Infinite horizon impulse control problem with jumps and continuous switching costs

Purpose – The purpose of this paper is to show the existence results for adapted solutions of infinite horizon doubly reflected backward stochastic differential equations with jumps. These results are applied to get the existence of an optimal impulse control strategy for an infinite horizon impulse control problem. Design/methodology/approach – The main methods used to achieve the objectives of this paper are the properties of the Snell envelope which reduce the problem of impulse control to the existence of a pair of right continuous left limited processes. Some numerical results are provided to show the main results. Findings – In this paper, the authors found the existence of a couple of processes via the notion of doubly reflected backward stochastic differential equation to prove the existence of an optimal strategy which maximizes the expected profit of a firm in an infinite horizon problem with jumps. Originality/value – In this paper, the authors found new tools in stochastic analysis. They extend to the infinite horizon case the results of doubly reflected backward stochastic differential equations with jumps. Then the authors prove the existence of processes using Envelope Snell to find an optimal strategy of our control problem.


Introduction
The main motivation of this paper is to prove the existence of an optimal strategy which maximizes the expected profit of a firm in an infinite horizon problem with jumps. More precisely, let a Brownian motion ðW t Þ t≥0 and an independent Poisson measure μðdt; deÞ defined on a probability space ðΩ; A; ℙÞ and let F be the right continuous complete filtration generated by the pair ðW ; μÞ. Assume that a firm decides at stopping times to change its technology to determine its maximum profit. Let f1; 2g be the possible technologies set. A right continuous left limited stochastic process X models the firm log value and a process ðξ t ; t ≥ 0Þ taking its values in f1; 2g models the state of the chosen technology. The firm net profit is represented by a function f, the switching technology costs are represented by c 1;2 and c 2;1 ; β > 0 is a discount coefficient. Then, the problem is to find an increasing sequence of stopping times b α :¼ ð b τ n Þ n≥−1 ; where b τ −1 ¼ 0; optimal for the following impulse control where A denotes the set of admissible strategies. The Snell envelope tools show that the problem reduces to the existence of a pair of right continuous left limited processes ðY 1 ; Y 2 Þ. This idea originates from Hamad ene and Jeanblanc [1]. Their results are extended to infinite horizon case and mixed processes (namely jump-diffusion with a Brownian motion and a Poisson measure). In [1] the authors considered a power station which has two modes: operating and closed. This is an impulse control problem with switching technology without jump of the state variable. They solved the starting and stopping problem when the dynamics of the system are the ones of general adapted stochastic processes. The existence of ðY 1 ; Y 2 Þ is established via the notion of doubly reflected backward stochastic differential equation. In this context, another interest of our work is to extend to the infinite horizon case the results of doubly reflected backward stochastic differential equations with jumps. Specifically, a solution for the doubly reflected backward stochastic differential equation associated to a stochastic coefficient g; a null terminal value and a lower (resp. an upper) barrier ðL t Þ t≥0 ðresp: ðU t Þ t≥0 Þ is a quintuplet of F-progressively measurable processes ðY t ; Z t ; V t ; K þ t ; K − t Þ t≥0 which satisfies 8 > > > > < > > > > : whereμ is the compensated measure of μ: Another specificity of this paper is to promote a constructive method of the solution of a BSDEs with two barriers. Specifically, we do not assume the so called Mokobodski's hypothesis. Indeed this one is not so easy to check (see e.g. [2] in finite horizon and continuous case). Our assumptions are more natural and easy to check on the barriers in practical cases.
The notion of backward stochastic differential equation (BSDE) was studied by Pardoux and Peng [3] (meaning in such a case L ¼ −∞; U ¼ þ∞ and K ± ¼ 0). To our knowledge, they were the first to prove the existence and uniqueness of adapted solutions, under suitable square-integrability and Lipschitz-type condition assumptions on the coefficients and on the terminal condition. Several authors have been attracted by this area that they applied in many fields such as Finance [1,[4][5][6], stochastic games and optimal control [7][8][9][10], and partial differential equations [11].
The existence and the uniqueness of BSDE solutions with two reflecting barriers and without jumps have been first studied by Cvitanic and Karatzas [4] (generalization of El Karoui et al. [5]) applied in Finance area by El Karoui et al. [6]. There is a lot of contributions on (1) a1 − dimensional Brownian motion W ¼ ðW t Þ t≥0 : (2) a point process N t :¼ R t 0 R E eμðds; deÞ associated with a Poisson random measure μ on ℝ þ 3 E; where E ¼ ℝnf0g; for some m ≥ 1 endowed with its Borel σ-algebra E, with compensator νðdt; deÞ ¼ dtλðdeÞ; for a σ-finite measure λ on ðE; EÞ; R E ð1∧jej 2 ÞλðdeÞ < ∞;μ :¼ μ − ν denotes the compensated measure associated with μ: Assume that a firm decides at random times to switch the technology in order to maximize its profit: the firm switches from the technology 1 to the technology 2 along a sequence of stopping times. An impulse control strategy is defined as a sequence α :¼ ðτ n Þ n≥−1 ; where ðτ n Þ n≥−1 is a sequence increasing to infinity of F-stopping times with τ −1 ¼ 0. The sequence ðτ n Þ models the impulse time sequence of the system as follows: for every n ≥ 0; τ 2n is the time when the firm moves from technology 1 to technology 2 and τ 2nþ1 is the time when the firm goes from 2 to 1. A c adl ag process ðξ t Þ taking its values in f1; 2g is defined by Given K > 0 and a measurable map γ : the firm value is defined as S t :¼ exp X t ; t ≥ 0; where ðX t Þ is the c adl ag process where X 0 ∈ ℝ is the initial condition, b : ℝ → ℝ and σ : ℝ → ℝ are two measurable functions satisfying the K-Lipschitz condition (thus the sublinear growth condition). The instantaneous net profit of the firm is given in terms of a positive function f, depending on the technology in use and the value of the firm. Let c 2;1 and c 1;2 be the positive switching technology costs, c i;j if one passes from technology i to technology j; with regular enough assumptions which will be specified later. One considers a discount coefficient β > 0 then, the profit associated with a strategy α is defined as and the expected profit of the firm is defined by

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Here, the impulse control problem is to prove the existence of an admissible strategy b α which maximizes the expected profit: The following notations will be used: (1) T :¼ fθ : F − stopping timeg; T t :¼ fθ ∈ T : θ ≥ tg: (2) P :¼ f F − progressively measurable c adl ag processesg: (3) C 2 :¼ fðX t Þ t≥0 ∈ P : such that E½sup t≥0 jX t j 2 < ∞g: (6) P d the σ algebra of F-predictable sets on Ω 3 ½0; þ ∞½: (10) Class [D] : fprocesses U : ðU θ ; θ ∈ T Þ uniformly integrableg: 3. The impulse control problem Section 5 shows that the problem reduces to the existence of a pair of c adl ag processes ðY 1 ; Y 2 Þ using the Snell envelope tools: this idea originates from Hamad ene and Jeanblanc [1]. The existence of ðY 1 ; Y 2 Þ is established in Section 5 via the reflected BSDEs tools. Indeed, the solution of the reflected BSDE corresponds to the value function of an optimal stochastic control problem and these processes allow to build an optimal switching strategy. We based on [17] to use the fundamental optimal control concepts. Proposition 3.1. Assume that there exist two right continuous left limited, regular (meaning that the predictable projection coincide with the left limit) ℝ-valued processes and satisfying the properties is optimal for the impulse control problem (6). The proof is based on the properties of the Snell envelope. The scheme of the proof is similar to the one in [18] and also [14, Appendix A, p. 246] as soon as the processes Y i are regular. As a consequence of (7) and (8), remark that almost surely

Reflected BSDE in case of a single barrier, infinite horizon
In this subsection, the case of infinite horizon reflected BSDE with one barrier and general jumps is considered.

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(3) almost surely L t ≤ Y t ; (4) ðK t Þ is a non-decreasing process satisfying E½ð R ∞ 0 dK s Þ 2 < ∞; K 0 ¼ 0; and for any t Z t 0 ðY s− À L s ÞdK s ¼ 0; ℙ-a:s: We then prove the following: Proof: (1) As a first step, the uniqueness of the solution is insured: if there exist two solutions, the proof of uniqueness is a standard one. For instance, look at Theorem 4.8 proof.
(2) Under the hypothesis E½sup t ðL t Þ 2 < ∞; Theorem 2.1 [10] can be applied: there exists a quadruplet ðY T ; Z T ; K T ; V T Þ verifying Y T ∈ C 2 ; Z T ∈ ℍ 2 ; V T ∈ L 2 ; (actually restricted to t ∈ ½0; T) and ∀t ≤ T : Considering T ≤ S; S; T ∈ ℝ þ ; one has ∀s ≤ T : Applying Itô's formula to the process s → ðY S s − Y T s Þ 2 between t and T yields Using Considering the decomposition: the Lipschitz property of the function g, the Cauchy-Schwarz inequality and the nonincreasing property of the map y↦gðt; y; z; vÞ for any ðt; z; vÞ and α > 0 lead to

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Remark that Δ s ðY T − Y S Þ includes the jumps of the Poisson measure μ. So Then, since Y S ; Y T ∈ C 2 ; Z S ; Z T ∈ ℍ 2 and V S ; V T ∈ L 2 ; the third line in (16) is a martingale; thus taking the expectation of both sides with α ¼ 2C yields for any t ≤ T On the one hand, Lemma 7.2 , we obtain for any ε > 0; T and any S: where wðTÞ : On the other hand, we have and from the Lipschitz property, we get T e −βs f ðs; 0; 0; 0Þds: Using estimation (19), there exists a constant M, such that for any T; S : If we subtract from wðTÞ the term εEð

Infinite horizon impulse control problem
This implies that the expectation on the left tends to zero uniformly when ε is chosen small enough: indeed, since sup s L þ s ∈ L 2 ; by Lebesgue's monotone convergence E sup s≥T ðL þ s Þ 2 tends to 0 when T tends to infinity. Globally wðTÞ → 0 when T tends to infinity and we obtain using (19) that the sequence ðY T Þ is a Cauchy sequence which converges in L 2 ðΩÞ to the process Y. Thus Lemma 7.2 concludes that, t being fixed, ðY T t ; T ≥ tÞ is a Cauchy sequence in L 2 ðΩ; F t ; ℙÞ, its limit defines the F t -measurable random variable Y ðt; :Þ: It is a family of random variables. We later prove that actually the limit Y is a process.
(3) Turning to Z and V, to deal with the convergence in ℍ 2 respectively in L 2 , An argument similar to (63) shows that the sequence ðZ T Þ is a Cauchy sequence in ℍ 2 , its limit defines a process Z which belongs to ℍ 2 and ðV T Þ is a Cauchy sequence in L 2 , its limit defines a process V which belongs to L 2 .
(4) We now prove that there exists a process Y ∈ C 2 which is the limit of a Cauchy sequence in C 2 .
(a) Coming back to (16), for all α > 0; we get The Burkholder-Gundy-Davis inequality gives the existence of a constant C 1 > 0 such that Choosing α and γ such that 1 − Cα β − 2C 1 γ > 0; using Lemma 7.2 and the facts that ðZ T ; T ≥ tÞ is a Cauchy sequence in ℍ 2 , and ðV T ; T ≥ tÞ is a Cauchy sequence in L 2 , then E½sup t≤T ðY S t − Y T t Þ 2 goes to 0 when S and T go to infinity.
(5) Now one proves the other items of the proposition: Item (2) According to (4.1) for all and due to the almost sure convergence of a subsequence of ðY T ; Z T ; V T Þ and the continuity of the function g, the right hand side of Eqn (22) converges almost surely. Thus R t 2 t 1 dK s is defined as the L 2 and almost sure limit of the right hand side of (22). Hence, for almost sure limit, we get the reflected BSDE (10).
Item (3) For any T ≥ t; one has L t ≤ Y T t , and using almost convergence of a subsequence, one deduces Item (3).
Item (4) The L 2 convergence in (22) proves that R ∞ t dK s ∈ L 2 : Moreover, for all T using (12), we get: On the one hand, for fixed ðω; tÞ ∈ Ω 3 ½0; T the left continuous and right limited function s → Y s− − L s is the uniform limit on ½0; t of a sequence ðf k ðωÞ; kÞ of step functions: We now deal with the successive bounds For ðω; tÞ fixed above, for any ε > 0 there exists Nðω; tÞ such that Infinite horizon impulse control problem so the first and third terms in (24) are bounded Remark that lim T→∞ ðK T t þ K t ÞðωÞ ¼ 2εK t ðωÞ: We now fix k ≥ N ðω; tÞ; and we remark that for any step function h : Thus when T goes to infinity the second term in (24) satisfies For any ε, using (25) and (26) the limit of (24) when T goes to infinity is bounded by 2εK t ðωÞ: This yields the fact that lim which goes to 0 when T goes to infinity according to the convergence of Y T to Y in C 2 and of K T in L 2 : So the proof of (4) is done.
-In case of a deterministic function g, meaning g is defined on ℝ þ 3 ℝ 3 ℝ d , an alternative proof of Theorem 4.3 (under the same hypotheses) can be provided using penalization method, as for instance Section 6 in [6] concerning continuous case, but here directed by a pair Brownian motion-Poisson measure. We associate to ðg k ðs; y; zÞ :¼ e −βs gðs; y; zÞ þ kðy − L s Þ − Þ where the function g k satisfies Assumption ðH 1 Þ, since g k is obviously non decreasing and uniformly Lipschitz, the solution ðY k ; Since k → g k is non decreasing, the standard comparison theorem proves that actually, for any fixed t ðY k t Þ is a non-decreasing sequence in L 2 , so it is almost surely and in L 2 convergent to the random variable Y t :¼ lim k→∞ Y k t : Using similar arguments as those ones in (4.1) ðY k Þ is a Cauchy sequence in C 2 so the limit defines the C 2 process Y : Now it is standard [19] to prove the existence of a non decreasing process K such that and the existence of Z ; V ∈ ðℍ 2 ; L 2 Þ such that This alternative method allows us to prove the following result.
Proof: The uniqueness of the solution (step (i) in the proof of Theorem 4.2) insures that this solution is the limit of the penalized Eqn (27): Y is the limit of the non-decreasing sequence ðY k Þ: Reproducing Step 2 in the proof of Theorem 3.1 [16] leads for any k to Remark that both J k and J are of class ½D since both are uniformly bounded with R ∞ 0 e −βs jgðsÞjds þ sup t jL t j ∈ L 1 : Let us denote as SN ðY Þ the Snell envelope of process Y. Then Lemma A.1 in Appendix [10,12] allows to commute the increasing limit and the essential supremum: on the left hand side, Y k t ↑Y t almost surely, on the right hand side SNðJ k Þ t ↑SNðJ Þ t which achieves the proof.
-From now on, we consider a function g defined on ℝ þ 3 Ω satisfying Assumption ðH 0 1 Þ: The following is an extension of Lemma 2.4 in [20]: in our case g is defined only on ℝ þ 3 Ω but the BSDE is directed by a mixed Brownian-Poisson process: Then almost surely for all t ≥ 0; Proof: The proof is similar to the one in [18]. The next step follows from Theorem 3.2 [20] or Proposition 4.12 [18].
Infinite horizon impulse control problem and for all t : E Â Proof: (1) By definition, we have dρ s ¼ ðe −βs jgðsÞj À βc 2;1 e −βs Þds À dΠ s þ θ s dW s þ Z EṼ s ðeÞμðde; dsÞ: Using Itô's formula, one has The last term on the right hand side of (32) is bounded: for any Gathering these bounds and using Assumption ðH Let Using extended Gronwall's Lemma 7.1 one has Let us denote ψðtÞ :¼ fðtÞ þ εE½ð R ∞ t dΠ s Þ 2 ; ψ being a decreasing function.

Comparison theorem in case of a single barrier
The following proposition is an extension of Theorem 2.2 in [10] to infinite horizon.
and assume in addition that (1) The F-progressively measurable process ðY n ; Z n ; V n ; K n Þ which is the unique solution of the reflected BSDE associated with ð−e −βt jgðtÞj − nðy − U t Þ þ ; LÞ satisfies (2) The F-progressively measurable process ðρ; θ;Ṽ ; ΠÞ which is the unique solution of the reflected BSDE associated with ð−e −βt jgðtÞj − E½sup s≥t ju s jjF t ; LÞ satisfies Thank to Lemma 4.4, one has the following inequalities: So as a consequence of Proposition 4.6, one has where Y n and K n are introduced in (27). Finally, Lemma 4.5 proves that for all t and all n;

Double barrier reflected BSDE with jumps and infinite horizon
Now one considers the problem of reflection with respect to two barriers L and U in the case of drift g being defined on ℝ þ 3 Ω and satisfying ðH 0 1 Þ.
Definition 4.7. Let ðe −β: g; L; U Þ be given. A solution of the double reflected BSDE associated to ðe −β: g; L; U Þ is a quintuplet of processes ðY ; Z ; V ; K þ ; K − Þ satisfying for any t ≥ 0: (1) Y ∈ C 2 and Z ∈ ℍ 2 ; V ∈ L 2 , (2) almost surely (3) almost surely L t ≤ Y t ≤ U t ; (4) ðK ± t Þ are non-decreasing processes satisfying E½ð  Proof: The proof of uniqueness is detailed, even if it is really standard, for stressing the role of the assumption L s < U s : One assumes that there exist two solutions ðY i ; Z i ; V i ; K ±i Þ; i ¼ 1; 2: Then they satisfy Ãμ ðds; deÞ: One has So according to Theorem 4.2, Hypothesis (H2) still being in force, for each n ∈ ℕ Ã ; there exists a unique solution ðY n ; Z n ; V n ; K n Þ of the reflected BSDE associated with ðe −βt g ðt; ωÞ − nðy − U t Þ þ ; LÞ, meaning Infinite horizon impulse control problem From Proposition 4.6, the sequence ðY n ; n ≥ 1Þ (resp K n ; n ≥ 1) is non increasing (resp. non-decreasing), let us denote Y ; K þ their almost sure limits, consequence of monotonicity.
The proof of Theorem 4.8 is done in five steps.
Step 1: There exists a constant C ≥ 0 such that ∀n ≥ 0 and ∀t ≥ 0; one has Using the Cauchy-Schwarz inequality, for any e > 0, one has for any t and n

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On the other hand, with (38), Lemmas 4.4 and 4.5, for any t and n one has: Similarly for any c 1 > 0 one has Note that the last line in the right hand side of (4.3) admits a zero expectation, and embedding the inequalities (44), (45) and (17) in the expectation of (4.3): where k is the function defined as follows: E½e −βs βc 1;0 ds 2 < ∞: Gronwall's Lemma 7.1 is now used with Infinite horizon impulse control problem Then one has a bound for (46) This bound and (44) end the proof.
Let T < ∞ and ν be a stopping time such that: t ≤ ν < ∞. Itô's formula is applied to the process ðe −nsỸ n s ; s ≥ 0Þ between ν and T ∨ ν: This last bound R ∞ ν e −nðs−νÞ dΠ s goes to 0 when n goes to infinity using Lebesque monotonous convergence Theorem. Consequentlỹ Y n ν → U ν 1 ν<∞ in L 2 ðΩ; ℙÞ as n → ∞: Therefore lim n Y n ν ≤ lim nỸ n ν ≤ U ν P-a.s. From this and "Section Theorem" [21, p. 220], it follows that, ℙ − a:s:; Y t ≤ U t ; ∀t and then ðY n t − U t Þ þ a0 ℙ − almost surely. We now denote by p X the predictable projection for any X. Since Y n ≥ Y, then p Y n ≥ p Y and p Y ≤ p U. So we deduce that p Y n a p Y ≤ p U, the semi-martingale U is regular and Lemma 7.3 proves that the processes Y n are regular so Y n Consequently, from a weak version of the Dini theorem [22, p. 202], one deduces that sup t≥0 ðY n t − U t Þ þ a0 ℙ − a:s: as n → ∞: Finally Lebesgue dominated convergence Theorem implies Step 3: There exist an F-adapted process Z ¼ ðZ t Þ t≥0 and an F-predictable process By Itô's formula one has for any p ≥ n ≥ 0 and for all t, where K n− t denotes n R t 0 ðY n s − U s Þ þ ds: Since p ≥ n; then Y p ≤ Y n ; dK n ≤ dK p ; so According to (7) in [20] Look at sup s ðY p s − U s Þ þ R ∞ 0 nðY n s − U s Þ þ ds, product of sup s ðY p s − U s Þ þ going to 0 when Step 2) and of R ∞ 0 nðY n s − U s Þ þ ds which is for all n bounded by the integrable random variable R ∞ 0 ½e −βs βc 1;0 ds (see Lemma 4.4): The second term in (49) is symmetrical and the sum is going to 0 in L 1 : Finally, taking the expectation of the left hand side in (48) and using (17) lim n;p→∞ It follows that ðZ n Þ n≥0 and ðV n Þ n≥0 are Cauchy sequences in complete spaces then there exist processes Z and V, respectively F-progressively measurable and P ⊗ E-measurable such that the sequences ðZ n Þ n≥0 and ðV n Þ n≥0 converge respectively toward Z in ℍ 2 and V in L 2 .
Step 4: lim n;p→∞ E½sup t≥0 Y n t − Y p t 2 ¼ 0 so lim n Y n defines a process in C 2 : Using Y n and Y p definitions, n ≥ p (so dK n ≥ dK p ) and applying Itô's formula between 0 and t to the process t → ðY n t − Y p t Þ 2 one has: (1) First look at For any c > 0; the right hand side of this inequality is smaller than

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(2) Using (49), the expectation of the last term in (52) is bounded: which actually goes to 0 when n and p go to infinity using (50).
Concerning the supremum with respect to t of the absolute value of second line in (52) the Burkholder-Davis-Gundy and Cauchy-Schwarz inequalities are used: there exists a universal constant C 1 such that for any constant c > 0: deÞ is an F-martingale (see [8], p. 4) and once again the Burkholder-Davis-Gundy and Cauchy-Schwarz inequalities are used: Using that sup s≤t jY s− j ≤ sup s≤t jY s j and gathering all these bounds, it yields for any t: Choosing c such that cð1 þ 2C 1 Þ < 1 and using the limit (50), the processes Z n , V n are Cauchy sequences respectively in ℍ 2 ; L 2 and the almost surely convergent monotonous sequences ðY n 0 Þ; ð R 0 dK n s Þ, ð R 0 dðK c Þ n s Þ are Cauchy sequences in L 2 so is the sequence Thus the sequence ðY n Þ is a Cauchy sequence in C 2 : This concludes Step 4 and proves item (1): Moreover, since for all t Y t is an almost sure limit of Y n t and ðY n Þ is C 2 Cauchy sequence, one has two progressively measurable cadlag processes which are modification of each other so that Y ¼ ðY t Þ t≥0 is an F-adapted right continuous left limited process belonging to C 2 .
Step 5: Existence of K − ; Item (4), Item (3) By definition of K n− ; for any n ≥ 0 and t ≥ 0: So, the right hand side of (53) converges almost surely and in L 2 to and the non-decreasing process K − can be defined almost surely and in L 2 : This proves Item (2) and the existence of the non-decreasing process K − in L 2 such that R t 0 dK − s ∈ L 2 . Then, using the differential of Equation (53) and multiplying by Y s− − U s yield almost sure convergence: The right hand side is almost surely finite since it is equal to Remark that the sequence ðY s− − U s ÞðY n s − U s Þ þ goes almost surely to ðY s− − U s Þ ðY s − U s Þ þ ; and multiplied by n the limit cannot be finite unless ðY s − U s Þ þ ¼ 0; thus Item (4) is proved: AJMS Finally Item (3) is a consequence of (1) the fact L t ≤ Y n t for any n and t; and the almost sure convergence of sequence ðY n t Þ, so L t ≤ Y t , (2) above (55) gives Y t ≤ U t .

Application to the impulse control problem with infinite horizon
In this section we use Proposition 3.1, and Theorem 4.8 with g : ðt; ωÞ → f ð1; X t ðωÞÞ − f ð2; X t ðωÞÞ satisfying Assumption ðH 0 1 Þ, a null terminal value, and barriers L t ¼ −c 1;2 e −βt ≤ 0; U t ¼ c 2;1 e −βt ≥ 0; satisfying Assumptions ðH 2 Þ. There exists a progressively measurable process ðY ; Z ; K þ ; K − ; V Þ such that: So the main result can be proved: the existence of processes ðY 1 ; Y 2 Þ introduced in Proposition 3.1. This is the extension of Theorem 3.2 [1, p. 186] to the infinite horizon set up with jumps.
Theorem 5.1. Assume that f ð1; X t Þ and f ð2; X t Þ are positive, t → f ði; X t Þ; i ¼ 1; 2; satisfy ðH 0 1 Þ, L t :¼ −e −βt c 1;2 and U t :¼ e −βt c 2;1 satisfies ðH 2 Þ. Then there exists a couple of ℝ-valued processes ðY 1 t ; Y 2 t Þ t≥0 satisfying the assumptions in Proposition 3.1, in particular (7) and (8) meaning: Since the random variables R þ∞ t dK ± s are integrable and f ði; X s Þ; i ¼ 1; 2 satisfy ðH 0 1 Þ, the following processes will be checked to satisfy Proposition 3.1 assumptions: Y i are positive right continuous left limited regular processes of class [D] satisfying (7) and (8). The following processes are proposed: (1) First one remarks that Y i t ≥ 0 as conditional expectation of non-negative random variables.
(2) Second e −βs f ði; X s Þds À K ± t are sum of an F-martingale minus a right continuous left limited finite variation process so these processes are right continuous left limited.
(3) Third one has E½sup t≥0 jY i t j 2 < ∞; i ¼ 1; 2: indeed, using the facts that f ði; :Þ and R t 0 dK ± s are positive, The facts that R ∞ 0 dK ± s ∈ L 2 , Assumption ðH 0 1 Þ and ð R ∞ 0 e −βs f ði; X s ÞdsÞ 2 ≤ 1 β R ∞ 0 e −βs f 2 ði; X s Þds belongs to L 1 , proves that the martingale M i which bounds Y i is uniformly square integrable. Thus Burkholder-Davis-Gundy inequality applied to this square integrable martingale M proves that E½sup t≥0 jY i t j 2 < ∞: As a byproduct, the process Y i is of class [D] since for any stopping time θ, 0 ≤ Y i θ ≤ sup t≥0 jY i t j ∈ L 2 : (4) Fourthly Y i are regular using the same argument as in [12]: the regularity of Y i is equivalent to the regularity of K ± ; and this one is equivalent to the regularity of Y defined by the system ðSÞ: Lemma 7.3 in Appendix insures this property.
t ; t ≥ 0; according to Proposition 3.1, an optimal strategy b α ¼ ðτ n Þ n≥0 is defined by

Numerical resolution
Recall that the optimal strategy b α ¼ ðb τ n Þ n≥0 is completely defined by the process Y and is obtained when Y reached successively the barriers L and U. As a result, solving numerically this strategy amounts to simulating sample path trajectories of the process Y. In recent years, several techniques have been proposed for the numerical solution of the process Y (for example the quantization algorithm, Malliavin calculus). Here the approximation by regression is chosen, which is well explained in [24,25]. Our method is totally different from the method used in [26] which is based on the approximation of the Brownian and Poisson processes by a random walk. Recall once again that here the process X is the diffusion (4). For this application, a simple case of stochastic differential equation with jump is considered: Let b; σ are constant drift and diffusion coefficients;μðds; deÞ gives an information about the jump: the probability of the jump happening at time t and the relative amplitude of the jump. It will be represented by a log-normal random variables, λ is the yearly average of the number of jumps. Thus the firm log-value is modeled as By using the classical Euler scheme for sample path trajectories of the process X where λ ¼ 3; x 0 ¼ 1 and T ¼ 1, one has: (see Figures 1 and 2). Let us now focus on our problem: namely, how to simulate the process Y, and therefore the optimal strategy. Recall that e −βt gðtÞ ¼ e −βt ðf ð1; X t Þ À f ð2; X t ÞÞ; L t ¼ −c 1;2 e −βt ; U t ¼ c 2;1 e −βt ; c 1;2 and c 2;1 > 0 which satisfy Hypotheses ðH 1 Þ and ðH 2 Þ.
First of all, when t tends to infinity, Y t goes to 0, so a finite horizon T should be fixed such that t i ¼ i T n ; i ¼ n; . . . ; 0: More specifically, below the numerical samples show that as soon as t ≥ 1; the length of interval ðL t ; U t Þ is negligible. b ¼ 1; σ ¼ 2 AJMS Y t ∈ ðL t ; U t Þ so the error is bounded by U t − L t , the order of which being e −βt : To approximate the backward component Y ; the following discretization approximation scheme is introduced, for 0 ¼ t 0 < t 1 < . . . < t n ¼ T: where E t i ¼ E½:jF t i : To approximate the conditional expectation, here is adopted the Longstaff-Schwarz algorithm [25] which uses a regression technique (Least-Square Monte Carlo method). Taking the parameters β ¼ 0:5 , X 0 ¼ 1; b ¼ 1; σ ¼ 2; and the profits/costs functions f ð1; xÞ ¼ 3 þ 2x − ; f ð2; xÞ ¼ 2x þ ; so f ð1; xÞ À f ð2; xÞ ¼ −2x þ 3; the evolution of Y is observed. Previously all the assumptions have to be checked: (1) One notes that with X t ¼ bt þ σW t þ N t − λt; : Assumption ðH 0 1 Þ is satisfied since f ði; xÞ ¼ a þ x ± ; so E½ða þ X ± t Þ 2 ≤ 2a 2 þ 2E½ðX 2 t Þ ≤ 2a 2 þ 6ðb 2 t 2 þ σ 2 t þ λtÞ thus E½ R ∞ 0 e −βt f 2 ði; X s Þds ≤ R ∞ 0 e −βt ½2a þ 6ðb 2 t 2 þ σ 2 t þ λtÞdt < ∞: Interpretation: Recall once again that the optimal strategy b α ¼ ðb τ n Þ n≥0 is obtained when Y reached successively the barriers L and U. In Figures 3 and 4, the costs are higher than in Figures 5 and 6. In Figures 3 and 4, it could be not interesting to switch the technology. It is preferable that the firm takes the precaution of keeping long enough the technology 1, which will enable to obtain suitable expected profit.
In the case of reasonable costs, as in Figures 5 and 6, the firm can switch the technology more often: actually at times τ 0 ∼ 0:15 and τ 1 ∼ 0:97 ( Figure 5), respectively in Figure 5, the firm can switch the technology at times τ 0 ∼ 0:05 and τ 1 ∼ 0:23.   Infinite horizon impulse control problem