# Branching Processes in a Lévy Random Environment

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## Abstract

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1–13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.

### Keywords

Continuous state branching processes in random environment Stochastic differential equations Strong solution Immigration Competition### Mathematics Subject Classification

60G17 60G51 60J80## 1 Introduction

In many biological systems, when the population size is large enough, many birth and death events occur. Therefore, the dynamics of the population become difficult to describe. Under this scenario, continuous state models are good approximations of these systems and sometimes they can be simpler and computationally more tractable. Moreover, the qualitative behaviour of the approximate models may be easier to understand.

Recently there has been a lot of interest in extending this model, in the sense that one would like to include immigration, competition or/and random environment. This interest comes from the fact that these new models arise as limits of discrete population models where there are interactions between individuals or where the offspring distribution depends on the environment (see for instance Lambert [26], Kawasu and Watanabe [22], Bansaye and Simatos [4] and the references therein).

Branching processes in random environment (BPREs) were first introduced and studied in Smith and Wilkinson [37] and have attracted considerable interest in the last decade (e.g. [1, 10, 27] and the references therein). BPREs are interesting since they are more realistic models compared with classical branching processes and, from the mathematical point of view, they have new properties such as another phase transition in the subcritical regime. Scaling limits in the finite variance case were conjectured by Keiding [24] who introduced Feller diffusions in random environment. This conjecture was proved by Kurtz [25] and more recently by Bansaye and Simatos [4] in more general cases.

In this paper, one of our aims is to construct a continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a SDE. In order to do so, we study a particular class of non-negative SDEs driven by a white noise and Poisson random measures which are mutually independent. The existence and uniqueness of strong solutions are established under some general conditions that allows us to consider the case when the strong solution explodes at a finite time. This result is of particular interest on its own.

The remainder of the paper is structured as follows. In Sect. 2, we study strong solutions of SDEs which are driven by a white noise and Poisson random measures which are mutually independent. Section 3 is devoted to the construction of CBI-processes with competition in a Lévy random environment which is an extension of the models introduced in Bansaye et al. [3] and Palau and Pardo [32]. In particular, we study the long term behaviour of CB-processes in a Lévy random environment. We finish our exposition by studying a population model with competition in a Lévy random environment which can be considered as an extension of the model of Evans et al. [17]. In particular, we study its long time behaviour and the Laplace transform of its first passage time below a level under the assumption that the environment has no negative jumps.

## 2 Stochastic Differential Equations

Stochastic differential equations with jumps have been playing an ever more important role in various domains of applied probability theory such as financial mathematics or mathematical biology. Under Lipschitz conditions, the existence and uniqueness of strong solutions of SDEs with jumps can be established by arguments based on Gronwall’s inequality and results on continuous-type equation, see for instance the monograph of Ikeda and Watanabe [21]. In view of the results of Fu and Li [18] and Dawson and Li [14] weaker conditions would be sufficient for the existence and uniqueness of strong solutions for one-dimensional equations.

Motivated by describing CBI processes via SDEs, Fu and Li [18] studied general stochastic equations that describes non-negative processes with jumps under general conditions. More precisely, the authors in [18] provided criteria for the existence and uniqueness of strong solutions of SDEs which are driven by a Brownian motion and Poisson random measures. Dawson and Li [14] studied similar SDEs but instead of considering a Brownian motion in the diffusion term, they replaced it by a white noise. The main idea of their criteria is to assume monotonicity conditions on the kernel which is associated with the compensated Poisson integral term so that the continuity condition can be weakened. In this setting, Li and Pu [29], weakened the monotonicity assumption but only in the case when the white noise is replaced by a Brownian motion. Nonetheless, their criteria do not include the case where the branching mechanism of a CBI process has infinite mean. This excludes some interesting models that can be of particular interest for applications.

Our goal in this section is to describe a general one-dimensional SDE driven by a white noise and Poisson random measures that may relax the moment condition of the previous works. The main idea is first to combine the works of Dawson and Li [14] and Li and Pu [29] and later on weaken the moment condition. This may imply that the process has the possibility of explosion at finite time. This will allow us to define branching processes in more general random environment that those considered by Bansaye et al. [3] and Palau and Pardo [32].

*admissible*if

- (i)
\(b: \mathbb{R}_{+}\rightarrow \mathbb{R}\) is a continuous function such that \(b(0)\ge 0\),

- (ii)
\(\sigma : \mathbb{R}_{+}\times E\rightarrow \mathbb{R}_{+}\) is a Borel function, continuous in the first entry such that \(\sigma (0,x)=0\), for all \(x\in E\),

- (iii)
\(g:\mathbb{R}_{+}\times U\rightarrow \mathbb{R}\) is a Borel functions such that \(g(x,u)+x\geq 0\) for \(x\geq 0\) and \(u\in U\),

- (iv)
\(h:\mathbb{R}_{+}\times V\rightarrow \mathbb{R}\) is a Borel function such that \(h(0,v)=0\) and \(h(x,v)+x\geq 0\) for \(x>0\) and \(v\in V\).

- (a)For each \(n\), there is a positive constant \(A_{n}\) such that$$ \int_{\widetilde{U}}\bigl(|g(x,u)|\wedge 1\bigr)\mu (\mathrm{d}u)\leq A_{n}(1+x), \quad \mbox{for every } x\in [0,n]. $$
- (b)Let \(b(x)=b_{1}(x)-b_{2}(x)\), where \(b_{1}\) is a continuous function and \(b_{2}\) is a non-decreasing continuous function. For each \(n\geq 0\), there is a non-decreasing concave function \(z\mapsto r_{n}(z)\) on \(\mathbb{R}_{+}\) satisfying \(\int_{0+} r_{n}(z)^{-1} \mathrm{d}z=\infty \) andfor every \(0\leq x,y\leq n\).$$ |b_{1}(x)-b_{1}(y)|+ \int_{\widetilde{U}}|g(x,u)\wedge n-g(y,u)\wedge n|\mu (\mathrm{d}u)\leq r_{n} (|x-y|) $$
- (c)For each \(n\ge 0\) and \(v\in V\), the function \(x\mapsto x+ h(x,v)\) is non-decreasing and there is a positive constant \(B_{n}\) such that for every \(0\leq x,y\leq n\),where \(l(x,y,v)=h(x,v)-h(y,v)\).$$ \int_{E}|\sigma (x,u)-\sigma (y,u)|^{2}\pi ( \mathrm{d}u)+ \int_{V} \bigl(|l(x,y,v)|\wedge \bigl(l(x,y,v) \bigr)^{2} \bigr)\nu (\mathrm{d}v)\leq B_{n} |x-y| $$

*solution*of

*strong solution*if, in addition, it is adapted to the augmented natural filtration generated by \(W\), \(M\) and \(N\). (See for instance Definition IV.2.1 of Ikeda and Watanabe [21] for further details of solutions that admits explosions.)

### Proposition 1

*Suppose that*\((b, \sigma , g, h)\)

*are admissible parameters satisfying conditions*(a), (b)

*and*(c).

*Then*,

*the stochastic differential equation*(5)

*has a unique non*-

*negative strong solution*.

*The process*\(Z=(Z_{t}, t\geq 0)\)

*is a Markov process and its infinitesimal generator*ℒ

*satisfies*,

*for every*\(f\in C^{2}_{b}(\overline{ \mathbb{R}}_{+})\),

^{1}

### Proof

Observe that Theorem 6.1 in Li and Pu [29] still holds if we replace the Brownian motion integral term by a white noise, i.e. in the setting of Dawson and Li [14], the result is true if we consider a white noise \(W\) in \((0,\infty )\times E\) and a function \(\sigma (x,u)\) on \(\mathbb{R}_{+}\times E\) such that (c) is satisfied. That is to say, following the steps of Li and Pu [29], we need to replace \(\sigma (x)\) by \(\int_{E}\sigma (x,u)\pi (\mathrm{d}u)\) and \(|\sigma (x)-\sigma (y)|^{2}\) by \(\int_{E}|\sigma (x,u)-\sigma (y,u)|^{2} \pi (\mathrm{d}u)\).

Now, we proceed to show that there is a unique non-negative strong solution to the SDE (5). In order to do so, we first define \(\tau_{m}=\inf \{t\geq 0: Z^{(m)}_{t} \geq m\}\), for \(m\geq 0\), and then we prove that the sequence \((\tau_{m}, m\ge 0)\) is non-decreasing and that \(Z_{t}^{(m)}=Z_{t}^{(n)}\) for \(m\leq n\) and \(t< \tau_{m}\).

It is important to note that the SDEs (5) can be driven by a finite number of Poisson random measures which are mutually independent. By rewriting the measures, we could apply the previous theorem.

## 3 CBI-Processes with Competition in a Lévy Random Environment

In this section, we construct a branching model in continuous time and space that is affected by a random environment as the unique strong solution of a SDE that satisfies the conditions of Theorem 1. In this model, the random environment is driven by a general Lévy process.

Our model is a natural extension of the CB-processes in random environment studied by Bansaye et al. [3], in the case where the Lévy process has paths of bounded variation, and by Palau and Pardo [32], in the case where the random environment is driven by a Brownian motion with drift.

*branching, immigration*and

*environmental*parts. Recall that the associated branching mechanism \(\psi \), satisfies

*competition mechanism*is given by a continuous non-decreasing function \(\beta \) on \([0,\infty )\) with \(\beta (0)=0\). For the

*branching part*, we introduce \(B^{(b)}=(B^{(b)}_{t},t \geq 0)\) a standard Brownian motion and \(N^{(b)}(\mathrm{d}s, \mathrm{d}z,\mathrm{d}u)\) a Poisson random measure with intensity \(\mathrm{d}s\mu (\mathrm{d}z)\mathrm{d}u\). We denote by \(\widetilde{N}^{(b)}\) the compensated measure of \(N^{(b)}\). The

*immigration*term is given by a Poisson random measure \(M^{(im)}(\mathrm{d}s,\mathrm{d}z)\) with intensity \(\mathrm{d}s\nu ( \mathrm{d}z)\). Finally, for the

*environmental*term, we introduce \(B^{(e)}=(B^{(e)}_{t},t\geq 0)\) a standard Brownian motion and \(N^{(e)}(\mathrm{d}s, \mathrm{d}z)\) a Poisson random measure in \(\mathbb{R}_{+}\times \mathbb{R}\) with intensity \(\mathrm{d}s\pi ( \mathrm{d}z)\), where \(\pi \) is a measure concentrated on \(\mathbb{R} \setminus \{0\}\) such that

*A CB-process in a Lévy random environment with immigration and competition*is defined as the solution of the SDE

### Theorem 1

*The stochastic differential equation*(8)

*has a unique non*-

*negative strong solution*.

*The process*\(Z=(Z_{t}, t\geq 0)\)

*has the Markov property and its infinitesimal generator*\(\mathcal{A}\)

*satisfies*,

*for every*\(f\in C^{2}_{b}(\overline{\mathbb{R}}_{+})\),

*and*\(x\in \mathbb{R}_{+}\)

### Proof

While this paper was under review, we were aware that CBILRE-processes under a first moment condition were introduced by He et al. [20]. The authors provide a direct construction of their model and gave sufficient and necessary conditions for the process to be absorbed at 0.

It is important to note that when \(\phi = 0\), conditioned on the environment K, the process Z satisfies the branching property. This is inherited from the branching property of the original CB-process and the fact that the additional jumps are multiplicative. The proof of this fact is the same as in the Brownian environment case (see Theorem 1 in [32]).

### Proposition 2

*Suppose that*\((\mathbf{H})\)

*holds*.

*Then for every*\(z,\lambda ,t>0\),

*a*.

*s*.

*where for every*\(t,\lambda \geq 0\),

*the function*\((v_{t}(s,\lambda , K), s\leq t)\)

*is the a*.

*s*.

*unique solution of the backward differential equation*

*and*

### Proof

The first part of the proof follows similar arguments as those used in Bansaye et al. [3]. The main problem in proving our result is finding the a.s. unique solution of the backward differential equation (12) in the general case. In order to do so, we need an approximation technique based on the Lévy-Itô decomposition of the Lévy process \(K\). The proof of the latter can be found in Lemma 2 in the Appendix.

The asymptotic behaviour of these probabilities has been computed recently and independently by Li and Xu [28] and by Palau et al. [33]. In particular, and similarly to the results obtained by Bansaye et al. [3] and Palau and Pardo [32], the authors in [28] and [33] obtained five different regimes for the probability of survival when \(\alpha \in (1,2]\) and the authors in [33] obtained three different regimes for the probability of non-explosion when \(\alpha \in (0,1)\). All the regimens depend on the characteristic of the Lévy process \(K^{(0)}\).

### 3.1 Long Term Behaviour of CB-Processes in a Lévy Random Environment

In the sequel, we exclude from our model the competition mechanism \(\beta \) and the immigration term \(M^{(im)}\). In this section, we are interested in determining the long term behaviour of CB-processes in a Lévy random environment (CBLRE for short). Our methodology follows similar arguments as those used in Proposition 2 in [32] and Corollary 2 in [3].

### Proposition 3

*Assume that*\((\mathbf{H})\)

*holds*.

*Let*\((Z_{t}, t\ge 0)\)

*be a CBLRE with branching mechanism given by*\(\psi \),

*and*\(z>0\)

- (i)
*If the process*\(K\)*drifts to*\(-\infty \),*then*\(\mathbb{P} _{z} (\lim_{t\rightarrow \infty } Z_{t}=0 |K )=1\),*a*.*s*. - (ii)
*If the process*\(K\)*oscillates*,*then*\(\mathbb{P}_{z} ( \liminf_{t \rightarrow \infty } Z_{t}=0 | K ) =1\),*a*.*s*.*Moreover if*\(\gamma >0\)*then*$$ \mathbb{P}_{z} \Bigl(\lim_{t\rightarrow \infty }Z_{t}=0 |K \Bigr)=1,\quad \textit{a.s.} $$ - (iii)
*If the process*\(K\)*drifts to*\(+\infty \),*so that*\(A(x)>0\)*for all*\(x\)*larger enough*.*Then if*$$ \int_{(a,\infty )}\frac{x}{A(x)} \big|\mathrm{d}\varPhi \bigl( e^{-x}\bigr)\big|< \infty , \quad \textit{for some }a>0, $$(15)*we have*\(\mathbb{P}_{z} (\liminf_{t\rightarrow \infty }Z _{t}>0 |K )>0 \)*a*.*s*.,*and there exists a non*-*negative finite r*.*v*. \(W\)*such that*$$ Z_{t}e^{-K_{t}}\mathop{\longrightarrow }_{t\rightarrow \infty }W,\quad \textit{a.s} \quad \textit{and}\quad \{W=0 \}= \Bigl\{ \lim_{t\rightarrow \infty }Z_{t}=0 \Bigr\} . $$*In particular*,*if*\(0< \mathbb{E}[K_{1}]<\infty \),*then the above integral condition is equivalent to*$$ \int^{\infty }x\ln x \mu (\mathrm{d} x)< \infty . $$

### Proof

Parts (i) and (ii) follow from the same arguments used in the proof of Proposition 2 in [32], so we skip their proofs. The key part is to note that via Ito calculus, \(Z_{t}e^{-K_{t}}\) is a non-negative local martingale. Therefore \(Z_{t}e^{-K_{t}}\) is a non-negative supermartingale that converges a.s. to a non-negative finite random variable, here denoted by \(W\).

Now, we derive a central limit theorem in the supercritical regime which follows from Theorem 3.5 in Doney and Maller [15] and similar arguments as those used in Corollary 3 in [3], so we skip its proof.

### Corollary 1

*Assume that*\(K\)

*drifts to*\(+\infty \), \(T(x)>0\)

*for all*\(x>0\),

*and*(15)

*is satisfied*.

*There are two measurable functions*\(a(t), b(t)> 0\)

*such that*,

*conditioned on*\(\{ W>0 \}\),

*if and only if*

*where*\(\xrightarrow{d}\)

*means convergence in distribution and*\({\mathcal{N}}(0,1)\)

*denotes a centred Gaussian random variable with variance equals*1.

### 3.2 Population Model with Competition in a Lévy Random Environment

From Theorem 1, there is a unique non negative strong solution of (16) satisfying the Markov property. Moreover, we have the following result that in particular says that the process \(Z\) is the inverse of a generalised Ornstein-Uhlenbeck process.

### Proposition 4

*Suposse that*\((Z_{t},t\geq 0)\)

*is the unique strong solution of*(16).

*Then*,

*it satisfies*

*where*\(K\)

*is the Lévy process defined in*(10).

*Moreover*,

*if*\(Z_{0}=z>0\)

*then*, \(Z_{t}>0\)

*for all*\(t\geq 0\)

*a*.

*s*.

*and it has the following asymptotic behaviour*:

- (i)
*If the process*\(K\)*drifts to*\(-\infty \),*then*\(\lim_{t\rightarrow \infty }Z_{t}=0\)*a*.*s*. - (ii)
*If the process*\(K\)*oscillates*,*then*\(\liminf_{t\rightarrow \infty }Z_{t}=0\)*a*.*s*. - (iii)
*If the process*\(K\)*drifts to*\(\infty \),*then*\((Z_{t}, t \ge 0)\)*has a stationary distribution whose density satisfies for*\(z>0\),$$ \mathbb{P}_{z}(Z_{\infty }\in \mathrm{d}x)=h \biggl( \frac{1}{kx} \biggr) \frac{ \mathrm{d}x}{x^{2}}, \quad x> 0, $$*where*$$ \int_{t}^{\infty }h(x)\mathrm{d}x= \int_{\mathbb{R}} h\bigl(te^{-y}\bigr)U( \mathrm{d}y),\quad \textit{a.e. }t\textit{ on }(0,\infty ), $$*and*\(U\)*denotes the potential measure associated to*\(K\),*i*.*e*.$$ U(\mathrm{d}x)= \int_{0}^{\infty }\mathbb{P}(K_{s}\in \mathrm{d}x) \mathrm{d}s,\quad x\in \mathbb{R}. $$*Moreover*,*if*\(0<\mathbb{E} [ K_{1} ] <\infty \),*then*$$ \lim_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t} Z_{s}\mathrm{d}s= \frac{1}{k}\mathbb{E} [ K_{1} ] , \quad \textit{a.s.} $$*and for every measure function*\(f:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\)*we have*$$ \lim_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t} f(Z_{s})\mathrm{d}s= \mathbb{E} \biggl[ f \biggl( \frac{1}{kI_{\infty }(-K)} \biggr) \biggr] , \quad \textit{a.s.} $$*where*\(I_{\infty }(-K)=\int_{0}^{\infty }e^{-K_{s}}\mathrm{d}s\).

### Proof

By Itô’s formula, we see that the process \(Z\) satisfies (17). Moreover, since the Lévy process \(K\) has infinite lifetime, then we necessarily have \(Z_{t}>0\) a.s.

The asymptotic behaviour of the positive moments of \(Z_{t}\) has been studied in Palau et al. [33] using a fine study of the negative moments of exponential functional of Lévy processes. In particular four different regimes appears that depends on the characteristic of the Lévy process \(K\).

### Lemma 1

*Suppose that*\(\lambda \geq 0\)

*and that*\(\kappa (\lambda )>1\),

*then for all*\(0< b\leq z\),

*where*

### Proof

## Footnotes

- 1.
\(\mathbb{R}_{+}=[0,\infty )\), \(\overline{\mathbb{R}}_{+}=[0,\infty ]\) and \(C^{2}_{b}(\overline{\mathbb{R}}_{+})=\{\textit{twice differentiable functions such that }f( \infty )=0\}\).

## Notes

### Acknowledgements

Both authors acknowledge support from the Royal Society and SP also acknowledge support from CONACyT-MEXICO Grant 351643.

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