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Scaling limit for stochastic control problems in population dynamics

Going from a scaling approach for birth/death processes, we investigate the convergence of solutions to BSDEs driven a sequence of converging martingales. We apply our results to non-Markovian stochastic control problems for discrete population models. In particular we describe how the values and optimal controls of control problems converge when the models converge towards a continuous population model.


Introduction
The sustainability of natural resources has become a major subject of interest in the last decades for public institutions. For instance, in 1983 the European Union has launched its common fisheries policy to manage European fish stocks. In August 2010, a report of European commission named Water Scarcity and Drought in the European Union, has emphasized that "an adequate supply of good-quality water is a pre-requisite for economic and social progress, so we need to do two things: we must learn to save water, and also to manage our available resources more efficiently". A large part of academic literature has dealt with such issues. For example, Reed in [Ree79] or Clarke and Kirkwood in [CK86] have studied the exploitation of a natural resource under uncertainty on its evolution in a multi-period model. May, Beddington, Horwood and Sherpherd in [MBHS78] have considered the problem by assuming that the intrinsic population growth rate is given by the difference between recruitment and mortality for general recruitment functions. These models have been extended to stochastic differential equations driven by a Brownian motion (see for instance the work of Saphores [Sap03]). Evans, Hening and Shreiber in [EHS15] or more recently Kharroubi, Lim and Ly Vath in [LKLV18] have modelled the dynamic of the natural resource as the solution of the logistic stochastic differential equation to solve a control problem under interaction between species and delayed renewal of the resource. All the models mentioned above use a Brownian motion to describe the uncertainty of the system evolution. We refer to this class of model as continuous models. On the other side of the literature, Getz in [Get75] has studied control problems related to a birth/death process. This work has been extended more recently by Claisse in [Cla18] to branching processes. We refer to those models as discrete models.
It is well known that some continuous population models can be seen as scaling limits of discrete models, see for example the work of Bansaye and Méléard in [BM15]. Hence continuous models can be considered as good approximations of the macroscopic evolution of a population size. Therefore it is relevant to consider continuous models for resources management purposes. Moreover those models are attractive from a tractability viewpoint compared to discrete models. Indeed solving control problems in Brownian driven model essentially boils down to solve a partial differential equation. Whereas for discrete models it leads to integral-partial differential equation, which are often more complex to solve. Yet the remaining question is the relevancy of designing a management policy based on a continuous modeling while the controlled population (or resource) is naturally discrete.
To try to give an answer to this question we are going to consider a sequence of discrete population models that converges towards a continuous population model. For each of those models we consider a control problem. Each of them are the natural adaptations of the same control problem to the different models. Therefore we expect the solutions of the discrete control problems to converge towards the solution of the continuous limit problem. From Γ-convergence results adapted to stochastic control problems as in for instance the articles of Buttazzo and Del Maso [BDM82] and Belloni, Buttazzo and Freddi [BBF93], we expect to have convergence of value functions (see also for instance [DM12,Theorem 10.22]) and a kind of weak convergence of optimal controls (see for instance [BBF93,Proposition 2.8]). This is emphasized in a toy model (see Section 2) where besides convergence of the value functions we also get convergence in law of the state process under the optimal control. In this paper we prove the convergence of the controls as sequence of processes. This is stronger than Γ−convergence. Since we aim at dealing with non-Markovian stochastic control problems our problematic is to prove the convergence of solutions to a sequence of Backward Stochastic Differential Equations (BSDE for short) driven by a sequence of converging martingales.
We know from the seminal paper of Donsker [Don51] that a scaling in time procedure leads to the weak convergence of a random walk to a Brownian motion. Extending this result to the theory of BSDE with fixed time horizon T > 0, Briand, Delyon and Mémin in [BDM01] have provided a time discretization of the Brownian motion to get the convergence of a time discretized BSDE. More precisely, they consider a sequence of random walks (W n ) n≥0 converging towards a Brownian motion W . Then they prove the convergence of the solutions of a sequence of BSDEs driven the (W n ) n≥0 towards the solution of a BSDE driven by W . The main idea is to prove the convergence of the terms involved in the martingale representation with respect to W n when n goes to infinity. For this they use the convergence, in the sense of Coquet, Mémin and Slominski [CMS01], of the filtrations associated to each of the (W n ) n≥0 towards the natural filtration associated to W . Those results have been extended in [BDM02] to a more general situation, without assuming that W n has a predictable representation property, but assuming that the brackets of the martingales (W n ) n≥0 are uniformly bounded.
In the present paper we aim to extend the results of [BDM01] to a wider class of martingale convergence. Starting from a scaling result in [BM15] showing that a sequence of scaled birth/death processes (X K ) K≥0 with scaling parameter K > 0 converges in law to the solution X of a stochastic Feller diffusion, we begin to extend it to more general birth and death intensities. We then consider a sequence of BSDEs of the form where ξ K is some general terminal random condition, g K the generator of the BSDE, M K a two dimensional martingale associated to the population model X K and φ K dA K denotes the measure associated to the angle bracket of M K . We also consider the continuous counterpart of (B) K , where ξ is some terminal condition, g is the generator of the BSDE, M X is a one dimensional martingale related to the diffusion term of X and A is its angle bracket. The existence and uniqueness of solutions to such BSDEs driven by general martingales have been studied, for instance, by El Karoui and Huang in [EKH97], Confortola and Fuhrman in [CF13] or more recently by Papapantoleon, Possamaï and Saplaouras in [PPS18] in a general framework. Inspired by [BDM02], we prove that the solution of (B) K converges to the solution of (B) when (ξ K ) K≥0 converges toward ξ and (g K ) K≥0 towards g. The difficulty pointed at this step, compared to [BDM02], is that the brackets of the (M K ) K≥0 are not bounded. Therefore we need a stronger assumption on the convergence of the sequence of terminal conditions. The methods used are related to the so-called martingale problem as stated by Jacod and Shiryaev in [JS13] and to the double-Picard iterations craftily used in [BDM02].
Finally we would like to point that after having published the first version of this paper online, we have learnt that a paper of Papapantoleon, Possamaï and Saplaouras dealing with convergence of BSDES for a very general class of martingales is in progress and preliminary results are written in the PhD Thesis of Alexandros Saplaouras. Our study, made independently of their, has to be seen as a particular case of their works more focused on applications in stochastic control theory for population models.
The structure of the paper is the following. In Section 2 we study the convergence of a rescaled birth/death process to the solution of a stochastic Feller type SDE by extending [BM15] to more general dynamics (see Theorem 1). We also provide fundamental properties of our state processes such as exponential moments (see Proposition 1 and Corollary 1). Section 3 introduces a toy model motivating our study and illustrated with numerical simulations. In Section 4 we first provide a convergence result for a sequence of martingale representations (see Proposition 2). Then in Theorem 2 we extend the convergence result of [BDM02] by showing that the solutions to (B) K converge to the solution of (B). In Section 5 from our BSDE approach we deduce convergence of the values and optimal controls to a sequence of control problems. Our results go beyond Γ−convergence since we obtain a strong form of convergence for the optimal controls. Section 6 gives the main proofs of our results. Minor proofs are given in the appendix.
The technical spaces considered related to discrete and continuous models are defined in Appendix A. We provide below the notations for classical spaces used in this paper.
Classical spaces.
• L p the set of real valued random variable Z such that 2 From a discrete to a continuous population model In this section we define a sequence of discrete population models. We show that this sequence converges in law towards a continuous Feller population model by extending [BM15, Theorem III-3.2] to more general population dynamic models.

Definition of the discrete population models
We consider positive continuous functions f b , f d and σ defined from R into R + that satisfy the following standing assumption.
(i) The functions f b , f d and σ are null on R − and there exists non negative constants ν, µ, η and η such that for any We note Ω d the set of piecewise continuous increasing positive functions with jumps equal to 1. We consider F the natural filtration associated to the canonical process For a fixed K ≥ 0 and n ≥ 0 we define a population model on the stochastic basis (Ω 2 d , F). The initial population is Kn and the processes N b and N d represent respectively the number of birth and death in the population. This means that when the process N b jumps there is a new individual in the population and when N d jumps there is one individual less in the population. Therefore at time t the population size is Kn As we are interested in the large population limit (which corresponds to K large) we consider the rescaled population process We define the birth intensity in the model with parameter K and initial population n as and the intensity of death Remark 1. Note that f b (x) = µx, f d (x) = νx and σ 2 (x) = σ 2 x satisfy Assumption 1. Consequently, the model studied in Theorem III-3.2 in [BM15] is included in the scope of this paper.
Following Theorem 3.6 in [Jac75] there exists a unique probability measure P K,n on (Ω 2 d , F) such that the processes are local martingales. It means that under the probability P K,n the process N b (resp. N d ) has intensity λ K,n,b (resp. λ K,n,d ). Note that if m > n ≥ 0 then P K,n is absolutely continuous with respect to P K,m and we have: dP K,n dP K,m = L n,m We justify this change of measure in Appendix B.
For the rest of this work we fix an initial population x 0 and do not write anymore the superscript x 0 to lighten the notations. We write E instead of E P K when there is no ambiguity on the probability used. For any K we consider the processes The rescaled population process is now noted 2.2 Scaling limit of the sequence (X K ) K≥0 Intuitively, and having in mind [BM15, Theorem 3.2], a continuous version of the processes (X K ) K≥0 , denoted by X, would be an Ito diffusion with drift equal to f (X) and volatility given by σ 2 (X). We formalize this intuition in the following result which extends [BM15, Theorem 3.2]. The proof is given in Section 6.1.
the process X is the unique strong solution of (S) : Moreover, there exists a probability space (Ω, F, P) and a sequence (N K,d , N K,b ) K≥0 such that for any K ≥ 0, (N K,d , N K,b ) has the law of (N d , N b ) under P K . Moreover on this space the sequence Before going to the next section, we define some processes that we will extensively use in the rest of the paper. We note for i ∈ {b, d} We also consider the processes Note that under the probability P K the random measure m associated to the process (N b , N d ) interpreted as a compound jump process with values in E = {b, d} admits as predictable compensator measure )K 2 and where δ i denotes the Dirac measure at point i ∈ {b, d}. This point of view is introduced in order to draw a parallel with the framework of [CF13] to which we will refer extensively in Section 4.2.

Uniform exponential moments
Finally we show that the sequence of processes (X K , · 0 X K s ) admits exponential moments uniformly in K if σ 2 is linear. The proof of this result is postponed in Appendix F. Proposition 1. If there exists a positive constant σ such that σ 2 (x) = σ 2 x there exists some positive constants β 0 , K 0 and T such that for any s ≤ T we have Without loss of of generality we assume that K 0 = 0. From now we fix a positive constant β strictly smaller than β 0 . As a consequence of Proposition 1, for any integer q we have for any We deduce from Fatou's Lemma together with Proposition 1 that X inherits from the exponential moments of X K as stated in the following corollary.
Corollary 1. If there exists σ positive such that σ 2 (x) = σ 2 x there exists some positive constants β 0 and T such that for any s ≤ T we have In order to benefit from those exponential moments we now assume that σ 2 (x) = σ 2 x for some positive constant σ fixed.

Illustration of the study on a toy model
In this section, we illustrate the Γ−convergence result applied to optimization problems in population dynamics. We consider specific parameters f d , f b and a sequence of control problems for which we are able to make explicit computations. Then, we show that the sequence of optimal controls converges in law to the optimal control of a continuous problem. In this section, we aim at providing the general main ideas of the paper rather than being perfectly accurate. Rigorous statements will be given in Section 5.

Discrete populations models
We consider a discrete birth/death model as studied in [BM15] by choosing: -the initial population x 0 ∈ N, -the birth rate f b (x) = νx for some ν > 0, -the death rate f d (x) = µx for some µ > 0.
Recall from Remark 1 and from [BM15, Theorem 3.2] that (X K t ) t∈[0,T ] converges in law for the Skorokhod topology towards the continuous diffusion process (X t ) t∈[0,T ] solution of the Feller stochastic differential equation for W a Brownian motion.
In this toy model, we assume that a resource manager regulates the population X K through an • there exists a unique law P K,α under which the death intensity of the population is and the birth intensity is λ K,b . When this probability exists, it is the law of the population under the control α.
• λ K,d,α is a non negative process P K,α almost surely.
We denote by A K the set of admissible controls.
The agent is assumed to be penalized if he fails at reaching a fixed level x > 0 of the resource at time T determined by a regulator. We model this penalization by the square of the difference between the effective population size at time T and the targetx. So that the manager pays γ(X K T − x) 2 at time T where γ is a positive constant. The manager payoff is also assumed to be penalized by the instantaneous amount |αtX K t | 2 2 per unit of time when its effort is α. The problem of the resource manager is thus to solve where E K,α denotes the expectation taken under the probability P K,α . We assume that σ 2 > 2γx and γ < µ.
To solve this problem, as usual in stochastic control theory, we study the corresponding Hamilton-Jacobi-Bellman (HJB for short) equation and use a verification argument. The HJB equation associated to the control problem (TM) K is with Hamiltonian H K given by and where The maximizer of the Hamiltonian is α K, * = Kp− x 1 x>0 , hence Note that we do not actually care about the value of the control when x = 0 since if the population reaches 0, it is stuck at this value. The partial differential equation (PDE for short) (HJB) K is quadratic, so we search for a solution under the form Identifying the monomials, we get that U K is solution of (HJB) K if and only if (a K , b K , c K ) is solution of the following systems of ODEs: By Cauchy-Lipschitz theorem this system admits a unique solution. Thus the optimal effort of the agent is and the corresponding death intensity is given by Note that in view of (ODE) K and since a K (T ) is negative and b K (T ) positive, there exists a T small enough, independent of K, such that for any K the control α K, * is in A K . We refer to Appendix C for more details on this point. We assume that we are considering such short enough time horizon here.

Continuous populations model
We now turn to the continuous version of the control problem. We assume that the manager controls the drift term in (2) through an F X −predictable process α. We say that α is an admissible control when the following SDE admits a unique weak solution When such solution exists we note P α its law that is the law of the population under the control α. We denote by A the set of admissible controls.
The control problem in the continuous framework is written The associated HJB equation is given by where the Hamiltonian H is The maximizer of the Hamiltonian is As previously, we are looking for a quadratic solution of the form Identifying the monomials, we get that U is solution of (HJB) if and only if (a, b, c) is solution of the following system of ODEs.
Hence, the optimal control is given by Note that in view of (ODE) and since a(T ) is negative and b(T ) positive, there exists a T small enough such that the control α * is in A, see Appendix C for details. We assume that we are considering such time horizon here. We also note that a K = a and as consequence of Grönwall Lemma (b K , c K ) K≥0 converges to (b, c) when K goes to +∞. Consequently we get the convergence of the value of the control problems, lim Moreover a direct adaption of the proof of Proposition 1 gives the convergence in law of the optimally controlled population: Those convergences are illustrated in Figures 1 and 2 respectively.

Convergence of BSDEs
In this section we prove the main results of this paper. We first prove 1 a result on the convergence of a sequence of martingale representations. Then we extend it to the convergence of a sequence of BSDEs driven by the sequence of martingales (M K ) K≥0 .

Convergence of martingale representations
From Theorem 2 in [Dav76] we know any F K −martingale has the representation property with respect to M K (in the sense of Definition III-4.22 in [JS13]). Moreover we prove in Appendix E that any F X − martingale has the representation property relative to M X .
For any K ≥ 0 we consider ξ K ∈ L 2 an F K T -measurable real random variable and ξ ∈ L 2 an F X T -measurable real random variable. We define the closed martingale Q K by , P−a.s.. Since Q K is an F K -martingale and ξ K ∈ L 2 , we know that there exists a unique process We prove the following fundamental convergence result in Appendix 6.2 Proposition 2 (martingale representations convergence). If the sequence (ξ K ) K≥0 and ξ are in L 2+ε and (ξ K ) K≥0 converges towards ξ in L 2+ε for ε > 0 then Compared to Theorem 5 in [BDM02] we have assumed that the convergence of (ξ K ) K≥0 takes place in L 2+ε instead in L 2 . This is in order to extend the convergence of ( Q K , M K ) K≥0 beyond S 1 2 . Indeed if we only assume that Q K ∈ S 1 1 , Q K , M K is not squared integrable a priori. In [BDM02] the authors do not face this issue since they assume that the brackets of the martingales they consider are bounded, see Hypothesis (H1). In our framework the sequence ( M K T ) K≥0 is not bounded in general. However, if we instead consider a sequence of models with a bounded population then ( M K T ) K≥0 would be bounded and we could get the same result assuming only the convergence of (ξ K ) K≥0 in L 2 only.

Convergence of BSDEs
We now extend the previous result to convergence of a sequence of BSDEs driven by M K .
For any K ≥ 0 we consider an F K T random variable ξ K and two continuous functions Note that in the above equation, we implicitly use the decomposition z = (z b , z d ). We will always assume such convention when we are dealing with a pair of elements such that one element of the pair is related to the birth in the population and the other is related to the death.
We introduce the BSDE with generator g K and terminal value ξ K by setting As a consequence of Theorem 3.4 in [CF13] we have the following result.
Lemma 1. Assume that (ii) there exists a positive constant L such that β > L 2 + 2L and for any x, y, y , z, z and K ≥ 0 we have for j ∈ {b, d} We also introduce a class of BSDE driven by the martingale M X . For an F X T real valued random variable ξ and a continuous function g from R 3 into R we consider the BSDE We get the following result on existence and uniqueness of solution to (B) which is a consequence of Theorem 6.1 in [EKH97] or Theorem 2.1 in [CFS08].
Lemma 2. Assume that (ii) there exists a positive constant L such that β > L 2 + 2L and for any x, y, y , z, z and K ≥ 0 we have: We are interested in the convergence of the solutions to (B) K when (ξ K ) K≥K and (g K ) K≥0 converge. Therefore we make the following converging assumptions on the drivers of the BSDEs (B) K .
(i) The sequence (ξ K ) K≥0 converges towards ξ ∈ T in L 2+ε for ε > 0, (ii) there exists a positive constant C such that for any x, x , y, z, K ≥ 0 and j ∈ {b, d} there exists a pair of continuous functions (g b , g d ) from R 3 and a positive sequence (υ K ) K≥0 converging towards 0 such that for any K ≥ 0, x, y and z we have for j ∈ {b, d}: Remark 2. Under Assumption 2 (iii) if for any K the pair (g K b , g K d ) satisfies assumptions (ii) and (iii) in Lemma 1 then the function g = (g b + g d )/2 satisfies the assumptions (ii) and (iii) in Lemma 2.
For any K ≥ 0 we consider (Y K , Z K ) ∈ S K the unique solution of (B) K . We have the following convergence result for the sequence (Y K , Z K ) K≥0 whose proof is given in Section 6.3.
Theorem 2. Under Assumption 2 if the assumptions of Lemma 1 are satisfied for any K then the BSDE driven by M X with generator g := g b +g d 2 and terminal value ξ has a unique solution (Y, Z) and we have the following convergence:

Application to a control problem
In this section we apply the results of Section 4 to the convergence of a sequence of controls problems.

The discrete problem
We first focus on the discrete control problem in the same spirit than Section 3. We consider that a resource manager monitors his harvesting intensity through a control α, which is assumed to be bounded with bounds a, a > 0. We assume that his harvesting modifies the death rate of the natural resource according to a continuous function h K : R + × [−a, a] −→ R which satisfies the following assumption.
Assumption 3. There exists a positive constant C < 2β such that for any The set of admissible controls is defined by For any α ∈ A K we define where E denotes the Doleans-Dade exponential process. We deduce from Assumption 3, Proposition 1 together with [Sok13, Corollary 2.6] that (L K,α t ) t∈[0,T ] is a true martingale. Hence the law of the population process under the control α is given by P K,α characterized by Under the probability P K,α the death intensity of the population becomes α t ) and the birth intensity is unchanged.
We assume that the manager receives at maturity T a lump sum random compensation ξ K ∈ T K for his action. In addition, the manager receives continuous incomes along the time depending on the size of the population and on his control that is given by a function c K from R + × [−a, a] into R. This gain can be negative which corresponds to a cost related to the effort of the manager. This is what we have considered in Section 3. Therefore, the goal of the manager is to solve the following maximization problem where E K,α denotes the expectation taken under the probability P K,α . Using the notations of Section 4.2 the BSDE associated to this control problem is .
We need to assume that the functions c K and h K are chosen such that g K satisfies the assumptions (ii) and (iii) of Lemma 1 and that the maximizer in the above equation is unique. Formally we make the following assumption.
(i) g K,d (X K , 0) ∈ H K 1 , (ii) there exists a positive L satisfying β > L 2 + 2L and such that for any x, z, z , α and K ≥ 0 (iii) For any x, z there exists a unique α K, * (x, z) such that We thus have the following characterization of the optimal control (we refer to Appendix 6.4 for the proof).
Theorem 3 (Verification for (P) K ). Let (Y K , Z K ) ∈ S K be the unique solution of (BSDE) K .
solves the problem (P) K .
We now define the continuous version of this control problem.

The continuous problem
As previously the resource manager monitors his harvesting intensity through a control α, assumed to be bounded with bounds a, a > 0. We assume that his harvesting modified the death rate of the natural resource according to a continuous function h : R + × [−a, a] −→ R which is assumed to satisfy the following assumption.
Assumption 5. There exists a positive constant C < 2β such that for any ( with equality if x = 0.
The set of admissible control is Considering the process where we recall that X is given by (S). We deduce from Assumption 5 and Corollary 2.6 in [Sok13] that L α is a true martingale. Hence, we define the probability P α by dP α dP = L α T which is the probability measure corresponding to the control α. Under P α the process X is a strong solution of Xs ds is a P α −Brownian motion.
As in the discrete case we assume that the manager receives at maturity T a lump sum random compensation ξ ∈ T for his action. In addition, the manager receives continuous incomes term depending on the size of the population and his control. This term is given by a function c from R + × [−a, a] into R. Therefore, the goal of the manager is to solve the following maximization problem where E α denotes the expectation taken under the probability P α . The BSDE associated to this control problem is We need to assume that the functions c and h are chosen such that g satisfies the assumptions (ii) and (iii) of Lemma 2 and that the maximizer in the above equation is unique. Formally we make the following assumption.
(i) g(X, 0) ∈ H 1 , (ii) there exists a positive L satisfying β > L 2 + 2L such that for any x, z, z and α we have (iii) For any x, z there exists a unique α * (x, z) such that We thus have the following characterization of the optimal control (we refer to Appendix 6.5 for the proof).

Convergence of the value functions and of the optimal controls
In this section, we show that under some natural assumptions the sequences of value functions (V K 0 ) K≥0 and of controls (α K, * ) K≥0 converge respectively towards V 0 and α * . More precisely we consider the following assumptions.
(i) (ξ K ) K≥0 converges to ξ in L 2+ε for some ε > 0, (ii) there exists a positive sequence (η K ) K≥0 that converges towards 0 such that for any x, α, z and K we have (iii) There exists a positive constant C > 0 such that for any x, x , z and K we have and for any x, x , z, z , α, α and K we have Assumption 7 contains the natural assumptions ensuring that the problem (P) is the version of the problems (P) K in the framework of the continuous population model X.
Using a slight abuse we note P K, * the law of X K under the control α K, * and P * the law of X under the control α * . We have the following convergence result which proof is given in Appendix 6.6.
Since Y K 0 = V K 0 and Y 0 = V 0 a consequence of Theorem 5 (i) is that (V K 0 ) K≥0 converges towards V 0 . The point (i) also implies that the sequence of controls (α K ) K≥0 converges towards the control α. But this convergence is in a weak sense and we do not get the convergence of (α K, * ) K≥0 towards α * in law for the Skorohod topology.
Remark 3. Note that the sequence of control problems considered in Section 3 when α ∈ [−ν, a] (for a positive) satisfy any of the assumptions of Section 5.

Proof of Theorem 1
We introduce the process The proof is divided in four main steps detailed below.
1. We prove that (S) admits a unique strong solution.
2. We show that the sequence ( 3. We show that for any limit point (Y, M, N b , N d , Λ b , Λ d ) of the above sequence we have Λ b = N b = Λ d = N d and Y is almost surely differentiable with derivative X weak solution of (S).
4. Finally, we prove that up to a probability space (Ω, F, P) such that the above convergence holds in probability, the process (X to (X, M, A, A, A, A) when K goes to +∞ in S 2 1 × S 2 2 × S 1 1 × S 1 1 × S 1 1 × S 1 1 .
In order to show tightness we first show that the sequence sup Hence, according to (A) − (iii), there exists a positive constant C (independent of K) such that By using Grönwall's inequality we deduce that sup is bounded uniformly with respect to K.
We have that converges towards 0. We conclude using Markov inequality.
In view of the tightness result obtained in Step 1, we denote by (Y, M, N According to Corollary IX-1.19 in [JS13] we have that M is a local martingales. Moreover we have T ] is uniformly bounded in K, so according to Fatou's lemma Λ i is integrable, therefore M i is a true martingale.
We recall that Then, X K converges almost surely and uniformly on [0, T ] towards f (X s )ds.
Since we have we get from Theorem V-3.9 in [RY13] that there exists a bi-dimensional Brownian motion (B b , B d ) such that So finally we have shown that This concludes the proof of the first part of Theorem 1 since ( converges in law for the Skorohod topology to (X, M, A, A, A, A).
Step 4: convergence of a copy (X K , M In view of the conclusion of Step 3, and by the Skorohod representation theorem, there exists a probability space (Ω, F, P) and a copy in law of the sequence of processes ( that converges in probability toward a copy of (X, M, A, A, A, A) when K goes to +∞. To prove that the convergence actually holds in S 2 1 × S 2 2 × S 1 1 × S 1 1 × S 1 1 × S 1 1 we show that: Then we will get the convergence using dominated convergence. Proof of (i). We write Therefore we have for a positive constant C independent of K such that Hence to conclude it is enough to show that (E[N So we get Therefore by Proposition 1 and Grönwall lemma we get point (i) (since the same results follows for N K,d ).
Proof of (ii). We have therefore point (ii) follows from Proposition 1. Same proof holds for N K,d .
Proof of (iii). Using the Burkholder-Davis-Gundy inequality we get Therefore because of point (i) we get point (iii). Same proof holds for M K,d .
Proof of (iv). We write So we have taking the supremum over t ∈ [0, T ] and then the expectation we obtain point (iv) as corollary of Proposition 1 and point (iii).

Proof of Proposition 2.
The proof of Proposition 2 is inspired by that of Theorem 5 in [BDM02].
It is easy to check using (1) that for any K the processes X K is a càd-làg Feller process. For details on this point see Appendix D. Moreover, by Theorem 1.1 in [Küh18] X is a continuous Feller process. Hence from Proposition 4.B) in [CMS01] we get the weak convergence of F K towards F X . So by Remark 1.2) in [CMS01] we have the convergence in probability of (Q K ) K≥0 towards Q for the Skorohod topology. Since Q is a Brownian martingale, it is a continuous martingale and therefore (Q K ) K≥0 converges in probability towards Q for the topology of the uniform convergence.
Since (|M K | 2 ) K≥0 is uniformly integrable it converges in S 2 1 towards M . From Proposition 2 in [BDM02], we obtain that We then extend this convergence and the one of (Q K ) K≥0 using dominated convergence theorem. For this we show that for any ε < ε we have: Proof of (i): By Doob's maximal inequality we have for a positive constant C Therefore we get (i) since (ξ K ) K≥0 is bounded in L 2+ε .
Proof of (ii): By Equation (100.2) p.183 in [DM80] and using BDG inequality we have thus we get (ii).

Proof of Theorem 2.
The proof of Theorem 2 is inspired from the proof of Theorem 12 in [BDM02].
We proceed in 3 steps: (i) We show that there exists α ∈ (0, 1) and some α−contracting functions (F K ) K≥0 and F such that for any K, the unique solution of (B) K is the fixed point of F K and the fixed point of F is solution to (B).
(ii) We introduce a double indexed sequence and prove a convergence result by induction.

Step (i)
For any K we define the function where (Y , Z ) is the unique solution of the BSDE: Since (Y, Z) ∈ S K and because of assumptions (ii) and (iii) in Lemma 1 we have g K (X K , Y, Z) ∈ H K 2 . So we can properly define and Z is the unique process in H K 2 satisfying Consider two pairs (Y 1 , Z 1 ), (Y 2 , Z 2 ) ∈ S K and noting (Y Taking the expectation we get Therefore using the assumptions of Lemma 1 together with Young's inequality we get for any positive α and γ that Inspired by the proof of Theorem 3.4 in [CF13] we choose γ = α/L and α ∈ (0, 1) such that We can make such choice since β − L 2 − L > L. Therefore we obtain Therefore for any K the function F K is an α−contraction on S K for the norm equivalent to · S K and defined by In the continuous case we consider where (Y , Z ) is the unique solution of the BSDE: Since (Y, Z) ∈ S and because of Remark 2 we have g(X, Y, Z) ∈ H 2 . So we can properly define and Z is the unique process in H 2 satisfying Similarly we obtain that F is an α−contraction for the equivalent norm on S:

Step (ii)
For any K ≥ 0 we define the sequence (Y K,p , Z K,p ) p≥0 satisfying We similarly consider the sequence (Y p , Z p ) p≥0 defined by Since for any K ≥ 0, F K is a contraction. For any K ≥ 0 the sequence (Y K,p , Z K,p ) p≥0 converges towards (Y K , Z K ) in S K . In the same way (Y p , Z p ) p≥0 converges towards (Y, Z) in S.
We use the following notation: So that we can write: and We prove by induction that the following convergence holds for any p: Obviously the result holds for p = 0. We assume that the converge holds for p and show that it implies the convergence for p + 1.

Step (iii)
Note that a consequence of Step (i) is that for a certain positive constant C we have We write Notice that according to the BDG inequality there exists a positive constant C such that for any which converges towards 0 uniformly in K when p → +∞ by Equation (5). Hence (Q K ) K≥0 converges in S 2 1 towards Q.
Similarly we write We proved in the previous section that the last term goes to 0 when K → +∞. Remark that we have so using Jensen and Doob's inequality we get Therefore Y K − Y K,p K K goes to 0 when p → +∞. In the same way we get that Y − Y p K goes to 0 when p → +∞. So Y K − Y 2 converges towards 0.
Finally notice that, in S 1 2 × S 1 1 follows from Proposition 2 in [BDM02] and from the convergence of (Q K , M K ) K≥0 in S 2 1 × S 2 2 towards (Q, M ).
For any n we note Z p,n = Z p 1 |Z p |<n . We write where for i ∈ b, d, we recall that g = (g b + g d )/2, For i ∈ {b, d} the sequence (T n,p i,4 ) n≥0 obviously converges to 0 in probability by almost sure convergence as n goes to infinity.
The sequence ( · 0 K −2 φ K,i s dA K s ) K≥0 converges towards A/2 in probability for the Skorohod topology and satisfy the P-UT condition by Proposition VI-6.12 in [JS13]. So for any n, (T n,K,p i,3 ) K≥0 converges towards 0 in probability (for the Skorohod topology) as a consequence of Theorem VI-6.22 in [JS13].
For the second term we write So by assumptions of Lemma 1 and Assumption 2 we get: Thus there exitsC > 0 which obviously goes to 0 in probability when K → +∞ according to Slutsky's theorem, in view of the induction hypothesis and since (υ K ) K≥0 goes to 0.
Finally we write: So we have Taking the average and going to the upper limit in K we get by induction hypothesis and from Theorem VI-6.22 in [JS13] that The RHS converges to 0 when n → +∞ by dominated convergence. Hence we have shown that (χ K,p ) K≥0 converges to χ p in probability for the uniform convergence.
To conclude we show that (|χ K, ] or equivalently Taking the expectation with respect to P K,α we get the result.

Proof of Theorem 4
From Assumption 6 (i)-(ii) we get that the generator g satisfies the conditions of Lemma 2. Therefore (BSDE) admits a unique solution (Y, Z) ∈ S. We consider α * t = α * (X t , Z t ), and show that α * solve the optimal control problem (C). Since α * is admissible according to Assumption 6 (iii) we have J α * 0 = Y 0 .
We now take any α ∈ A and show that J α * 0 ≥ J α 0 . We write: By definition the first integrand term is almost surely non negative and therefore we have Taking the expectation with respect to P α we get the result.

Proof of Theorem 5
According to Assumption 7 the sequences (ξ K ) K≥0 and (g K ) K≥0 satisfy the assumptions of Lemma 4 for any K and Assumption 2. So from Theorem 2 we have in S 2 1 × S 1 6.6.1 Proof of point (i) The second term converges towards 0 by Theorem 1. The last one terms goes to 0 from Assumption 7 (ii). Using Assumption 7 (iii) we can dominate the first term by that goes to 0 according to Theorem 1 and to the convergence (6).
In the same way, using that the control is bounded, we get that in probability We then extend the convergences to S 1 2 by uniform integrability since the control is bounded. Thus we get the first statement of Theorem 5.

Proof of point (ii)
We consider (t i ) 1≤i≤n ∈ [0, T ] n and a bounded continuous function f defined from R n into R. We show that . . , X tn )] as K → +∞ where E K, * (resp. E * ) denotes the expectation under the control α K, * (resp. α * ). We write and Then we conclude noticing that: We finally prove the convergence of (L K,α K, * T ) K≥0 towards L * T in probability. We introduce the following sequences and show that they all converges to 0 in probability.
For some C > 0 independent of K we have The first term of the RHS converges towards 0 in probability according to Markov inequality. The second one since sup converges almost surely towards 0 from Assumption 7-(i).
Remark that Consequently using Assumption 7 (iii) and Tchebychev inequality we get that (ε K 2 ) K≥0 converges towards 0 in probability. Notice that we have The second and last terms go to 0 in probability by Theorem 1, Assumption 7 (ii) and from Proposition VI-6.12 and Theorem VI-6.22 in [JS13]. As we did in the proof of Theorem 5 (i), the first term goes to 0 from Cauchy Schwarz inequality Assumption 7 (iii) together with the convergence (6).

Finally we write
The second and last terms converge towards 0 by Assumption 7 (ii), Theorem 1, Proposition VI-6.12, Theorem VI-6.22 in [JS13] and Theorem 5. Using Ito's isometry, Cauch-Schwarz inequality, Assumption 7 (iii) together with Theorem 1 and Theorem 5 (i) we get that the first term goes to 0 in probability. Therefore (ε K 4 ) K≥0 converges towards 0 in probability.
Thus we conclude that (L K,α K, * T ) K≥0 converges toward L α * T in probability.

B Change of measure for initial population
We consider m ∈ R * + and n ∈ [0, m) and define the process We have |∆Q K,n,m | ≤ 1 and therefore ∆Q K,n,m ≥ 1. Moreover from Assumption 1 we have for some constant C positive where X m,K min > 0 is the lowest positive value that the process X m,K can take. Therefore by Theorem 2.4 in [Sok13] the process L n,m is a uniformly integrable martingale.
Moreover according to Theorem III-3.11 in [JS13]  C Admissibility of the controls in the toy model

C.1 Discrete models
We show that the control α K, * is admissible. We have By [Jac75] the probability P K,α K, * exists. We recall that we have chosen T small enough such that a K is negative and b K positive on [0, T ]. Hence we have We can always assume that T is small enough so that we can assume that for any t ∈ [0, T ], σ 2 − b K (t) > 0. So λ K,d,α K, * is P K,α K, * almost surely non negative and the control α K, * is admissible.

C.2 Continuous models
We have Obviously this SDE admits a unique strong solution given by Y t 1 inf Ys>0 where Y is the unique strong solution of

D Feller property of the model
We consider a non negative real x. We obviously have that when t → 0 the X K,x t converges almost surely towards x. Now we consider a non negative sequence (x n ) n≥0 that converges towards x and show that for any t > 0, (X K,xn t ) n≥0 converges in law towards X K,x t . We fix x 0 larger than x and any of the x n and f a bounded continuous function on R + .
We write The first term of the right hand side (RHS for short) goes to 0 by dominated convergence. We can dominate the second on by f ∞ |L K,xn,x0 t − L K,x,x0 t | that converges towards 0 according to Scheffé's lemma. Therefore our model has the Feller property.

E Martingale representation with respect to M X
We show in this section that any (F X , P)−martingale has the representation property relative to M X .
We set H = F X 0 and P 0 = ε X0=x0 , i.e. the probability measure on H such that that P 0 (X 0 = x 0 ) = 1. For X a càdlàg process adapted to the filtration F X and B and C two F X -predictable processes with finite variation such that B 0 = C 0 = 0 we recall the definition of the martingale problem associated with (H, X) and (P 0 , B, C).
Definition 3 (Definition III-2.6 in [JS13]). A solution to the martingale problem associated with (H, X) and (P 0 , B, C) is a probability measure Q on (Ω, F X ) such that • the restriction of Q to H equals P 0 , • X is a semi-martingale on (Ω, F X , Q) with characteristics (B, C).
We denote by s(H, X|P 0 ; B, C) the set of solutions to this martingale problem.
From this definition we see that the projection of P on F K is a solution of s(H, X|P 0 ; D, A) where We have M X = X t − D t so that M X is a F X -adapted process and it makes sense to consider the set s(H, M X |P 0 ; 0, A). We show that and that s(H, X|P 0 ; D, A) is reduced to a singleton. This will be enough to conclude according to Theorem III-4.29 in [JS13].
Consider Q ∈ s(H, M |P 0 ; 0, C). We have X = M X + A and since D and C are continuous process with finite variation, we deduce that under Q the characteristics of X are (A, C). Conversely, if Q ∈ s(H, X|P 0 ; D, A) then by recalling that M X = X − D we obtain Q ∈ s(H, M |P 0 ; 0, A). Hence, (7) holds.
Since (S) admits a unique strong solution it admits a unique solution in law (see Theorem IX-1.7 [RY13]). Therefore from Theorem III-2.26 in [JS13] the set s(H, X|P 0 ; D, C) is reduced to a singleton. As a consequence of (7), the set s(H, M X |P 0 ; 0, C) is also reduced to a singleton. Therefore, we deduce from Theorem III-4.29 that all (F X , P)-martingales have the representation property relative to M X .

F Proof of Proposition 1
For two non negative reals ν and µ we say that N is a linear branching process with birth rate ν and intensity µ if it can be written as N = N b − N d where N b and N d are two counting processes with respective intensity νN and µN . This corresponds to a branching process as defined in Section III-3.3.1 in [Mél16] with parameters a = ν + µ, p 0 = µ ν+µ and p 2 = ν ν+µ .
To prove Proposition 1 we proceed in two steps: • Step 1: We prove a result similar to Proposition 1 for linear branching process. • Step 2: We show that a under some assumption a population processes is almost surely dominated by a linear branching process. • Step 3: We conclude using the previous steps.

F.1 Step 1: exponential moments for linear branching processes
We consider N a linear branching process with birth and death rate given by ν and µ.
Note that since (0, 0) satisfies the above conditions, they also hold for β 1 and β 2 small enough.
Proof of Lemma 4. Consider a population starting with one individual. We call τ the lifetime of this particle and C = 1 or 2 the size his offspring. Since all particles are independent and follow the same law we can consider that: where (N (i) ) 1≤i≤2 are independent copies of N .
Finally if we consider a population N starting with n individual we can consider that where (N (i) ) 1≤i≤n are independent copies of the branching process starting with one individuals. Therefore for t < t * ν,µ,β we get F (n, β, t) = F (1, β, t) n which concludes the proof.
We now consider a sequence of branching process (N K ) K≥0 with initial condition Kn and parameters µ K = µ + aK and ν K = ν + aK.
To lighten the notations we use the under-script K instead of (ν K , µ K , β K ). One can easily show the following convergence or equivalence: The convergence of the sequence (t K ) K≥0 implies that for any t < t * ∞ and K large enough F (nK, β K , t) is finite. Moreover from (10) and (11) we get that the sequence F (nK, β K , t) K≥0 converges. More precisely it is easy to check that for any t < t * ∞ we have lim Therefore we deduce that there exists K 0 ∈ N, T > 0 and β ∈ R 2 + such that for any s ∈ [0, t) we have