Suppose that ρn→ρ in D(ℝ+,Ω) holds. We need to show ∫0t‍lsnds→∫0t‍lsds in D(ℝ+,ℝ) as n→∞. Indeed, since λh∈Cb(S) and ∫S‍h(μ,λ)ρt(dμ,dλ):D(R+,Ω)→D(ℝ+,ℝ) is continuous for any h∈Cb(S), we have ∫S‍λh(μ,λ)ρtn(dμ,dλ)→∫S‍λh(μ,λ)ρt(dμ,dλ) in D(ℝ+,ℝ) and ∫S‍ρtn(dμ,dλ)→∫S‍ρt(dμ,dλ) in D(ℝ+,ℝ), as n→∞. Hence, from the assumptions (A3) and (A4), we obtain ltn→lt in D(ℝ+,ℝ). It follows from [18, Equation (12.14), page 124] that ltn→lt for all but countably many t. The dominated convergence theorem then yields ∫0t‍lsnds→∫0t‍lsds in supremum norm on finite time intervals and hence in D(R+,R).Lemma 6.
The solution to (5) is unique in D(ℝ+,Ω).Proof.
Suppose thatρ and ρ~ are two solutions to (5), and they are the limits of two weakly convergent subsequences of ρn, respectively. By the assumption (A2), ρ0=ρ~0. We need to show that ρt=ρ~t for all t∈ℝ+.
From (3), we have
(37)∫S‍h(μ,λ)ρtn(dμ,dλ)≤∫S‍h(μ,λ)σtn(dμ,dλ),
for any h∈Cb(S). Therefore, by the assumption (A2),
(38)∫S‍h(μ,λ)ρt(dμ,dλ)≤∫S‍h(μ,λ)σt(dμ,dλ),(39)∫S‍h(μ,λ)ρ~t(dμ,dλ)≤∫S‍h(μ,λ)σt(dμ,dλ)
hold. Define d(ρt,ρ~t)=sup|h|≤1,h∈Cb(S)|ρt(h)-ρ~t(h)|. In view of (5), we obtain
(40)|ρt(h)-ρ~t(h)|=∫0t‍Gρs(h)-Gρ~s(h)ds≤∫0t‍∫S‍λh(μ,λ)|f(∫S‍ρs(dμ′,dλ′)+g-)-f(∫S‍ρ~s(dμ′,dλ′)+g-)|×σs(dμ,dλ)ds+∫0t‍∫S‍λh(μ,λ)|f(∫S‍ρs(dμ′,dλ′)+g-)-f(∫S‍ρ~s(dμ′,dλ′)+g-)|×ρs(dμ,dλ)ds+∫0t‍|∫S‍λh(μ,λ)ρs(dμ,dλ)-∫S‍λh(μ,λ)ρ~s(dμ,dλ)|×f(∫S‍ρ~s(dμ′,dλ′)+g-)ds+∫0t|∫S‍μh(μ,λ)ρs(dμ,dλ)‍-∫S‍μh(μ,λ)ρ~s(dμ,dλ)|ds.
The assumption (A4) indicates that
(41)|f(∫S‍ρs(dμ′,dλ′)+g-)-f(∫S‍ρ~s(dμ′,dλ′)+g-)|≤Cf|∫S‍ρs(dμ′,dλ′)-∫S‍ρ~s(dμ′,dλ′)|≤Cfd(ρs,ρ~s),
for some constant Cf>0. Therefore, using (38) we can derive
(42)|ρt(h)-ρ~t(h)|≤2Cf∫0t‍(∫S‍λh(μ,λ)σs(dμ,dλ))d(ρs,ρ~s)ds+C(sup(μ,λ)∈Sλ)∫0t‍d(ρs,ρ~s)ds+(sup(μ,λ)∈Sμ)∫0t‍d(ρs,ρ~s)ds,
where C>0 is some constant. To see this, note that by the assumptions (A1), (A2), and (38), we have
(43)∫S‍ρ~s(dμ′,dλ′)+g-≤∫S‍σs(dμ′,dλ′)+g-<∞,
for any 0≤s≤t. The assumption (A4) then implies
(44)sups≤tf(∫S‍ρ~s(dμ′,dλ′)+g-)≤C<∞.
Again employing the assumption (A2), we have∫S‍λh(μ,λ)σs(dμ,dλ)<∞ for all 0≤s≤t. Hence, by using the assumption (A3), the inequality (42) becomes
(45)d(ρs,ρ~s)≤C′∫0t‍d(ρs,ρ~s)ds
for some constant C′>0. A simple application of Gronwall's lemma yields d(ρt,ρ~t)=0 for all t∈ℝ+, which concludes the proof.Proof of Theorem1.
By Lemma4 and Lemma 5, the limit of any weakly convergent subsequence of ρn must satisfy (5). By Lemma 6, we find that the sequence ρn must itself converge weakly to that unique solution (see the corollary in [18, page 59]). The proof of Theorem 1 is thus completed.Proof of Corollary2.
For any open setF⊆S, we obtain from (38) that
(46)∫S‍fn(μ,λ)ρt(dμ,dλ)≤∫S‍fn(μ,λ)σt(dμ,dλ),
where fn is taken as a continuous function upwardly converging to the indicator function of F. By using the dominated convergence theorem, we know that ρt(F)≤σt(F). A regularity property (see, e.g., [17, page 18, Lemma  1.34]) implies that ρt(A)≤σt(A) for all A∈S. This means that ρt is absolutely continuous with respect to σt for any t≥0. An application of the Radon-Nykodym theorem yields the existence of φt such that (8) holds with 0≤φt≤1. Now the result follows straightforwardly from Theorem 1.



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