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Long time asymptotics for fully nonlinear Bellman equations: a Backward SDE approach

We study the large time behavior of solutions to fully nonlinear parabolic equations of Hamilton-Jacobi-Bellman type arising typically in stochastic control theory with control both on drift and diffusion coefficients. We prove that, as time horizon goes to infinity, the long run average solution is characterized by a nonlinear ergodic equation. Our results hold under dissipativity conditions, and without any nondegeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments relying on new backward SDE representation for nonlinear parabolic, elliptic and ergodic equations.


Introduction
Let us consider the fully nonlinear parabolic equation of the form with v(0, .) = h on R d , where L a is the second-order differential operator: Here A is some Borel subset of R q , b, σ are continuous functions on R d × R q , and f = f (x, a, y) is a measurable function on R d × R q × R satisfying some conditions to be specified later on. When the generator f = f (x, a) does not depend on y, equation (1.1) is the dynamic programming equation, also called Hamilton-Jacobi-Bellman (HJB) equation, associated to the stochastic control problem: where X α is the controlled diffusion process driven by a d-dimensional Brownian motion W on a probability space (Ω, F, P) equipped with the natural filtration of W , and given a control α ∈ A, i.e., an A-valued adapted process. In the general case f = f (x, a, y), we shall see that under suitable conditions, there exists a unique viscosity solution v = v(T, x) to the generalized parabolic HJB equation (1.1), and our aim is to investigate the large time behavior of v(T, .) as T goes to infinity. It turns out that this asymptotic problem is related to the generalized ergodic HJB equation: Asymptotics for stochastic control and related HJB equation have been studied in various settings by many authors since the works [4] and [2]. In the PDE literature, we refer for instance to [3] in a periodic setting, [28] under Dirichlet conditions, or [12] in the whole space. In these cited papers, the HJB equation is semi-linear, i.e., the nonlinearity appears only in the first order derivative. Recently, by combining PDE and stochastic analysis arguments, the papers [15], [16] and [24] studied large time behavior of semilinear HJB equations with quadratic nonlinearity in gradients. We would like also to point out the recent paper [14], which studied large time behavior of solutions to semi-linear HJB equations by a probabilistic approach relying on ergodic BSDE introduced in [11]. Interestingly, the authors are able to prove in their context a rate of convergence for the solution to the parabolic equation towards the ergodic equation under weak dissipativity conditions. Long time asymptotics of solutions to HJB equations has been also considered in the context of risk-sensitive stochastic control and utility maximization problem, see e.g. [10] and [21].
Our motivation is to develop a systematic study applicable to a large class of fully nonlinear HJB type equation, and to give natural conditions on the dynamics of the control system ensuring ergodicity. The principal novelty of this paper is to consider control on both drift and diffusion coefficients b(x, a), σ(x, a), possibly degenerate, and satisfying dissipativity conditions, instead of periodicity condition. In this case, we do not have in general smooth solution to the HJB equation. Another original feature of our framework is the dependence of f (x, a, y) on y, which occurs for example in stochastic control with recursive utility functions. Our first main result is to prove the existence of a viscosity solution pair (λ, φ) ∈ R × C(R d ), with φ Lipschitz, to the ergodic fully nonlinear equation (1.3). We adopt the following approach. We consider the sequence of fully nonlinear elliptic HJB equation for β > 0: and obtain the existence and uniqueness of a solution v β to (1.4) by combining analytical and probabilistic methods. More precisely, following the randomization approach of [19] for representing parabolic HJB equations, we introduce a class of Backward Stochastic Differential Equations (BSDEs) with nonpositive jumps over an infinite horizon, and supported by a forward regime switching process (X, I) where and I is a pure jump process valued in A. The minimal solution Y β to this class of elliptic BSDEs is shown to exist and to provide the unique (viscosity) solution v β to (1.4), and the key point is to derive uniform Lipschitz estimate for the sequence (v β ) β . This is achieved by ergodicity properties on the forward process X, and suitable estimation on the minimal solution Y β . Then, by standard analytical approximation procedures (when β goes to zero) as in [15] or [11], we obtain the existence of a pair (λ, φ) solution to (1.3). Moreover, the function φ admits a probabilistic representation in terms of a new class of BSDEs, namely ergodic BSDEs with nonpositive jumps. Ergodic BSDEs have been introduced in [11] and then in [23], and related to optimal control on the drift of diffusions. We extend this connection to the case of controlled diffusion coefficient by imposing a nonpositive jump constraint on the ergodic BSDE. Next, our main theorem is to prove that for any solution (λ, φ) to (1. Here, convergence "in C(R d )" stands for locally uniform convergence in R d . This shows as a byproduct that λ in (1.3) is unique. The main difficulty with respect to the semi-linear HJB case is that we do not have in general a smooth solution and an optimal control for the finite horizon and ergodic stochastic control, and the classical arguments as in [15] or [16] do not work anymore. Moreover, when f = f (x, a, y) depends also on y, we do not even have a stochastic control representation of the function v. for some constant c. The paper is organized as follows. Section 2 introduces some notations, formulates the dissipativity conditions on b, σ, and assumptions on f . We then state ergodicity properties on the regime switching process (X, I) in (1.5) as well as on the controlled diffusion process X α in (1.2). In Section 3, we prove the existence and uniqueness of a solution to the fully nonlinear elliptic HJB equation (1.4) and its relation to BSDE with nonpositive jumps over an infinite horizon. Section 4 is concerned with the ergodic equation (1.3) and its probabilistic representation in terms of ergodic BSDE with nonpositive jumps. Convergence results (1.6) and (1.7) are studied in Section 5. We collect in the Appendix some proofs and technical estimates needed in the paper.

Ergodicity properties 2.1 Notations and assumptions
Let (Ω, F, P) be a complete probability space on which are defined a d-dimensional Brownian motion W = (W t ) t≥0 and an independent Poisson random measure µ on R + ×A, where A is a compact subset of R q , endowed with its Borel σ-field B(A). We assume that the random measure µ has the intensity measure ϑ(da)dt for some finite measure ϑ on (A, B(A)). We setμ(dt, da) = µ(dt, da) − ϑ(da)dt the compensated martingale measure associated to µ, and denote by F = (F t ) t≥0 the completion of the natural filtration generated by W and µ. We also denote, for any T > 0, P T the σ-field of F-predictable subsets of [0, T ] × Ω. Let us introduce some additional notations. We denote by: We also define S 2 loc := ∩ T >0 S 2 (0, T).
We are given some continuous functions b : and consider the forward regime switching process (X, I) governed by the stochastic differential We note that the fact that σ is a square matrix does not involve any loss of generality, since we are not going to assume any nondegeneracy condition. In particular, some rows or columns of σ may be equal to zero. In the following we use the notation M ⊺ for the transpose of any matrix M , and M 2 = tr(M M ⊺ ) for the Hilbert-Schmidt norm. We shall make the following assumptions on the coefficients b and σ.
(H1) (i) There exists a positive constant L 1 such that for all x, (ii) There exists a constant γ > 0 such that for all x, x ′ ∈ R d , a ∈ A, It is well-known that under (H1)(i), there exists a unique solution (X x,a t , I a t ) t≥0 to (2.1) starting from (x, a) ∈ R d × R q at time t = 0. Notice that when a ∈ A, then I a t ∈ A for all t ≥ 0. Condition (H1)(ii) is called dissipativity condition and will ensure the ergodicity of the process X, as stated in the next paragraph.
In this case, (H1) is satisfied, and this example corresponds to a controlled Ornstein-Uhlenbeck process with uncertain mean-reversion and volatility. ✷ We also consider some real-valued continuous function f on R d × R q × R satisfying the following assumption: We end this paragraph of notations by introducing the following set of probability measures, which shall play an important role in the sequel for establishing estimates. Let V n be the set of P ⊗ B(A)-measurable maps valued in [1, n + 1], V = ∪ n∈N V n , and consider for ν ∈ V, the probability measure P ν equivalent to P on (Ω, F T ), for any T > 0, with Radon-Nikodym density: is the Doléans-Dade exponential. Actually, since ν ∈ V is essentially bounded, it is shown in Lemma 2.4 in [19] that (ζ ν ) 0≤t≤T is a uniformly integrable P-martingale, with ζ ν T ∈ L 2 (F T ), for any T > 0, and so it defines a probability measure P ν via (2.3). We shall denote by E ν the expectation under P ν . Moreover, by Girsanov theorem, the compensator of µ under P ν is ν t (a)ϑ(da)dt, while W remains a Brownian motion independent of µ under P ν . We denote byμ ν (dt, da) = µ(dt, da) − ν t (a)ϑ(da)dt the compensated martingale measure of µ under P ν .

Ergodicity
We now use the dissipativity condition in (H1)(ii) to state moment estimates and stability results on the component X of (2.1).
The proof relies on rather standard arguments based on Itô's formula and Gronwall's lemma, and is reported in the Appendix.

Remark 2.1
We shall need the following generalization of estimate (2.4), for all x ∈ R d and a ∈ A, which is valid with the same constant C = C b,σ , independent of t, as in (2.4), when Assumption (H1) holds. ✷ Let α : R d → A be a feedback control and let X = X α be the associated diffusion process governed by (2.7) Suppose that the functions are Lipschitz. Then, equation (2.7) defines a time-homogeneous Markov process {X α t , t ≥ 0}, and we denote by (P α t ) t≥0 the associated semigroup, which acts on B(R d ), the set of bounded measurable functions ϕ, by Notice that (P α t ) t≥0 has the Feller property, i.e., for any f ∈ C b (R d ), the space of continuous and bounded functions on R d , we have that P α t f ∈ C b (R d ). The next result shows the ergodicity of X α . Proposition 2.1 Let α : R d → A be a feedback control such that b, σ in (2.8) satisfy Assumption (H1). Then X α is ergodic, i.e., the following assertions are valid: (i) There exists a unique invariant probability measure ρ = ρ α on R d : (ii) X α t converges weakly to ρ as t → ∞: Moreover, |x| 2 ρ(dx) < ∞, and the convergence (2.9) holds for all continuous ϕ satisfying a linear growth condition.
The proof is based on the "pullback" method (see, e.g., Theorem 6.3.2 in [8]) and is detailed in the Appendix.

Elliptic HJB equation
For any β > 0, let us consider the fully nonlinear elliptic equation of HJB type: Notice that in the particular case where f = f (x, a) does not depend on y, the equation (3.1) is the dynamic programming equation associated to the stochastic control problem on infinite horizon: In this case, it is easy to see from the Lipschitz and growth condition on f in (H2)(i), and the estimates (2.4), (2.5) that the sequence of functions (v β ) β satisfies the uniform estimates: for some positive constant C independent of β.
In the general case f = f (x, a, y), this section is devoted to the existence and uniqueness of a viscosity solution v β to (3.1), and to uniform estimate on (v β ) β as in (3.2). To this purpose, we introduce the following class of BSDE with nonpositive jumps over an infinite horizon, for any β > 0: and BSDEs driven by Brownian motion over an infinite horizon have been introduced in [9], [22], studied also in [5] and extended in [25], and related to elliptic semi-linear PDEs.
for all 0 ≤ t ≤ T , P-almost surely. Then, we see that the right-hand side is a finite variation process, while the left-hand side has not finite variation, unless Z =Z. Now, from (3.5), we obtain the identity where the right-hand side is a predictable process, therefore it has no totally inaccessible jumps (see, e.g., Proposition 2.24, Chapter I, in [18]); on the other hand, the left-hand side is a pure-jump process with totally inaccessible jumps, unless U =Ũ . As a consequence, we must have U =Ũ , from which it follows that K =K. ✷ In the sequel, we prove by a penalization approach the existence of the minimal solution to (3.3)-(3.4), which shall provide the solution to the elliptic nonlinear HJB equation (3.1). Then, by using this probabilistic representation of v β , we shall state uniform Lipschitz estimate on (v β ) β .
We first state an a priori estimate on the above elliptic penalized BSDE.

✷
The next result states the existence and uniqueness of a solution to (3.6), and uniform estimate on the solution.
Proposition 3.1 Let Assumptions (H1) and (H2) hold. Then, for any (x, a, β, n) ∈ R d × A× (0, ∞)× N, there exists a solution (Y x,a,β,n , Z x,a,β,n , U x,a,β,n ) ∈ S 2 loc × L 2 loc (W) × L 2 loc (μ) to (3.6), with (X, I) = (X x,a , I a ), and satisfying: for some positive constant C b,σ,f depending only on b, σ, f . Moreover, this solution is unique in the class of triplets (Y, , for all t ≥ 0 and for some positive constant C (possibly depending on x, a, β and n).
Let us now investigate some key properties of the function v β,n . We first state a uniform Lipschitz estimate on (v β,n ).
for all x, x ′ ∈ R d and a ∈ A.
Proof. See Appendix. ✷ As expected, for fixed (β, n), the function v β,n is related to the elliptic integro-differential equation: for any ϕ ∈ C(A). More precisely, we have the following result.
Proposition 3.2 Let Assumptions (H1) and (H2) hold. Then, the function v β,n in (3.9) is a continuous viscosity solution to (3.12), i.e., it is continuous on R d × A and it is a viscosity supersolution (resp. subsolution) to (3.12), namely Proof. See Appendix. ✷

Elliptic BSDE with nonpositive jumps
We can now prove the existence of the minimal solution to the elliptic BSDE with nonpositive jumps (3.3)-(3.4).

Proposition 3.3 Let Assumptions (H1) and (H2) hold. Then, for any
for some positive constant C independent of β, x, a, t.
, for any T > 0, is the strong (resp. weak) limit of (Z x,a,β,n , for all t ≥ 0 and for some positive constant C (possibly depending on x, a, and β). (3.14) From point (i) of Proposition 3.3, it follows that (v β,n ) n converges increasingly to v β . Then, the identification Y x,a,β,n issue is to prove that v β does not depend actually on a. However, as we do not know a priori that the function v β is continuous in both arguments, we shall rely on discontinuous viscosity solutions arguments as in [19], and make the following assumptions on the set A and the intensity measure ϑ: (HA) The interiorÅ of A is connected, and A = Cl(Å), the closure of its interior.
(Hϑ) The measure ϑ supports the whole setÅ, and the boundary of A: ∂A = A\Å, is negligible with respect to ϑ.
Notice that equation (3.1) does not depend on ϑ, and this intensity measure only appears in order to give a probabilistic representation of v β . Therefore, we have the choice to fix an intensity measure ϑ satisfying condition (Hϑ), which is anyway a fairly general condition easy to realize. In the sequel, we shall make the standing assumption that (Hϑ) holds.
We can now state the main result of this section.
for all x ∈ R d , and we set by misuse of notation: Then, v β is the unique continuous viscosity solution to equation (3.1), i.e., it is continuous on R d and it is a viscosity supersolution (resp. subsolution) to (3.1), namely: for any x ∈ R d and any ϕ ∈ C 2 (R d ) such that Moreover, there exists some positive constant C independent of β such that: Proof. We use the corresponding result for the parabolic case in [19] to prove the non dependence of v β on a, and then the viscosity property to the elliptic equation. More precisely, we start by observing that, for any β > 0 and To prove this, for any (t, . , it follows from Proposition 3.2 that (3.18) is satisfied. Then, from Theorem 3.1 in [19], we deduce by sending n to infinity that the function v β does not depend on the variable a, and so (3.15) holds. We should point out that in [19] the terminal condition of the parabolic PDE solved by v β,n does not depend on n, contrary to our case. However, the part of the proof of Theorem 3.1 in [19] regarding the independence with respect to the variable a does not involve the terminal condition, so the result still holds. Next, we obtain again from Theorem 3.1 in [19] that v β solves in the viscosity sense the parabolic PDE on [0, T ) × R d × A (as before, we do not call in the terminal condition): Since this equation holds for any T , and v β does not depend on t, we obtain that v β is a viscosity solution to the elliptic equation (3.1). The uniqueness follows from Theorem 7.4 in [17]. Finally, the linear growth and Lipschitz properties (3.16)-(3.17) of v β are direct consequences of the corresponding properties (3.13) for Y x,a,β 0 and (3.11) for v β,n , respectively. ✷ Remark 3.2 Notice that, from the identification Y x,a,β = v β (X x,a ), for any x ∈ R d and for some a ∈Å, and the Lipschitz property (3.17), it follows that Y x,a,β is a continuous process, so that U x,a,β ≡ 0, while K x,a,β is also a continuous process. ✷

Ergodic HJB equation and ergodic BSDE with nonpositive jumps
This section is devoted to the existence of a solution pair (λ, φ) to the ergodic HJB equation and to its probabilistic representation in terms of ergodic BSDE with nonpositive jumps. We first give the definition of viscosity solution to equation (4.1).
, with λ a real number and φ : R d → R a lower (resp. upper) semicontinuous function, is called a viscosity supersolution (resp. viscosity subsolution) to equation for any x ∈ R d and any ϕ ∈ C 2 (R d ) such that (ii) A pair (λ, φ), with λ a real number and φ : R q → R a continuous function, is called a viscosity solution to equation (4.1) if it is both a viscosity supersolution and a viscosity subsolution to (4.1).
Proof. We follow the approximation procedure as in [15] or [11]: for any β > 0, let v β be the solution of (3.1) given in Theorem 3.1, and define λ β ∈ R, and the function φ β : R d → R by: Then, the family of functions (φ β ) β is equicontinuous and uniformly bounded on every compact subset of R d . Thus, by means of Ascoli-Arzelà theorem (for more details, see (4.4) and (4.5) in [11]), we can construct a sequence (β k ) k decreasing monotonically to zero such that, for all x ∈ R d , for some real constant λ, and a Lipschitz function φ : R d → R. Moreover, the convergence of φ β k towards φ is uniform on compact sets. Now, from the elliptic equation (3.1) satisfied by v β , and by definition of (λ β , φ β ), we see that φ β is a viscosity solution to: Let us denote by so that by (4.3), φ β k is a viscosity solution to: . As a consequence, from the method of half-relaxed limits of Barles and Perthame, see Remark 6.3 in [6], we deduce by (4.2) that φ is a viscosity solution to: i.e., (λ, φ) is a viscosity solution to equation (4.1). ✷ We postpone the uniqueness problem of the ergodic equation (4.1) to the next section, and conclude this section by providing a probabilistic representation formula for a solution to the ergodic equation. Let us introduce the ergodic BSDE with nonpositive jumps, P-a.s., Here, in addition to the components (Y, Z, U, K), the real number λ is part of the unknowns of the ergodic BSDE. We recall that ergodic BSDEs driven by Brownian motion have been defined in [11] for the study of optimal control problems on the drift of a diffusion process, which are related to ergodic semilinear HJB equations. In this paper, we extend this definition by considering the jump constraint (4.5), and our first purpose is to introduce a notion of "minimal" solution to (4.4)-(4.5). However, we notice that the "natural" definition of minimal solution as the solution (Ȳ ,Z,Ū , we haveȲ ≤Ỹ , is not relevant in this case, since (Ȳ −c,Z,Ū ,K, λ), with c > 0, would be another solution to (4.4)-(4.5), contradicting the minimality of (Ȳ ,Z,Ū ,K, λ). For this reason, we give the following definition of minimal solution to the ergodic BSDE with nonpositive jumps (4.4)-(4.5).
together with the jump constraint In other words, for any other solution and for t ≥ 0, and, in particular, for some a ∈Å.
Proof. We start by observing that for any T > 0, φ is a viscosity solution to the following parabolic equation, in the unknown ψ, (4.10) It follows from Theorem 7.4 in [17] that φ is the unique uniformly continuous viscosity solution to (4.10). From Theorem 3.1 in [19], we see that this viscosity solution admits a representation in terms of the minimal solution to a BSDE with nonpositive jumps on [0, T ]. More precisely, for any T > 0, let (Y x,a,T , Z x,a,T , U x,a,T , K x,a,T ) ∈ S 2 (0, T)×L 2 (W; 0, T)× L 2 (μ; 0, T) × K 2 (0, T) be the minimal solution to the BSDE with nonpositive jumps on [0, T ], P-a.s., and Then, from Theorem 3.1 in [19] we see that Y x,a,T t ) also implies that Y x,a,T does not depend on T . Moreover, using the fact that the Y component remains the same, reasoning as in Remark 3.1, we can prove that Z x,a,T , U x,a,T , and K x,a,T do not depend on is the minimal solution to (4.11)-(4.12), from which we deduce that (Y x,a , Z x,a , U x,a , K x,a , λ) is a minimal solution to the ergodic BSDE (4.4)-(4.5). ✷ For any x ∈ R d , consider a ∈Å and a minimal solution (Y x,a , Z x,a , U x,a , K x,a , λ) ∈ S 2 loc × L 2 loc (W) × L 2 loc (μ) × K 2 loc × R to the ergodic BSDE with nonpositive jumps (4.8)-(4.9), such that Y x,a t = φ(X x,a t , I a t ), t ≥ 0. Then, φ does not depend on the variable a and (λ, φ) is a viscosity solution to the ergodic equation (4.1). Indeed, fix T > 0, then by Definition 4.2 we know that (Y x,a , Z x,a , U x,a , K x,a ) is the unique minimal solution to the BSDE with nonpositive jumps on [0, T ] in (4.6)-(4.7). It follows from Theorem 3.1 in [19] that φ does not depend on a (this last property follows from Proposition 3.2 in [19], which does not call in the terminal condition; therefore, even if in our case, as opposite to [19], the terminal condition depends on a, the result is still valid) and it is a viscosity solution to equation (4.10). As a consequence, (λ, φ) is a viscosity solution to the ergodic equation (4.1). ✷

Convergence of solutions
Let us consider the parabolic fully nonlinear equation of HJB type: Consequently, this will show the uniqueness of the component λ in (4.1). We shall end this part of the paper by proving a stronger result than (5.2) under additional assumptions. More precisely, suppose that φ belongs to C 2 (R d ) so that (λ, φ) is a classical solution to the ergodic equation (4.1), and assume that in the ergodic equation, the supremum is attained at a = α(x), for every x ∈ R d , for some locally Lipschitz function α : A → R d . Then, the following convergence holds for some constant c. In particular, φ is uniquely determined up to a constant.

Wellposedness of the parabolic equation (5.1)
We shall build a solution to equation (5.1) through BSDE methods, as this construction will be useful in the sequel. More precisely, from Theorem 3.1 in [19], under (H1)(i), (H2)(i), (HA), and (Hϑ), there exists a uniformly continuous viscosity solution v to equation (5.1), which admits the following probabilistic representation formula Moreover, from the viscosity properties of v T it follows that v is the unique uniformly continuous viscosity solution to equation (5.1). Then, from the probabilistic representation formula for v T we deduce the representation formula (5.4) for v. ✷ In conclusion, we have proved the following result.
The aim is to state an upper and lower estimate for w, uniformly in time T , so that by dividing by T , we obtain the convergence of the long run average v(T, .)/T to λ. Classical PDE arguments (in the case where f does not depend on y) rely on the smoothness of v and φ in order to prove that w is a sub and supersolution to some PDE without cost or gain function. Then by comparison principle, and under ergodicity conditions, one would obtain for w a lower and upper bound function which does not depend on time. In our general framework, the major difficulty is due to the non-regularity in general of v and φ, especially when there is singularity of the diffusion coefficient. In this case, it is not clear, even with the notion of viscosity solution, how to derive an equation for w involving the difference of v and φ. We circumvent this issue by adopting an alternative approach where we use probabilistic representations formulae for v and φ. We are also interested in the case where f (x, a, y) depends on y, that we shall actually tackle by using the nondecreasing feature of f in y and imposing the following additional assumption.

(H3)
The function f can be written as f (x, a, y) = f 0 (x, a) + f 1 (x, a, y), where f 1 can be either the zero function or it satisfies, for all x ∈ R d , a ∈ A, y, y ′ ∈ R, for some constant κ > 0.
where A is the set of adapted control processes valued in A, and (X x,α t ) t≥0 is the unique solution to the controlled equation (1.2) starting from x at time 0. Similarly, we know from the proof of Theorem 4.2 that φ is the unique uniformly continuous viscosity solution to equation (4.10), so that φ is given by From the definition of w, we have Proceeding in a similar way, we obtain Since h and φ are Lipschitz, from estimate (2.4) we deduce (5.9). ✷ Proof of Theorem 5.1. We recall from (5.4) and Theorem 4.2 the nonlinear Feynman-Kac formulae for all (T, x) ∈ [0, ∞) × R d , and any a ∈Å. Fix then a ∈Å, and define, for (T, x) ∈ [0, ∞) × R d , the process: Then, by definition of w in (5.8), we have Moreover, from the BSDE (5.5)-(5.6) for Y x,a,T and (4.8)-(4.9) for Y x,a , we derive the BSDE forỸ x,a,T : We shall now prove suitable upper and lower bounds forỸ x,a,T , and thus for w(T, x).
We know from Theorem 2.1 in [19] that there exists a unique minimal solution (Ŷ x,a,T , Z x,a,T ,Û x,a,T ,K x,a,T ) in S 2 (0, T) × L 2 (W; 0, T) × L 2 (μ; 0, T) × K 2 (0, T) to equation (5.12)-(5.13). SetȲ x,a,T t =Ŷ x,a,T t +λ(T −t)+Y x,a t , t ∈ [0, T ], and recall that Y x,a t = φ(X x,a t ). Then, from the BSDEs satisfied byŶ x,a,T and Y x,a , we easily see that (Ȳ x,a,T ,Ẑ x,a,T + Z x,a ,Û x,a,T + U x,a ,K x,a,T + K x,a ) is a solution to (5.5)-(5.6). From the minimality of (Y x,a,T , Z x,a,T , U x,a,T , K x,a,T ), we get: Y x,a,T t ≤Ȳ x,a,T t , and by subtracting to both sides (5.14) Let us now derive an upper bound forŶ x,a,T 0 . To this end, we introduce the associated penalized BSDE with jumps on [0, T ], for n ∈ N: From the uniform Lipschitz condition on f (x, a, y) with respect to y, together with Assumptions (H2)(ii) and (H3), we have f X x,a s , I a s ,Ŷ x,a,T,n s with (we suppose here that f 1 in (H3) is not the zero function; otherwise, ρ n can be taken equal to zero everywhere and the proof becomes easier) Now, from Proposition 2.1 in [19] we have the following dual representation formula for the right-hand side of (5.18): From the definition of ρ n in (5.17), we find Recalling that h and φ are Lipschitz, from estimate (2.4) we obtain for some positive constant C, independent of x, a, T , and n. Since from Theorem 2.1 in [19] we have thatŶ x,a,T,n 0 converges toŶ x,a,T 0 , as n goes to infinity, we get the same estimate: Y x,a,T 0 ≤ C(1 + |x|), and therefore, from (5.11) and (5.14), we deduce that w(T, x) ≤ C(1 + |x|).
• Step 2. Lower bound: w(T, x) ≥ −C(1 + |x|). As in step 1, where we built an upper bound forỸ x,a,T using the minimality of Y x,a,T , here we shall construct a lower bound forỸ x,a,T exploiting the minimality of Y x,a in the sense of Definition 4.2. In particular, we fix T > 0 and we recall that is the minimal solution to (4.6)-(4.7) on [0, T ] with terminal condition φ(X x,a T ). Now, let us consider the BSDE with nonnegative jumps on [0, T ]: Theorem 2.1 in [19] gives the existence of a unique maximal solution (Y x,a,T ,Ž x,a,T , U x,a,T ,Ǩ x,a,T ) ∈ S 2 (0, T) × L 2 (W; 0, T) × L 2 (μ; 0, T) × K 2 (0, T) to (5.19)-(5.20). Actually, Theorem 2.1 in [19] is designed for minimal solutions, while here we deal with maximal solutions; however, it is easy to show that −Y x,a,T is a minimal solution to a certain BSDE with nonpositive jumps, therefore we can apply Theorem 2 From the Lipschitz property of h and φ, and estimate (2.4), we have Moreover, from the uniform Lipschitz condition on f (x, a, y) with respect to y, and the nondecreasing property of y → f (x, a, y) in (H2), there exists some adapted, nonpositive, and bounded process ζ such that From step 1 and the Lipschitz property of φ, it follows that v(T, x) − λT ≤ C(1 + |x|), and where we used the fact that ζ is bounded by L 2 , the Lipschitz constant of f . Plugging the above estimate into (5.
is called a Tauberian theorem, see, e.g., [1]. In our paper, Theorem 4.1 together with Theorem 5.1 can be thought as a "robust" Tauberian theorem. Indeed, in Theorem 4.1 we proved that the convergence of t , α t )dt] towards λ, and also of v β k (x) − v β k (0) towards φ, allows to construct a viscosity solution (λ, φ) to the ergodic equation (4.1). Then, Theorem 5.1 implies the convergence of v(T,x)

Further convergence result via verification: the proof of (5.3)
We conclude this section by presenting, in the form of a verification theorem, the following result, which shows the validity of the convergence (5.3).
(ii) In the ergodic equation (4.1), the supremum is attained at a = α(x), for every x ∈ R d , for some function α : A → R d such that b, σ in (2.8) satisfy Assumption (H1).
Consider the unique (viscosity) solution v to (5.4). Then, there exists a real constant c for all x ∈ supp ρ, the support of the invariant measure ρ given by Proposition 2.1. In particular, when supp ρ = R d we deduce that φ is uniquely determined up to a constant. Notice that, thanks to the presence of the term "δψ(x)" in equation (5.24), we can apply comparison Theorem 3.3 in [17], from which it follows that φ is the unique uniformly continuous viscosity solution to equation (5.24)-(5.25). Let us now impose the following uniform ellipticity condition: there exists ν ∈ (0, 1], possibly depending on R, such that for all x ∈ B R and a ∈ A. Then, as explained in Remark 1.1 of [27], under the above assumption, Theorem 1.1 in [27] holds, and there exists a unique solution ψ ∈ C 2 (B R ) ∩ C(B R ) to equation (5.24)-(5.25). Theorem 3.3 in [17] implies that ψ coincides with our function φ, so that φ ∈ C 2 (B R ). Since R is arbitrary, we conclude that φ ∈ C 2 (R d ). ✷ Proof.
Step 1. Notice that, for any T, S > 0 and for all x ∈ R d , we have where A is the set of adapted control processes valued in A, and (X x,α t ) t≥0 is the unique solution to the controlled equation (1.2) starting from x at time 0. As a matter of fact, to prove (5.26) we recall from Remark 5.
where v T +S is the unique uniformly continuous viscosity solution to the following Hamilton-Jacobi-Bellman equation in the unknownṽ T +S : As a consequence, v T +S is given, for all (t, x) ∈ [0, T ] × R d , by the stochastic control representation: which implies (5.26). In particular, we have On the other hand, applying Itô's formula to φ(X x,α s ) between 0 and T , and using the optimality of α in the ergodic equation (4.1), we obtain Therefore, w in (5.8) satisfies Step 2. Let us prove that there exists a positive constant C such that for all T ≥ 0 and x, x ′ ∈ R d . Recalling (5.8) and since φ is Lipschitz, it is therefore enough to prove that the function v satisfies for some positive constant C. We know that v(T, x) = Y x,a,T 0 is represented by the minimal solution to the BSDE with nonpositive jumps (5.5)-(5.6) on [0, T ]. We recall from Theorem 2.1 in [19] that Y x,a,T,n ↑ Y x,a,T , where (Y x,a,T,n , Z x,a,T,n , U x,a,T,n ) ∈ S 2 (0, T) × and U x,a,T,n t (a ′ ) ≤ 0, dt ⊗ dP ⊗ ϑ(da ′ )-a.e.
Then, (5.28) follows once we get: for a constant C that does not depend on x, a, T , and n. This can be done using Girsanov theorem and the dissipativity condition (2.2) in the same way as for (3.11).
Step 3. Now, we proceed as in [15] and we introduce the set Γ which contains all the ω-limits of the family {w(T, ·)} T >1 in C(R d ) (we endow C(R d ) with the topology for which and only if f j converges uniformly to f on any compact subset of R d ).
In other words, Γ is given by It follows from step 2 that the family {w(T, ·)} T >1 is relatively compact in C(R d ). In particular, Γ = ∅, and any w ∞ in Γ is Lipschitz. To conclude, it suffices to prove that every w ∞ ∈ Γ is equal to the same constant c ∈ R on supp ρ.
-Step 3a. We first show that any element of Γ is constant on supp ρ. Let w ∞ ∈ Γ, therefore there exists a sequence ( Therefore, sending j → ∞ in (5.29), and by the dominated convergence theorem, we obtain Moreover, choosing T := T j and letting j → ∞ in the above inequality, we obtain, from Proposition 2.1: Now, taking the infimum with respect to x ∈ R d , we end up with As a consequence w ∞ = inf R d w ∞ , ρ-a.s., therefore w ∞ is constant on supp ρ.
-Step 3b. We next prove that every w ∞ ∈ Γ is equal to the same constant c on supp ρ. Proceeding as in the derivation of (5.27), we have: for any T, S, R > 0 with R ≤ T . Suppose that there exist two real constants c 1 , c 2 and two diverging sequences (T j ) j∈N and (S j ) j∈N such that w(T j , ·) → c 1 and w(S j , ·) → c 2 on supp ρ as j → ∞. Let us take T := T j , R = T j − S, and S := S k in (5.30). Then, letting j → ∞ we obtain (notice that, by (5.30) and Lebesgue's dominated convergence theorem, the two integral terms in (5.30) simplify one with the other as j → ∞) Now, sending k → ∞, we find Therefore c 1 ≥ c 2 . Changing the role of (T j ) j∈N and (S j ) j∈N , we also find c 2 ≥ c 1 . Hence, c 1 = c 2 and the claim follows.
where X x,α is the controlled diffusion process satisfying (1.2), starting from x ∈ R d at time 0, and α ∈ A is a control process, i.e., an A-valued adapted process. More precisely, it is clear that J(x, α) depends only on the asymptotic behavior of the trajectories of X x,α . Therefore, from the ergodicity of X x,α , we expect that there exists a real number λ * , independent of x ∈ R d , such that namely, λ * is the value of the ergodic control problem. Let us prove that λ * = λ. Firtsly, observe that, since f does not depend on y, the function v in (5.4) admits the stochastic control representation From (5.10) we know that, for any x ∈ R d , where the last equality follows from the fact that lim T →∞ sup α∈A 1 T E[h(X x,α T )] = 0, which is a consequence of the Lipschitz property of h and estimate (2.4). From (5.31) we see that λ * ≤ λ. To prove the reverse inequality, fix x ∈ R d , then, applying Itô's formula to φ(X x,α t ) between 0 and T , and using the optimality of α in the ergodic equation (4.1), we obtain From the Lipschitz property of φ and estimate (2.4), we have 1 which implies that λ * = λ. ✷ e γs X x,a s .b(X x,a s , I a s )ds + t 0 e γs σ(X x,a s , I a s ) 2 ds + 2 t 0 e γs (X x,a s ) ⊺ σ(X x,a s , I a s )dW s .
Rearranging the terms in a suitable way so to exploit the dissipativity condition (H1)(ii), we obtain  (0, a) ). From the inequality |X x,a s | ≤ ε|X x,a s | 2 + 1/(4ε), for any ε > 0, we obtain We can find ε such that γ − 2M 1 ε − 2M 1 L 1 ε ≥ 0 (more precisely, if M 1 = 0 then ε can be any positive real number; otherwise we take ε ≤ γ/(2M 1 + 2M 1 L 1 )), therefore (also multiplying both sides in (A.1) by e −γt ) is indeed a P ν -martingale. Then, we have E ν [e −γt M t ] = 0. Therefore, taking the expectation E ν in (A.2), we find is indeed a P ν -martingale. Therefore, taking the expectation E ν with respect to P ν in (A.4) and using the dissipativity condition (2.2), we obtain ✷ A.2 Proof of Proposition 2.1 Step 1. Existence and uniqueness of ρ. LetW = (W t ) t≥0 be a d-dimensional Brownian motion, independent of W and µ. Then, we definē For any T ∈ R and x ∈ R d , we denote X T,x = (X T,x t ) t≥T the unique solution to the equation on [T, ∞): Taking the expectation and using the dissipativity condition (2.2), we find which implies It follows from (A.9) that (X −T,x 0 ) T >0 converges, as T → ∞, to some square integrable random variable η x , which a priori depends on x. Let x ′ ∈ R d , then applying Itô's formula to Taking the expectation and using the dissipativity condition (2.2), we obtain As a consequence, η x = η x ′ =: η. We denote ρ := L(η). Finally, using the law invariance property already recalled, and the fact that convergence in L 2 (P) implies convergence in law, we deduce weakly as T → ∞. From the square integrability of η we see that R d |x| 2 ρ(dx) < ∞. Let us now prove the invariance property. Let ϕ ∈ C b (R d ), then, from the Markov property we have P Sending t → ∞, using (A.10) and the Feller property, we obtain for all ϕ ∈ C b (R d ). By a monotone class argument, we see that (A.11) remains true for all ϕ ∈ B(R d ), which implies the invariant property of ρ. Concerning the uniqueness of ρ, let us consider another invariance probability measure ν and take ϕ ∈ C b (R d ), then Since the result holds for any ϕ ∈ C b (R d ), we deduce the uniqueness of ρ.
Step 2. (2.9) is valid for any continuous ϕ satisfying a linear growth condition. For any R > 0, consider a continuous function χ R : R d → [0, 1] which is equal to 1 on B R ⊂ R d (the open ball of radius R centered at the origin) and is equal to 0 on R d \B 2R . Then Since ϕχ R ∈ C b (R d ), the first term on the right-hand side of (A.12) goes to zero as t → ∞. If then, taking first lim sup t→∞ and then lim R→∞ in (A.12), we get the thesis. Therefore it remains to prove (A.13). From Cauchy-Schwarz inequality, the linear growth property of ϕ, and estimate (A.8), we have that there exists a positive constant C such that Similarly, we have where we recall that R d |x| 2 ρ(dx) < ∞. In conclusion, we obtain Notice that Let 0 ≤ t ≤ T < ∞ and apply Itô's formula to e −2βs |∆Y s | 2 between t and T , then Notice that, using the nonincreasing property of f 1 in y, we have Now, define the [1, n + 1]-valued map ν as follows Observe that ν is a P ⊗ B(A)-measurable map satisfying 1 ≤ ν s (a) ≤ n + 1, ds ⊗ dP ⊗ ϑ(da)a.e., then ν ∈ V n . Let us consider the probability measure P ν equivalent to P on (Ω, F T ) with Radon-Nikodym density given by (2.3). Recalling thatμ ν denotes the compensated martingale measure associated to µ under P ν , equation (B.14) can be rewritten as follows From Lemma 2.5 in [19], we see that the two stochastic integrals on the right-hand side of (B.15) are martingales. Hence, taking the expectation E ν , conditional on F t , with respect to P ν in (B.15), we end up with estimate (3.7). ✷
Step 4. Convergence of (Z T , U T ) T >0 . Let S, T, T ′ > 0, with S < T < T ′ . Then, applying Itô's formula to e −2βt |Y T ′ t − Y T t | 2 between 0 and S, and taking the expectation, we find Since the map y → f (x, a, y) is nonincreasing, we get (using also the inequality ab ≤ a 2 /2 + b 2 /2, for any a, b ∈ R) Multiplying the previous inequality by e 2βS , we obtain Then, we deduce that which implies (3.11). ✷
Denote by T 1 the first jump time of the marked point process (T n , α n ) n≥1 associated to the Poisson random measure µ. Notice that the two processes I a and I a ′ coincide after T 1 , while we have I a s = a and I a ′ s = a ′ before T 1 . In other words, |I a s − I a ′ s | = |a − a ′ |1 {s≤T 1 } ≤ |a − a ′ |. Therefore