Infinite horizon impulse control problem with jumps and continuous switching costs

Rim Amami, Monique Pontier, Hani Abidi

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Open Access. Article publication date: 16 February 2021

Issue publication date: 11 January 2022

2875

Abstract

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.

Keywords

Citation

Amami, R., Pontier, M. and Abidi, H. (2022), "Infinite horizon impulse control problem with jumps and continuous switching costs", Arab Journal of Mathematical Sciences, Vol. 28 No. 1, pp. 2-36. https://doi.org/10.1108/AJMS-10-2020-0088

Publisher

:

Emerald Publishing Limited

Copyright © 2020, Rim Amami, Monique Pontier and Hani Abidi

License

Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode


1. 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 (Wt)t0 and an independent Poisson measure μ(dt,de) defined on a probability space (Ω,A,) and let F be the right continuous complete filtration generated by the pair (W,μ). Assume that a firm decides at stopping times to change its technology to determine its maximum profit. Let {1,2} be the possible technologies set. A right continuous left limited stochastic process X models the firm log value and a process (ξt,t0) taking its values in {1,2} models the state of the chosen technology. The firm net profit is represented by a function f, the switching technology costs are represented by c1,2 and c2,1,β>0 is a discount coefficient. Then, the problem is to find an increasing sequence of stopping times α^:=(τn^)n1, where τ^1=0, optimal for the following impulse control problem

K(α^,i,x):=esssupαAEi,x[0+eβsf(ξs,Xs)dsn0{eβτ2nc1,2+eβτ2n+1c2,1}],
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 (Y1,Y2). This idea originates from Hamadène and Jeanblanc []. Their results are extended to infinite horizon case and mixed processes (namely jump-diffusion with a Brownian motion and a Poisson measure). In [] 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 (Y1,Y2) 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 (Lt)t0(resp.(Ut)t0) is a quintuplet of F-progressively measurable processes (Yt,Zt,Vt,Kt+,Kt)t0 which satisfies

(1){Yt=t+eβsg(s)ds+t+dKs+t+dKst+ZsdWstEVs(e)μ˜(ds,de),LtYtUt,t00t(YsLs)dKs+=0t(YsUs)dKs=0,a.s.
where μ˜ 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. [] 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 [] (meaning in such a case L=,U=+ and K±=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 [], stochastic games and optimal control [], and partial differential equations [].

The existence and the uniqueness of BSDE solutions with two reflecting barriers and without jumps have been first studied by Cvitanic and Karatzas [] (generalization of El Karoui et al. []) applied in Finance area by El Karoui et al. []. There is a lot of contributions on this subject since then, consisting essentially in weakening the assumptions, adding jumps and considering an infinite horizon.

The extension to the case of BSDEs with one reflecting barrier and jumps has been studied by Hamadène and Ouknine [] considering a finite horizon T=1. The authors show the existence and uniqueness of the solution using the penalization scheme and the Snell envelope tools. They stress the connection between such reflected BSDEs and integro-differential mixed stochastic optimal control. The authors' assumptions are: the terminal value is a square integrable random variable, the drift coefficient function g(t,ω,y,z,v) is uniformly Lipschitz with respect to (y,z,v) and the obstacle (St)t1 is a right continuous left limited process whose jumps are totally inaccessible. Hamadène and Ouknine [] deal with reflected BSDEs in finite horizon, the barrier being right continuous left limited and progressively measurable. Hamadène and Hassani [] proved existence and uniqueness results of local and global solutions for doubly reflected BSDEs driven by a Brownian motion and an independent Poisson measure in finite horizon. The authors applied these results to solve the related zero-sum Dynkin game.

Here the model is inspired from the papers []. But their results do not apply directly to the situation which here requires an infinite horizon. Moreover we connect the reflected BSDE with the impulse control problem. All these papers provide a solution to the reflected BSDE problem which are here extended to the case of infinite horizon by adding a discount coefficient and imposing admissibility conditions of strategies. In this paper, the drift function is assumed to be Lipschitz and non-increasing in y. It is proved that the reflected BSDE solutions are limit of Cauchy sequences in appropriate complete metric spaces. Another interesting area is the one of oblique reflections, meaning a multimodal switching problem, see for instance []. El Asri [] considers the same problem proposed by Hamadène and Jeanblanc [] and extends it to the infinite horizon case without jump of the state variable, namely a power station which produces electricity and has several modes of production (the lower, the middle and the intensive modes). Naturally, the switching from one mode to another induces costs. The optimal switching problem is solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. Moreover their proofs are based on the verification theorem and the system of variational inequalities that we do not use.

Our purpose is similar to the one in [], but instead of using Snell envelope and fixed point theorem as they do, here the two barriers case is solved using comparison theorem in one barrier case and adding some assumptions on the drift coefficient g.

This paper is composed of six sections. presents the impulse control problem and describes the corresponding model. introduces a pair of right continuous left limited processes (Y1,Y2) that allows one to exhibit an optimal strategy. extends the doubly reflected BSDEs tools in the infinite horizon setting with jumps: first the case of a single barrier with general Lipschitz drift is solved, then a comparison theorem is proved, finally the uniqueness and the existence of solution for the doubly reflected BSDE under suitable assumptions are proved in case of drift non depending on state (y,z,v). proves the existence of the required pair (Y1,Y2), and provides an application of these doubly reflected BSDE to a switching problem. Finally, with some simulations, the results allow to define an optimal strategy in . An appendix is devoted to an extension of Gronwall's lemma and some technical results.

2. Preliminaries and problem formulation

Let (Ω,F,) be a filtered complete probability space with a right continuous complete filtration F=(Ft)t0, generated by the two following mutually independent processes:

  1. a1dimensional Brownian motion W=(Wt)t0.

  2. a point process Nt:=0tEeμ(ds,de) associated with a Poisson random measure μ on +×E, where E=\{0}, for some m1 endowed with its Borel σ-algebra , with compensator ν(dt,de)=dtλ(de), for a σ-finite measure λ on (E,),E(1|e|2)λ(de)<; μ˜:=μν 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:=(τn)n1, where (τn)n1 is a sequence increasing to infinity of F-stopping times with τ1=0. The sequence (τn) models the impulse time sequence of the system as follows: for every n0,τ2n is the time when the firm moves from technology 1 to technology 2 and τ2n+1 is the time when the firm goes from 2 to 1. A càdlàg process (ξt) taking its values in {1,2} is defined by

(2)ξt:=n01[τ2n1,τ2n[(t)+2n01[τ2n,τ2n+1[(t).

Given K>0 and a measurable map γ:×E such that

(3)supeE|γ(0,e)|KandsupeE|γ(x,e)γ(x,e)|K|xx|x,x,
the firm value is defined as St:=expXt,t0, where (Xt) is the càdlàg process
(4)Xt:=X0+0tb(Xs)ds+0tσ(Xs)dWs+0tEγ(Xs,e)μ˜(ds,de),
where X0 is the initial condition, b: and σ: 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 c2,1 and c1,2 be the positive switching technology costs, ci,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

k(α):=0+eβsf(ξs,Xs)dsn0{eβτ2nc1,2+eβτ2n+1c2,1},
and the expected profit of the firm is defined by
(5)K(α,i,x):=Ei,x[0+eβsf(ξs,Xs)dsn0{eβτ2nc1,2+eβτ2n+1c2,1}].
Definition 2.1.

The strategy α:=(τn)n1 is admissible if:

0+eβsf(ξs,Xs)dsandn0{eβτ2nc1,2+eβτ2n+1c2,1}
belong to L1(Ω,F,). We denote by A the set of admissible strategies.

Here, the impulse control problem is to prove the existence of an admissible strategy α^ which maximizes the expected profit:

(6)K(α^,i,x):=supαAEi,x[0+eβsf(ξs,Xs)dsn0{eβτ2nc1,2+eβτ2n+1c2,1}].

The following notations will be used:

  1. T:={θ:Fstoppingtime},Tt:={θT:θt}.

  2. P:={ Fprogressivelymeasurablecàdlàgprocesses}.

  3. C2:={(Xt)t0P:suchthatE[supt0|Xt|2]<}.

  4. 1:={(Xt)t0P:suchthatE[0|Xt|2dt]<}.

  5. 2:={(Xt)t0P: such thatE[0|Xt|2dt]<}.

  6. Pd the σ algebra of F-predictable sets on Ω×[0,+[.

  7. L2:{V:Ω×[0,+]×E,Pdmeasurables.t.E[0E|Vs(e)|2λ(de)ds]<}

  8. Lp:=Lp(Ω,F,),p=1,2.

  9. p:=p(Ω,F,),p=1,2.

  10. Class [D] : {processesU:(Uθ,θT)uniformlyintegrable}.

3. The impulse control problem

shows that the problem reduces to the existence of a pair of càdlàg processes (Y1,Y2) using the Snell envelope tools: this idea originates from Hamadène and Jeanblanc []. The existence of (Y1,Y2) is established in 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 [] 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) -valued processes Y1=(Yt1)t0 and Y2=(Yt2)t0 of class [D] and satisfying the properties

(7)Yt1=esssupθTt E[tθeβsf(1,Xs)dseβθc1,2+Yθ2|Ft]
(8)Yt2=esssupθTtE[tθeβsf(2,Xs) dseβθc2,1+Yθ1|Ft]Y1=Y2=0,ci,j>0.
where f(i,.) are positive functions satisfying 0eβsf2(i,Xs)dsL1,i=1,2. Then Y01=supαAK(α,1,x). Moreover, the strategy α^=(τn)n0 defined as follows:
τ1:=0τ2n:=inf{t>τ2n1,Yt1eβtc1,2+Yt2}, n0τ2n+1:=inf{t>τ2n,Yt2eβtc2,1+Yt1}
is optimal for the impulse control problem .

The proof is based on the properties of the Snell envelope. The scheme of the proof is similar to the one in [] and also [, , p. 246] as soon as the processes Yi are regular. As a consequence of and , remark that almost surely

(9)eβtc1,2Yt1Yt2eβtc2,1.

4. Reflected BSDE with jumps and infinite horizon

In this section, the results from [] are extended to infinite horizon reflected backward stochastic differential equations with general jumps, showing existence and uniqueness of an infinite horizon solution, imposing additional assumptions on the drift function and using appropriate estimates of the process Y. The following assumptions are done:

  1. (H1): A map g:Ω×[0,+[×1+d×L2(E,,λ;) which is F-progressively measurable and:

{(t,z,v),yg(t,y,z,v)isnonincreasingalmostsurely,C>0suchthatforanyt0,y,y,z,zd,v,vL2(E,,λ;):|g(t,y,z,v)g(t,y,z,v)|C(|yy|+|zz|+||vv||)a.s.
where the norm of L2(E,,λ;) is defined as v2:=Ev2(e)λ(de).
  1. (H1): An F-progressively measurable map g:Ω×[0,+[ such that 0eβsg2(s)dsL1,

  2. (H2): Let the barriers (Lt)t0 and (Ut)t0 be F-progressively measurable continuous real valued processes satisfying

E[supt0(Lt+)2]<,limt+supLt0Ut, a.s.

To prove the existence of the solution for doubly reflected BSDE with jumps and infinite horizon, we first consider the case of a single barrier () then a comparison theorem is proved in .

4.1 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.

Definition 4.1.

Let (eβ.g,L) be given. A solution of the reflected BSDE associated to (eβ.g,L) is a quadruplet of processes (Y,Z,V,K) satisfying for any t0:

  1. YC2, Z2 and VL2,

  2. almost surely

(10)Yt=t+eβsg(s,Ys,Zs,Vs)ds+t+dKst+ZsdWstEVs(e)μ˜(ds,de),
  1. (3)

    almost surely LtYt,

  2. (4)

    (Kt) is a non-decreasing process satisfying E[(0dKs)2]<,K0=0, and for any t

0t(YsLs)dKs=0,ℙ-a.s.

We then prove the following:

Theorem 4.2.

Let (eβ.g,L) satisfy Hypotheses (Hi),i=1,2. Then there exists a unique process (Y,Z,K,V) solution to the BSDE associated to (eβ.g,L).

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 proof.

  1. (2)

    Under the hypothesis E[supt(Lt)2]<, [] can be applied: there exists a quadruplet (YT,ZT,KT,VT) verifying YTC2,ZT2,VTL2, (actually restricted to t[0,T]) and tT:

(11)LtYtT,YtT=tTeβsg(s,YsT,ZsT,VsT)ds+tTdKsTtTZsTdWstTEVsT(e)μ˜(ds,de),
(12)E(0TdKsT)2<;t0,0t(YsTLs)dKsT=0a.s.

Considering TS,S,T+, one has sT:

(13)d(YsSYsT)=eβs[g(s,YsS,ZsS,VsS)g(s,YsT,ZsT,VsT)]ds[dKsSdKsT]+[ZsSZsT]dWs+E[VsS(e)VsT(e)]μ˜(ds,de).

Applying Itô's formula to the process s(YsSYsT)2 between t and T yields

(14)(YTS)2=(YtSYtT)2+tT(ZsSZsT)2ds+t<sT[Δs(YTYS)]22tTeβs(YsSYsT)[g(s,YsS,ZsS,VsS)g(s,YsT,ZsT,VsT)]ds+2tT(YsSYsT)[ZsSZsT]dWs+2tTE[(YsSYsT)(VsS(e)VsT(e))]μ˜(ds,de)2tT(YsSLs+LsYsT)(dKsSdKsT).

Using (YsSLs)dKsS=(YsTLs)dKsT=0 and LsYsS and YsT, one has

tT(YsSLs+LsYsT)(dKsSdKsT)=tT[(LsYsS)dKsT+(LsYsT)dKsS]0,
so we get
(15)(YtSYtT)2+tT(ZsSZsT)2ds+t<sT[Δs(YTYS)]2(YTS)2+2tTeβs(YsSYsT)[g(s,YsS,ZsS,VsS)g(s,YsT,ZsT,VsT)]ds2tT(YsSYsT)[ZsSZsT]dWs2tTE[(YsSYsT)(VsS(e)VsT(e))]μ˜(ds,de).

Considering the decomposition: g(s,YsS,ZsS,VsS)g(s,YsT,ZsT,VsT)=g(s,YsS,ZsS,VsS)g(s,YsT,ZsS,VsS)+g(s,YsT,ZsS,VsS)g(s,YsT,ZsT,VsS)+g(s,YsT,ZsT,VsS)g(s,YsT,ZsT,VsT), the Lipschitz property of the function g, the Cauchy-Schwarz inequality and the non-increasing property of the map yg(t,y,z,v) for any (t,z,v) and α>0 lead to

2tTeβs(YsSYsT)[g(s,YsS,ZsS,VsS)g(s,YsT,ZsT,VsT)]ds2CtTeβs|YsSYsT|[|ZsSZsT|+VsSVsT]ds2CαtTeβs|YsSYsT|2ds+CαtT|ZsSZsT|2ds+CαtTE|VsS(e)VsT(e)|2λ(de)ds.

Thus for any α>0:

(16)(YtSYtT)2+tT(ZsSZsT)2ds+t<sT[Δs(YTYS)]2(YTS)2+2CαtTeβs|YsSYsT|2ds+CαtT|ZsSZsT|2ds+CαtTE|VsS(e)VsT(e)|2λ(de)ds2tT(YsSYsT)[ZsSZsT]dWs2tTE[(YsSYsT)(VsS(e)VsT(e))]μ˜(ds,de).

Remark that Δs(YTYS) includes the jumps of the Poisson measure μ. So

(17)E[t<sT[Δs(YTYS)]2]=E[tTE(VsT(e)VsS(e))2λ(de)ds].
Then, since YS,YTC2,ZS,ZT2 and VS,VTL2, the third line in is a martingale; thus taking the expectation of both sides with α=2C yields for any tT
(18)E(YtSYtT)2E(YtSYtT)2+12tTE(ZsSZsT)2ds+12EtTE|VsS(e)VsT(e)|2λ(de)dsE(YTS)2+4C2EtTeβs|YsSYsT|2ds.

On the one hand, , we obtain for any ε>0, T and any S:

(19)E(YTS)2exp(4C2+2C+1β)ϕ(T),
where ϕ(T):=1βg2eβT+1εEsupsT(Ls+)2+εE(TSdKsS)2 as defined in . Using Gronwall's lemma, inequality becomes
E(YtSYtT)2ϕ(T)exp(2C2+2C+1β)exp(2C2β).

On the other hand, we have

TSdKsS=YTSTSeβsf(s,YsS,ZsS,VsS)ds+TSZsSdWs+TSVsSμ˜(ds,de),
and from the Lipschitz property, we get
TSeβsf(s,Ys,Zs)dsCTSeβs(|YsS|+|ZsS|+||VsS||)ds+TSeβsf(s,0,0,0)ds.

Using estimation , there exists a constant M, such that for any T,S:

14E(TSdKsS)2Mϕ(T)+1β||f||2eβT.

If we subtract from ϕ(T) the term εE(TSdKsS)2 we get

(20)(14εM)E(TSdKsS)2(1+M)1β||f2||eβT+MεEsupsT(Ls+)2.

This implies that the expectation on the left tends to zero uniformly when ε is chosen small enough: indeed, since supsLs+L2, by Lebesgue's monotone convergence EsupsT(Ls+)2 tends to 0 when T tends to infinity. Globally ϕ(T)0 when T tends to infinity and we obtain using that the sequence (YT) is a Cauchy sequence which converges in L2(Ω) to the process Y. Thus concludes that, t being fixed, (YtT,Tt) is a Cauchy sequence in L2(Ω,Ft,), its limit defines the Ft-measurable random variable Y(t,.). It is a family of random variables. We later prove that actually the limit Y is a process.

  1. (3)

    Turning to Z and V, to deal with the convergence in 2 respectively in L2, An argument similar to shows that the sequence (ZT) is a Cauchy sequence in 2, its limit defines a process Z which belongs to 2 and (VT) is a Cauchy sequence in L2, its limit defines a process V which belongs to L2.

  2. (4)

    We now prove that there exists a process YC2 which is the limit of a Cauchy sequence in C2.

    (a) Coming back to , for all α>0, we get

    suptT(YtSYtT)2(YTS)2+CαsupsT|YsSYsT|21β+Cα0T|ZsSZsT|2ds+Cα0TE|VsS(e)VsT(e)|2λ(de)ds+2suptT|tT(YsSYsT)[ZsSZsT]dWs|+2suptT|tTE[(Ys–SYs–T)(VsS(e)VsT(e))]μ(ds,de)|.

The Burkholder-Gundy-Davis inequality gives the existence of a constant C1>0 such that

2EsuptT|tT(YsSYsT)[ZsSZsT]dWs|2C1E(0T[YsSYsT]2[ZsSZsT]2ds)1/22C1EsupsT|YsSYsT|(0T[ZsSZsT]2ds)1/2C1[γEsupsT|YsSYsT|2+1γE(0T[ZsSZsT]2ds)],

for all γ>0. Similarly

E[suptT|tTE[(YsSYsT)(VsS(e)VsT(e))]μ˜(ds,de)|]C1[γEsupsT|YsSYsT|2+1γ0TEE[(VsS(e)VsT(e))2]λ(de)ds].

Gathering these bounds yields

(21)(1Cαβ2C1γ)[EsupsT(YsSYsT)2]E(YTS)2]+Cα0T|ZsSZsT|2ds+Cα0TE|VsS(e)VsT(e)|2λ(de)ds+C1γ[E(0T[ZsSZsT]2ds)+0TEE[(VsS(e)VsT(e))2]λ(de)ds].

Choosing α and γ such that 1Cαβ2C1γ>0, using and the facts that (ZT,Tt) is a Cauchy sequence in 2, and (VT,Tt) is a Cauchy sequence in L2, then E[suptT(YtSYtT)2] goes to 0 when S and T go to infinity.

  1. (5)

    Now one proves the other items of the proposition: Item (2) According to (4.1) for all t1<t2T

(22)t1t2dKsT=Yt2TYt1Tt1t2eβsg(s,YsT,ZsT,VsT)ds+t1t2ZsTdWs+t1t2EVsT(e)μ˜(ds,de),

and due to the almost sure convergence of a subsequence of (YT,ZT,VT) and the continuity of the function g, the right hand side of converges almost surely.

Thus t1t2dKs is defined as the L2 and almost sure limit of the right hand side of . Hence, for almost sure limit, we get the reflected BSDE .

Item (3) For any Tt, one has LtYtT, and using almost convergence of a subsequence, one deduces Item (3).

Item (4) The L2 convergence in proves that tdKsL2. Moreover, for all T using , we get:

(23)(YsTLs)dKsT=0.

On the one hand, for fixed (ω,t)Ω×[0,T] the left continuous and right limited function sYsLs is the uniform limit on [0,t] of a sequence (fk(ω),k) of step functions:

limksup0st|fsk(ω)(YsLs)|=0.

We now deal with the successive bounds

(24)|0t(YsLs)dKsT0t(YsLs)dKs|0t|YsLsfsk|dKsT+|0tfskdKsT0tfskdKs|+0t|YsLsfsk|dKs.

For (ω,t) fixed above, for any ε>0 there exists N(ω,t) such that

kN,sup0st|fsk(ω)(YsLs)|ε,
so the first and third terms in are bounded
(25)0t|YsLsfsk|dKsT+0t|YsLsfsk|dKsTε(KtT+Kt).

Remark that limT(KtT+Kt)(ω)=2εKt(ω).

We now fix kN(ω,t), and we remark that for any step function h:

0th(s)dKsTT0th(s)dKs.

Thus when T goes to infinity the second term in satisfies

(26)|0tfskdKsT0tfskdKs|0.

For any ε, using and the limit of when T goes to infinity is bounded by 2εKt(ω). This yields the fact that limT0t(YsLs)dKsT=0t(YsLs)dKs.

Finally using we get

0t(YsLs)dKsT=0t(YsLs)dKsT0t(YsTLs)dKsT=0t(YsYsT)dKsT.

Thus

|0t(YsLs)dKsT|sup0st|YsYsT||KtT|
which goes to 0 when T goes to infinity according to the convergence of YT to Y in C2 and of KT in L2. So the proof of (4) is done.

In case of a deterministic function g, meaning g is defined on +××d, an alternative proof of Theorem 4.3 (under the same hypotheses) can be provided using penalization method, as for instance Section 6 in [] concerning continuous case, but here directed by a pair Brownian motion-Poisson measure. We associate to (gk(s,y,z):=eβsg(s,y,z)+k(yLs)) where the function gk satisfies Assumption (H1), since gk is obviously non decreasing and uniformly Lipschitz, the solution (Yk,Zk,Vk) in (C2,2,L2) of the following BSDE

(27)Ytk=t(eβsg(s,Ysk,Zsk)+k(YskLs))dstZskdWstEVsk(e)μ˜(ds,de).

Since kgk is non decreasing, the standard comparison theorem proves that actually, for any fixed t (Ytk) is a non-decreasing sequence in L2, so it is almost surely and in L2 convergent to the random variable Yt:=limkYtk. Using similar arguments as those ones in (4.1) (Yk) is a Cauchy sequence in C2 so the limit defines the C2 process Y. Now it is standard [] to prove the existence of a non decreasing process K such that

tdKs:=limktk(YskLs)ds,
and the existence of Z,V(2,L2) such that
(28)Yt=teβsg(s,Ys,Zs)ds+tdKstZsdWstEVs(e)μ˜(ds,de),
YtLtandtYsdKs=tLsdKs.

This alternative method allows us to prove the following result.

Proposition 4.3.

Under Hypotheses (H1,Hi,i=2), g being defined on +×Ω, one has

(29)Yt=ess supθTtE[tθeβsg(s)ds+Lθ1{θ<}|Ft].

Proof: The uniqueness of the solution (step (i) in the proof of ) insures that this solution is the limit of the penalized : Y is the limit of the non-decreasing sequence (Yk).

Reproducing Step 2 in the proof of Theorem 3.1 [] leads for any k to

Ytk=ess supθTtE[tθeβsg(s)ds+YθkLθ1θ<|Ft],

so (Ytk+0teβsg(s)ds)t is the Snell envelope of the process Jk:t0teβsg(s)ds+YtkLt which is increasing almost surely towards the process J:t0teβsg(s)ds+YtLt. Remark that both Jk and J are of class [D] since both are uniformly bounded with 0eβs|g(s)|ds+supt|Lt|L1.

Let us denote as SN(Y) the Snell envelope of process Y. Then Lemma A.1 in [] allows to commute the increasing limit and the essential supremum: on the left hand side, YtkYt almost surely, on the right hand side SN(Jk)tSN(J)t which achieves the proof. ■

From now on, we consider a function g defined on +×Ω satisfying Assumption (H1).

The following is an extension of Lemma 2.4 in []: in our case g is defined only on +×Ω but the BSDE is directed by a mixed Brownian-Poisson process:

Lemma 4.4.

For n0,let (Yn,Zn,Vn) be the solution of the single barrier reflected BSDE associated to the barrier teβt|g(t)|n(yUt)+, where Ut=c2,1eβt and YTn=0,supt(Lt+)L2. Then almost surely for all t0,

n(YtnUt)+βc2,1eβt.

Proof: The proof is similar to the one in [].

The next step follows from Theorem 3.2 [] or Proposition 4.12 [].

Lemma 4.5.

Let (ρ,θ,V˜,Π) be the solution of the reflected BSDE associated to the barriers L and U:teβt|g(t)|βc2,1eβt. Then there exists a constant C such that

(30)E[(0dΠs)2]C,
(31) andforall t: E[ρt2]C and E(tθs2ds)+E(tEV˜s2(e)λ(de)ds)C.

Proof: (1) By definition, we have

dρs=(eβs|g(s)|βc2,1eβs)dsdΠs+θsdWs+EV˜s(e)μ˜(de,ds).

Using Itô's formula, one has

(32)E((ρt)2)+E(tθs2ds)+E(tEV˜s2(e)λ(de)ds)=2E[t(eβs|g(s)|βc2,1eβt)ρsds]+2E[tρsdΠs].

The last term on the right hand side of is bounded: for any ε>0

2E[tρsdΠs]=2E[tLsdΠs]2E[supst|Ls|tdΠs]1εE[(supst|Ls|)2]+εE[(tdΠs)2]

Gathering these bounds and using Assumption (H1) yield

(33)E(ρt2)+E(tθs2ds)+E(tEV˜s2(e)λ(de)ds)E[t(eβs)ρs2ds]+E[t[eβs|g(s)|2+(βc2,1)2eβs]ds]+1εE[(supstLs)2]+εE[(tdΠs)2].

Let

φ(t):=E[t(eβs|g(s)|2+(βc2,1)2eβs)ds]+1εE[(supstLs)2].

Using extended Gronwall's one has

(34)E[(ρt)2](φ(t)+εE[(tdΠs)2]exp(1β).

Let us denote ψ(t):=φ(t)+εE[(tdΠs)2], ψ being a decreasing function.

(2)Coming back to one has

E(tθs2ds)+E(tEV˜s2(e)λ(de)ds)E[t(eβs+eαs)ρs2ds]+ψ(t),
and from setting γ=1β:
(35) E(tθs2ds)+E(tEV˜s2(e)λ(de)ds)expγ.t(eβs)ψ(s)ds+ψ(t)ψ(t)(1+γexpγ).
Now one turns to the estimate of Π:
tdΠs=ρtt(eβs|g(s)|+βc2,1eβs)ds+tθsdWs+tEV˜s(e)μ˜(ds,de).
So
15E|tdΠs|2E|ρt|2+φ(t)+Etθs2ds+EtEV˜s2(e)λ(de)ds.
Using and
15E|tdΠs|2(φ(t)+εE[(tdΠs)2])(expγ+φ(t)+(1+γexpγ)).
This yields and , as soon as ε is chosen such that
1>5ε(expγ+1+γexpγ).

4.2 Comparison theorem in case of a single barrier

The following proposition is an extension of Theorem 2.2 in [] to infinite horizon.

Proposition 4.6.

Assume that (Y,Z,V,K) and (Y,Z,V,K) are solutions of the reflected BSDE with jumps associated with (g,L) and (g,L), satisfying Assumptions (Hi,i=1,2), g being defined on +×Ω××d, g being defined on +×Ω××d×L2, and assume in addition that

(H):{ almost surely t:g(t,Yt,Zt)g(t,Yt,Zt,Vt).
Then, YtYt -almost surely.

If moreover g is defined on +×Ω××d, then KtKtt0,p.s.

Proof: Theorem 2.2 in [] proves that for any T, almost surely, tT, YtTYtT and in the case where f does not depend on v, dKtTdKtT.

proof gives us the almost sure convergence of YT,KT,Y'T,KT, so the inequalities are preserved when T goes to infinity.■

Here we summarize the results concerning the reflected BSDEs: In case of a function g defined on +×Ω satisfying (H1), the functions

teβtg(t)n(yUt)+,eβt|g(t)|n(yUt)+,eβt|g(t)|βc2,1eβs satisfy Hypothesis (H1): Lipschitz property and non increasingness with respect to y.

  1. The F-progressively measurable process (Yn,Zn,Vn,Kn) which is the unique solution of the reflected BSDE associated with (eβt|g(t)|n(yUt)+,L) satisfies

(36)Ytn=teβs|g(s)|dstZsndWsnt(YsnUs)+ds+tdKsntEVsn(e)μ˜(ds,de),
  1. (2)

    The F-progressively measurable process (ρ,θ,V˜,Π) which is the unique solution of the reflected BSDE associated with (eβt|g(t)|E[supst|us||Ft],L) satisfies

(37)ρt=teβs|g(s)|dstθsdWstβc2,1eβsds+tdΠstEV˜s(e)μ˜(ds,de).

Thank to , one has the following inequalities:

eβtg(t)n(YtnUt)+eβt|g(t)|n(YtnUt)+eβt|g(t)|βc2,1eβt.

So as a consequence of , one has

(38)YnYnρ ;KnKnΠ,
where Yn and Kn are introduced in . Finally, proves that for all t and all n,
(39)E[(tdKsn)2]C.

4.3 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 +×Ω and satisfying (H1).

Definition 4.7.

Let (eβ.g,L,U) be given. A solution of the double reflected BSDE associated to (eβ.g,L,U) is a quintuplet of processes (Y,Z,V,K+,K) satisfying for any t0:

  1. YC2 and Z2,VL2,

  2. almost surely

(40)Yt=t+eβsg(s)ds+t+dKs+t+dKst+ZsdWstEVs(e)μ˜(ds,de)
  1. (3)

    almost surely LtYtUt,

  2. (4)

    (Kt±) are non-decreasing processes satisfying E[(0dKs±)2]< and for any t

0t(YsLs)dKs+=0t(YsUs)dKs=0,-a.s.
Theorem 4.8.

Let (eβ.g,L,U) satisfying Hypotheses (H1),(H2), then there exists a unique solution (Y,Z,K+,K,V) to .

The proof is given in the following subsections.

4.3.1 Uniqueness of the solution

As a first result, one proves the uniqueness of solution when it exists.

proposition 4.9.

If there exists a solution of satisfying Items (1) to (4), it is unique.

Proof: The proof of uniqueness is detailed, even if it is really standard, for stressing the role of the assumption Ls<Us. One assumes that there exist two solutions (Yi,Zi,Vi,K±i),i=1,2. Then they satisfy

d(Ys1Ys2)=[Zs1Zs2]dWs[dKs+1dKs+2]+[dKs1dKs2]E[Vs1(e)Vs2(e)]μ˜(ds,de).
One has
E(Yt1Yt2)2+E[t(Zs1Zs2)2ds+tE(Vs1(e)Vs2(e))2λ(de)ds]E[2t(Ys1Ls+LsYs2)(dKs+1dKs+2)2t(Ys1Us+UsYs2)(dKs1dKs2)].

Using Item (4) (YsiLs)dKs+i=0 and (YsiUs)dKsi=0, the last line satisfies

(41)EtT[(Ys1Ls)dKs+2+(LsYs2)dKs+1]+EtT[(Ys1Us)dKs2(UsYs2)dKs1]0
since LsYsiUs.

It follows that for any t

E(Yt1Yt2)2=tE(Zs1Zs2)2ds=EtE|Vs1(e)Vs2(e)|2λ(de)ds=0.

So Y1=Y2, Z1=Z2, V1=V2 and as a consequence K+1K1=K+2K2. Thus there exists a finite variation process h=K+1K+2=K1K2 satisfying h(0)=0, (YsLs)dhs=0 and (YsUs)dhs=0. But the assumption Ls<Us contradicts these equalities if h0 : indeed as soon as dhs0, (YsLs)=0 and (YsUs)=0 so Ls would be equal to Us. This concludes the proof of uniqueness.

4.3.2 Existence of the solution for double barrier reflected BSDE with jumps

Here one uses the so called penalization method: Let g satisfying (H1) be the drift parameter and introduce h(t,y)=eβtg(t)n(yUt)+ which obviously satisfies (H1).

So according to , still being in force, for each n*, there exists a unique solution (Yn,Zn,Vn,Kn) of the reflected BSDE associated with (eβtg(t,ω)n(yUt)+,L), meaning

(42)Ytn=teβsg(s)dstZsndWsnt(YsnUs)+ds+tdKsntEVsn(e)μ˜(ds,de).

From , the sequence (Yn,n1) (resp Kn,n1) is non increasing (resp. non-decreasing), let us denote Y,K+ their almost sure limits, consequence of monotonicity.

From the inequality LtYtYt1, it follows that Yt=limnYtn belongs to L2 for all t.

The proof of is done in five steps.

  • Step 1: There exists a constant C0 such that n0 and t0, one has

E[(Ytn)2+(nt(YsnUs)+ds+tdKsn)2+t(Zsn)2ds+tE(Vsn(e))2λ(de)ds]C.

Its formula yields

(43)(Ytn)2+t|Zsn|2ds+t<s[Δs(Yn)]2=+2teβsYsng(s)ds+2tYsndKsn2ntYsn(YsnUs)+ds2tYsnZsndWs2tTE(Ysn)Vsn(e)μ˜(ds,de).

By definition of the solution

tYsndKsn=tLsdKsn.

Then, since LsYsn,YsnLsnYsn(YsnUs)+dsnLs(YsnUs)+ds , so t,n:

ntYsn(YsnUs)+dstLsn(YsnUs)+ds.

Thus, one has

tYsndKsnntYsn(YsnUs)+dstLs(dKsnn(YsnUs)+ds).

Using the Cauchy-Schwarz inequality, for any ϵ>0, one has for any t and n

2tLs(dKsn(nYsnUs)+ds)ϵ(t(dKsnn(YsnUs)+ds)2+ϵ1supst(Ls)2.

On the other hand, with , and , for any t and n one has:

(44)t(dKsnn(YsnUs)+ds)2||tdKsn||2+||tn(YsnUs)+ds||2||tdΠs||2+||teβsβc1,0ds||2<.

Similarly for any c1>0 one has

(45)2|teβsYsng(s)ds|teβs{c1|Ysn|2+c11|g(s)|2}ds.

Note that the last line in the right hand side of (4.3) admits a zero expectation, and embedding the inequalities , and in the expectation of (4.3):

(46)E[(Ytn)2+t(Zsn)2ds+tE(Vsn(e))2λ(de)ds]E[c1teβs|Ysn|2ds+c11teβs|g(s)|2ds+ϵ(k(t))2+ϵ1supst(Ls)2],
where k is the function defined as follows:
k(t):=||tdΠs||2+||tE[eβsβc1,0]ds||2<.

So one has

E[(Ytn)2+t(Zsn)2ds+tE(Vsn(e))2λ(de)ds]E[c11teβs|g(s)|2ds+ϵ(k(t))2+ϵ1supst(Ls)2]+c1teβsE|Ysn|2 ds.

Gronwall's is now used with D=E[c110eβs|g(s)|2ds+ϵ(k(0))2+ϵ1sups0(Ls)2] and ψ(s)=c1eβs so

E[(Ytn)2]Dexpc1β
and
c1teβsE[|Ysn|2]dsc1Dexpc1βteβsds=Dc1βeβtexpc1β.

Then one has a bound for (46)

(47)E[(Ytn)2+t(Zsn)2ds+tE(Vsn(e))2λ(de)ds]E[c11teβs|g(s)|2ds+ϵ(k(t))2+ϵ1supst(Ls)2]+Dc1βeβtexpc1βD(1+c1βexpc1β).

This bound and end the proof.

  • Step 2: limnYtnUt and limnE[supt0|(YtnUt)+|]=0.

The proof is an adaptation of the one given in Step 3 [, p. 169].

Let (Y˜n,Z˜n,V˜n,K˜n) be the solution of the reflected BSDE with jumps associated to (eβsg(s)n(yUs),L): so since eβsg(s)n(yUs)+eβsg(s)n(yUs) and both applications (s,y)eβsg(s)n(yUs)+ and eβsg(s)n(yUs) satisfy obviously (H1), implies that -a-s YnY˜n and dK˜ndKn.

Let T< and ν be a stopping time such that: tν<. Itô's formula is applied to the process (ensY˜sn,s0) between ν and Tν:

nensY˜snds+ensdY˜sn=ens[(eβsg(s)+nUs)dsdK˜sn+Z˜sndWs+EV˜sn(e)μ˜(ds,de)].
This yields to
en(Tν)Y˜TνnenνY˜νn=νTνens[(eβsg(s)+nUs)dsdK˜sn+Z˜sndWs+EV˜sn(e)μ˜(ds,de)].

Using that, t,LtY˜tnY˜t1L1, one has limTen(Tν)Y˜Tνn=0. This yields for any n:

Y˜νn=E[νen(sν)(eβsg(s)+nUs)ds+νen(sν)dK˜sn|Fν].

Since U is right continuous then almost surely and in L1

nνen(sν)UsdsUν1ν<,as n.

In addition, one has

E|νen(sν)eβsg(s)ds|12n(E[νe2βsg2(s)ds])12
then due to Assumption (H1)
νen(sν)eβsg(s)ds0 in L1(Ω,) as n.

Finally with

0νen(sν)dK˜snνen(sν)dKsnνen(sν)dΠs.

This last bound νen(sν)dΠs goes to 0 when n goes to infinity using Lebesque monotonous convergence Theorem. Consequently

Y˜νnUν1ν< in L2(Ω,)  as  n.

Therefore limnYνnlimnY˜νnUν P-a.s.

From this and “Section Theorem” [, p. 220], it follows that, a.s.,YtUt,t and then (YtnUt)+0 almost surely.

We now denote by Xp the predictable projection for any X. Since YnY, then YnpYp and YpUp. So we deduce that Ynp Yp Up, the semi-martingale U is regular and proves that the processes Yn are regular so YnU= YnpUpYpUp0. It follows that limn(YtnUt)+=0 for all t almost surely.

Consequently, from a weak version of the Dini theorem [, p. 202], one deduces that supt0(YtnUt)+0 a.s. as n. Finally Lebesgue dominated convergence Theorem implies

E[supt0|(YtnUt)+|2]0 as n.
  • Step 3: There exist an F-adapted process Z=(Zt)t0 and an F-predictable process V=(Vt)t0 such that

limnE[0|ZsnZs|2ds+0E|Vsn(e)Vs(e)|2λ(de)ds]=0.

By Itô's formula one has for any pn0 and for all t,

(48)(YtnYtp)2+t|ZsnZsp|2ds+t<s[Δs(YnYp)]2=2t(YsnYsp)(dKsndKsp)2t(YsnYsp)(dKsndKsp)2t(YsnYsp)(ZsnZsp)dWs2tTE[(YsnYsp)(Vsn(e)Vsp(e))]μ˜(ds,de)
where Ktn denotes n0t(YsnUs)+ds.

Since pn, then YpYn, dKndKp, so

t(YsnYsp)(dKsndKsp)0.
According to (7) in [] (YspYsn)(YsnUs)+(YspUs)+(YsnUs)+, so
(49)2t(YspYsn)(dKsndKsp)=2t(YspYsn)n(YsnUs)+ds2t(YspYsn)p(YspUs)+ds2sups0(YspUs)+0n(YsnUs)+ds+2sups0(YsnUs)+0p(YspUs)+ds.

Look at sups(YspUs)+0n(YsnUs)+ds, product of sups(YspUs)+ going to 0 when p in L2 (Step 2) and of 0n(YsnUs)+ds which is for all n bounded by the integrable random variable 0[eβsβc1,0]ds (see ):

(50)limpsupnE[2sups0(YspUs)+0n(YsnUs)+ds]=0.

The second term in is symmetrical and the sum is going to 0 in L1.

Finally, taking the expectation of the left hand side in and using

(51)limn,p[E[0|ZsnZsp|2ds+0E|Vsn(e)Vsp(e)|2λ(de)ds]]=0.

It follows that (Zn)n0 and (Vn)n0 are Cauchy sequences in complete spaces then there exist processes Z and V, respectively F-progressively measurable and P-measurable such that the sequences (Zn)n0 and (Vn)n0 converge respectively toward Z in 2 and V in L2.

  • Step 4: limn,pE[supt0|YtnYtp|2]=0 so limnYn defines a process in C2.

Using Yn and Yp definitions, np (so dKndKp) and applying Itô's formula between 0 and t to the process t(YtnYtp)2 one has:

(52)(YtnYtp)2=(Y0nY0p)2+0t|ZsnZsp|2ds+st[Δs(YnYp)]2+20t(YsnYsp)(ZsnZs)dWs+20t(YsnYsp)E(Vsn(e)Vsp(e))μ˜(ds,de)20t(YsnYsp)(dKsndKsp)+20t(YsnYsp)(dKsndKsp).
  1. (1)

    First look at

2|0t(YsnYsp)(dKsndKsp)|2supst|YsnYsp|0(dKsndKsp),

For any c>0, the right hand side of this inequality is smaller than

csupst|YsnYsp|2+c1(0(dKsndKsp))2.
  1. (2)

    Using , the expectation of the last term in is bounded:

02E[0t(YsnYsp)(dKspdKsn)]supnE[2sups0(YspUs)+0n(YsnUs)+ds]+suppE[2sups0(YsnUs)+0p(YspUs)+ds]
which actually goes to 0 when n and p go to infinity using .

Concerning the supremum with respect to t of the absolute value of second line in the Burkholder-Davis-Gundy and Cauchy-Schwarz inequalities are used: there exists a universal constant C1 such that for any constant c>0:

E[supst|20s(YunYup)(ZunZup)dWu|]2C1E[0t(YsnYsp)2(ZsnZsp)2ds]2C1E[suput|YunYup|0t(ZsnZsp)2ds]cC1E[suput(YunYup)2]+c1C1E[0(ZsnZsp)2ds].

Similarly one has t0tE[(YsnYsp)(Vsn(e)Vsp(e))]μ˜(ds,de) is an F-martingale (see [], p. 4) and once again the Burkholder-Davis-Gundy and Cauchy-Schwarz inequalities are used:

E[supst|20sE(YunYup)(Vun(e)Vup(e))]μ˜(du,de)|]cC1E[suput(YunYup)2]+c1C1E[0E(Vsn(e)Vsp(e))2λ(de)ds].

Using that supst|Ys|supst|Ys| and gathering all these bounds, it yields for any t:

E[supst(YsnYsp)2]E[(Y0nY0p)2]+c(1+2C1)E[suput(YunYup)2]+E[s(Δs(KnKp))2]+c1C1(E[0(ZsnZsp)2ds]+E[0E(Vsn(e)Vsp(e))2λ(de)ds])+c1E[(0(dKsndKsp)])2)+supnE[2sups0(YspUs)+0n(YsnUs)+ds]+suppE[2sups0(YsnUs)+0p(YspUs)+ds].

Choosing c such that c(1+2C1)<1 and using the limit , the processes Zn, Vn are Cauchy sequences respectively in 2, L2 and the almost surely convergent monotonous sequences (Y0n), (0dKsn), (0d(Kc)sn) are Cauchy sequences in L2 so is the sequence (sΔsKn=0dKsn0d(Kc)sn). Thus the sequence (Yn) is a Cauchy sequence in C2. This concludes Step 4 and proves item (1):

E[sup0s(YsnYsp)2]0 as p,n.

Moreover, since for all t Yt is an almost sure limit of Ytn and (Yn) is C2 Cauchy sequence, one has two progressively measurable cadlag processes which are modification of each other so that Y=(Yt)t0 is an F-adapted right continuous left limited process belonging to C2 .

  • Step 5: Existence of K, Item (4), Item (3)

By definition of Kn, for any n0 and t0:

(53)0tdKsn=YtnY0n+0teβsg(s)ds+0tdKsn0tZsndWs0tEVsnμ˜(ds,de).

So, the right hand side of converges almost surely and in L2 to

(54)YtY0+0teβsg(s)ds+0tdKs+0tZsdWs0tEVs(e)μ˜(ds,de)
and the non-decreasing process K can be defined almost surely and in L2:
0tdKs:=limnn0t(YsnUs)+ds=limn(YtnY0n+0teβsg(s)ds+0tdKsn0tZsndWs0tEVsn(e)μ˜(ds,de)).

This proves Item (2) and the existence of the non-decreasing process K in L2 such that 0tdKsL2.

Then, using the differential of and multiplying by YsUs yield almost sure convergence:

t,n0t(YsUs)(YsnUs)+ds0t(YsUs)dKs.

The right hand side is almost surely finite since it is equal to

0t(YsUs)[dYs+eβsg(s)ds+dKs+ZsdWsEVs(e)μ˜(ds,de)].

Remark that the sequence (YsUs)(YsnUs)+ goes almost surely to (YsUs)(YsUs)+, and multiplied by n the limit cannot be finite unless (YsUs)+=0, thus Item (4) is proved:

(55)YsUs and (YsUs)dKs=0.

Finally Item (3) is a consequence of

  1. the fact LtYtn for any n and t, and the almost sure convergence of sequence (Ytn), so LtYt,

  2. above gives YtUt.

5. Application to the impulse control problem with infinite horizon

In this section we use , and with g:(t,ω)f(1,Xt(ω))f(2,Xt(ω)) satisfying Assumption (H1), a null terminal value, and barriers Lt=c1,2eβt0,Ut=c2,1eβt0, satisfying Assumptions (H2). There exists a progressively measurable process (Y,Z,K+,K,V) such that:

(S){YC2,Z2,VL2Yt=t+eβs(f(1,Xs)f(2,Xs))ds+t+dKs+t+dKst+ZsdWst+EVs(e)μ˜(ds,de)c1,2eβtYteβtc2,10+dKs±L2,0t(YsLs)dKs+=0t(YsUs)dKs=0

So the main result can be proved: the existence of processes (Y1,Y2) introduced in . This is the extension of Theorem 3.2 [, p. 186] to the infinite horizon set up with jumps.

Theorem 5.1.

Assume that f(1,Xt) and f(2,Xt) are positive, tf(i,Xt),i=1,2, satisfy (H1), Lt:=eβtc1,2 and Ut:=eβtc2,1 satisfies (H2). Then there exists a couple of -valued processes (Yt1,Yt2)t0 satisfying the assumptions in , in particular and meaning:

Yt1=ess supθTt E[tθeβsf(1,Xs) dseβsc1,2+Yθ2|Ft],
Yt2=ess supθTt E[tθeβsf(2,Xs) dseβsc2,1+Yθ1|Ft],Y2=Y1.

Proof: is applied with g(t)=f(1,Xt)f(2,Xt),Lt=c1,2eβt0Ut=eβtc2,1. Since the random variables t+dKs± are integrable and f(i,Xs),i=1,2 satisfy (H1), the following processes will be checked to satisfy assumptions: Yi are positive right continuous left limited regular processes of class [D] satisfying and . The following processes are proposed:

Yt1:=E[teβsf(1,Xs)ds+t+dKs+|Ft]
Yt2:=E[teβsf(2,Xs)ds+t+dKs|Ft].
  1. First one remarks that Yti0 as conditional expectation of non-negative random variables.

  2. Second

Yti=E[0eβsf(i,Xs)ds+0+dKs+|Ft]0teβsf(i,Xs)dsKt±
are sum of an F-martingale minus a right continuous left limited finite variation process so these processes are right continuous left limited.
  1. (3)

    Third one has E[supt0|Yti|2]<,i=1,2: indeed, using the facts that f(i,.) and 0tdKs± are positive,

0YtiMti:=E[0eβsf(i,Xs)ds|Ft]+E[0+dKs±|Ft].

The facts that 0dKs±L2, Assumption (H1) and (0eβsf(i,Xs)ds)21β0eβsf2(i,Xs)ds belongs to L1, proves that the martingale Mi which bounds Yi is uniformly square integrable. Thus Burkholder-Davis-Gundy inequality applied to this square integrable martingale M proves that E[supt0|Yti|2]<. As a byproduct, the process Yi is of class [D] since for any stopping time θ, 0Yθisupt0|Yti|L2.

  1. (4)

    Fourthly Yi are regular using the same argument as in []: the regularity of Yi is equivalent to the regularity of K±, and this one is equivalent to the regularity of Y defined by the system (S). in insures this property.

One now turns to the checking of and . Theorem 4.34 [, p. 189], applied to the semi martingale Ht:=Wt+Nt=Wt+0tEeμ(ds,de), with characteristics C=1,ν(ω;dt×de)=dtλ(de) and ΔBt(ω)=ΔtH(ω)=ΔtN(ω), there exists a couple of F-progressively measurable processes (Z1,V1)2×L2 such that for any t:

0tZs1dWs+0tEVs1(e)μ˜(ds,de)=0teβsf(1,Xs)ds+0tdKs++Yt1E[0eβsf(1,Xs)ds+0dKs+].

Using the third inequality of system (S), one has Ytc1,2eβt, and replacing Yt by Yt1Yt2, one has Yt1c1,2eβt+Yt2. Similarly, the fourth equality of system (S), meaning 0.(Yt+eβtc1,2)dKt+=0, replacing Yt by Yt1Yt2 shows

T,0T(Yt1Yt2+eβtc1,2)dKt+=0.

As a result, the quadruplet (Y1,Z1,V1,K+) satisfies the single barrier reflected BSDE:

{dYt1=eβtf(1,Xt)dt+dKt+Zt1dWtEVs1(e)μ˜(ds,de)Yt1c1,2eβt+Yt2 and (Yt1Yt2+eβtc1,2)dKt+=0.

Then Equality in is applied with Lt=c1,2eβt+Yt2. Since E[supt0|Yt2|2]<, the hypothesis E[supt0(Lt)2]< is satisfied and one has

Yt1=esssupθTtE[tθeβsf(1,Xs)dsc1,2eβθ+Yθ2|Ft].

Similarly, using the third inequality of system (S), one has Ytc2,1eβt, and once again Equality is used with Lt=c2,1eβt+Yt1. Since E[supt0|Yt1|2]<, the hypothesis E[supt0(Lt)2]< is satisfied and one has

Yt2=esssupθTtE[tθeβsf(2,Xs)dsc2,1eβθ+Yθ1|Ft],
hence the existence of the asked couple (Y1,Y2).
Remark 5.2.

Since Yt=Yt1Yt2,t0, according to , an optimal strategy α^=(τn)n0 is defined by

{τ1=0τ2n=inf{t>τ2n1,Ytc1,2eβt}, n0τ2n+1=inf{t>τ2n,Ytc2,1eβt}.

6. Numerical resolution

Recall that the optimal strategy α^=(τ^n)n0 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 []. Our method is totally different from the method used in [] 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 . For this application, a simple case of stochastic differential equation with jump is considered: Let b,σ are constant drift and diffusion coefficients; μ˜(ds,de) 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

Xt=x0+bt+σWt+Ntλt.

By using the classical Euler scheme for sample path trajectories of the process X where λ=3,x0=1 and T=1, one has: (see and ).

Let us now focus on our problem: namely, how to simulate the process Y, and therefore the optimal strategy. Recall that

eβtg(t)=eβt(f(1,Xt)f(2,Xt)),Lt=c1,2eβt,Ut=c2,1eβt,c1,2 and c2,1>0
which satisfy Hypotheses (H1) and (H2).

First of all, when t tends to infinity, Yt goes to 0, so a finite horizon T should be fixed such that ti=iTn,i=n,,0. More specifically, below the numerical samples show that as soon as t1, the length of interval (Lt,Ut) is negligible.

Yt(Lt,Ut) so the error is bounded by UtLt, the order of which being eβt.

To approximate the backward component Y, the following discretization approximation scheme is introduced, for 0=t0<t1<<tn=T:

(56){Y˜Tπ=YTπ=0Y˜tiπ=Eti[Yti+1π]+(ti+1ti) eβtif(Xtiπ)Ytiπ=(Y˜tiπLti)Uti,in1,
where Eti=E[.|Fti]. To approximate the conditional expectation, here is adopted the Longstaff-Schwarz algorithm [] which uses a regression technique (Least-Square Monte Carlo method). Taking the parameters β=0.5 , X0=1,b=1,σ=2, and the profits/costs functions
f(1,x)=3+2x,f(2,x)=2x+, so  f(1,x)f(2,x)=2x+3,
the evolution of Y is observed. Previously all the assumptions have to be checked:
  1. (1)

    One notes that with Xt=bt+σWt+Ntλt, : Assumption (H1) is satisfied since f(i,x)=a+x±, so E[(a+Xt±)2]2a2+2E[(Xt2)]2a2+6(b2t2+σ2t+λt) thus E[0eβtf2(i,Xs)ds]0eβt[2a+6(b2t2+σ2t+λt)]dt<.

Interpretation: Recall once again that the optimal strategy α^=(τ^n)n0 is obtained when Y reached successively the barriers L and U. In and , the costs are higher than in and . In and , 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 and , the firm can switch the technology more often: actually at times τ00.15 and τ10.97 (), respectively in , the firm can switch the technology at times τ00.05 and τ10.23.

Figures

b=0.01,σ=0.2

Figure 1

b=0.01,σ=0.2

b=1,σ=2

Figure 2

b=1,σ=2

Lt=−0.5e−2t, Ut=0.4e−2t

Figure 3

Lt=0.5e2t,Ut=0.4e2t

Lt=−0.5e−2t, Ut=0.4e−2t

Figure 4

Lt=0.5e2t,Ut=0.4e2t

Lt=−0.02e−2t, Ut=0.01e−0.2t

Figure 5

Lt=0.02e2t,Ut=0.01e0.2t

Lt=−0.02e−2t, Ut=0.01e−2t

Figure 6

Lt=0.02e2t,Ut=0.01e2t

Appendix

For sake of completeness, references being out of our knowledge, here is provided an extension of Gronwall's lemma.

Lemma 7.1. Let g and ψ be positive functions, let D be a positive constant satisfying t>0 f(t)D+tψ(s)f(s)ds, then

  1. if ψL1(+),t,f(t)Dexptψ(s)ds,

  2. if D=0, then f(t)=0.

Lemma 7.2. Assume that f and L satisfy respectively (H) and (H2). let (Y,Z,K) be the solution of the RBSDE: YT=0,

Yt=tTeβsg(s,Ys,Zs,Vs)ds+tTdKstTZsdWstTEVs(e)μ˜(ds,de),t[0,T],
where YC2, Z2, LtYt, dK. is a positive measure such that E(0TdKs)2< and tTeβs(YsLs)dKs=0,a.s. Then,
(57)E[Yt2]ϕ(t)exp(4C2+2C+1β),
where ϕ(t):=1β||f||2eβt+1εEsupst(Ls+)2+εE(tTdKs)2,
(58)EtT|Zs|2ds4ϕ(t).
and
E0E|Vs(e)|2λ(de)ds4ϕ(t).

Proof. Ito's formula and tT(YsLs)dKs=0 show

(59)(YtS)2+tT(Zs)2ds+t<sT[Δs(Y)]2=2tTeβs(Ys)[g(s,Ys,Zs,Vs)]ds2tTYsZsdWs2tTE[(Ys)(Vs(e))]μ˜(ds,de)+2tTLsdKs.

Using the Lipschitz property of g, we obtain

(60)E[Yt2]+EtT|Zs|2ds+E0E|Vs(e)|2λ(de)ds)E[2CtTeβs(Ys2+|Ys||Zs|+|Ys|Vs)ds+tTeβs2|Ys||g(s,0,0,0)|ds+2tTeβsLsdKs]E[(4C2+2C+1)tTYs2eβsds+tTeβsg2(s,0,0,0)ds+12tTZs2ds+2tTLsdKs+120E|Vs(e)|2λ(de)ds)]

It follows that

E[Yt2]E[(4C2+2C+1)tTYs2eβsds+tTeβsg2(s,0,0)ds+2tTLsdKs].

Moreover, for any ε>0:

2tTLsdKs2tTLs+dKs2supstLs+tTdKs1εsupst(Ls+)2+ε(tTdKs)2
(we use 2ab1εa2+εb2). Applying Gronwall's lemma (see ) to bound tE[Yt2] with ψ(t)=(4C2+2C+1)eβt and
(61)ϕ(t)=1β||g||2eβt+1εEsupst(Ls+)2+εE(tTdKs)2.

Since ϕ is decreasing, we get

(62)E[Yt2]ϕ(t)exp((2C2+2C+1)tTeβudu).

Using and , we get:

(63)12EtT|Zs|2ds(4C2+2C+1)EtTYs2eβsds+1β||g||2eβt+1εEsupst(Ls+)2+εE(tTdKs)2HtTϕ(s)eβsds+ϕ(t),
where H=(4C2+2C+1)exp(4C2+2C+1β). Since ϕ is decreasing, we get
EtT|Zs|2ds2ϕ(t)(1+Hβeβt)4ϕ(t).

Similarly we get

E0E|Vs(e)|2λ(de)ds4ϕ(t).

Lemma 7.3. The solutions of the reflected BSDE () and of the double reflected BSDE () are regular.

Proof: Let T be a finite stopping time and (Tn) be a non decreasing sequence of stopping times going to T. Using [VI 50 p. 125] [], a sufficient and necessary condition for Y to be “regular” (meaning Y=Yp) is

TnT,E(YT)=limnE(YTn).

If the process Y is a solution to reflected BSDE, we get

E[YTnYT]=E[TnTeβsg(s,Ys,Zs,Vs)ds]+E[KTKTn].

So a sufficient condition is: for any Fpredictable stopping time τ, E[ΔτK]=0. Under Assumption (H2) this condition is satisfied since under these hypotheses K± are continuous.

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Further reading

[27]Abidi H, Amami R, Pontier M. Infinite horizon impulse control problem with continuous costs, numerical solutions. Stochastics. 2017; Taylor & Francis, 89(6–7): 1039-60.

[28]Abidi H, Amami R, Pontier M. Infinite horizon impulse control problem with jumps using doubly reflected BSDEs: Preprint, HAL Archives-ouvertes.fr, Available from: https://hal.archives-ouvertes.fr/hal-01547004v1/document.

[29]Chesney M, Jeanblanc M, Yor M. Mathematical methods for financial Markets: Springer; 2009.

Acknowledgements

The authors would like to thank Monique Jeanblanc for her constructive comments and suggestions to improve the quality of our work.

Corresponding author

Rim Amami can be contacted at: rabamami@iau.edu.sa

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