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
The existence of
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
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 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 [8] considering a finite horizon
Here the model is inspired from the papers [5, 8–10, 12]. 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
Our purpose is similar to the one in [16], 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. Section 2 presents the impulse control problem and describes the corresponding model. Section 3 introduces a pair of right continuous left limited processes
2. Preliminaries and problem formulation
Let
dimensional Brownian motiona point process
associated with a Poisson random measure μ on where for some endowed with its Borel σ-algebra , with compensator for a σ-finite measure λ on denotes the compensated measure associated with
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
Given
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
The strategy
Here, the impulse control problem is to prove the existence of an admissible strategy
The following notations will be used:
the σ algebra of -predictable sets onClass [D] :
3. The impulse control problem
Section 5 shows that the problem reduces to the existence of a pair of càdlàg processes
Assume that there exist two right continuous left limited, regular (meaning that the predictable projection coincide with the left limit)
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
4. Reflected BSDE with jumps and infinite horizon
In this section, the results from [10] 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:
A map which is -progressively measurable and:
: An -progressively measurable map such that Let the barriers and be -progressively measurable continuous real valued processes satisfying
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 (Section 4.1) then a comparison theorem is proved in Section 4.2.
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.
Let
, and ,almost surely
- (3)
almost surely
- (4)
is a non-decreasing process satisfying and for any t
We then prove the following:
Let
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
Theorem 2.1 [10] can be applied: there exists a quadruplet verifying (actually restricted to ) and
Considering
Applying Itô's formula to the process
Using
Considering the decomposition:
Thus for any
Remark that
On the one hand, Lemma 7.2 , we obtain for any
On the other hand, we have
Using estimation (19), there exists a constant M, such that for any
If we subtract from
This implies that the expectation on the left tends to zero uniformly when
- (3)
Turning to Z and V, to deal with the convergence in
respectively in , An argument similar to (63) shows that the sequence is a Cauchy sequence in , its limit defines a process Z which belongs to and is a Cauchy sequence in , its limit defines a process V which belongs to . - (4)
We now prove that there exists a process
which is the limit of a Cauchy sequence in .(a) Coming back to (16), for all
we get
The Burkholder-Gundy-Davis inequality gives the existence of a constant
for all
Gathering these bounds yields
Choosing
- (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
Thus
Item (3) For any
Item (4) The
On the one hand, for fixed
We now deal with the successive bounds
For
Remark that
We now fix
Thus when T goes to infinity the second term in (24) satisfies
For any
Finally using (23) we get
Thus
■
In case of a deterministic function g, meaning g is defined on
Since
This alternative method allows us to prove the following result.
Under Hypotheses
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
Reproducing Step 2 in the proof of Theorem 3.1 [16] leads for any k to
so
Let us denote as
From now on, we consider a function g defined on
The following is an extension of Lemma 2.4 in [20]: in our case g is defined only on
For
Proof: The proof is similar to the one in [18].
The next step follows from Theorem 3.2 [20] or Proposition 4.12 [18].
Let
Proof: (1) By definition, we have
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
Let
Using extended Gronwall's Lemma 7.1 one has
Let us denote
(2)Coming back to (33) one has
■
4.2 Comparison theorem in case of a single barrier
The following proposition is an extension of Theorem 2.2 in [10] to infinite horizon.
Assume that
If moreover
Proof: Theorem 2.2 in [10] proves that for any
Theorem 4.2 proof gives us the almost sure convergence of
Here we summarize the results concerning the reflected BSDEs: In case of a function g defined on
The
-progressively measurable process which is the unique solution of the reflected BSDE associated with satisfies
- (2)
The
-progressively measurable process which is the unique solution of the reflected BSDE associated with satisfies
Thank to Lemma 4.4, one has the following inequalities:
So as a consequence of Proposition 4.6, one has
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
Let
and ,almost surely
- (3)
almost surely
- (4)
are non-decreasing processes satisfying and for any t
Let
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.
If there exists a solution of (40) 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
Using Item (4)
It follows that for any t
So
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
So according to Theorem 4.2, Hypothesis (H2) still being in force, for each
From Proposition 4.6, the sequence
From the inequality
The proof of Theorem 4.8 is done in five steps.
Step 1: There exists a constant
such that and one has
Its formula yields
By definition of the solution
Then, since
Thus, one has
Using the Cauchy-Schwarz inequality, for any
On the other hand, with (38), Lemmas 4.4 and 4.5, for any t and n one has:
Similarly for any
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):
So one has
Gronwall's Lemma 7.1 is now used with
Then one has a bound for (46)
This bound and (44) end the proof.
Step 2:
and
The proof is an adaptation of the one given in Step 3 [20, p. 169].
Let
Let
Using that,
Since U is right continuous then almost surely and in
In addition, one has
Finally with (38)
This last bound
Therefore
From this and “Section Theorem” [21, p. 220], it follows that,
We now denote by
Consequently, from a weak version of the Dini theorem [22, p. 202], one deduces that
Step 3: There exist an
-adapted process and an -predictable process such that
By Itô's formula one has for any
Since
Look at
The second term in (49) is symmetrical and the sum is going to 0 in
Finally, taking the expectation of the left hand side in (48) and using (17)
It follows that
Step 4:
so defines a process in
Using
- (1)
First look at
For any
- (2)
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
Similarly one has
Using that
Choosing c such that
Moreover, since for all t
Step 5: Existence of
Item (4), Item (3)
By definition of
So, the right hand side of (53) converges almost surely and in
This proves Item (2) and the existence of the non-decreasing process
Then, using the differential of Equation (53) and multiplying by
The right hand side is almost surely finite since it is equal to
Remark that the sequence
Finally Item (3) is a consequence of
the fact
for any n and and the almost sure convergence of sequence , so ,above (55) gives
.
5. Application to the impulse control problem with infinite horizon
In this section we use Proposition 3.1, and Theorem 4.8 with
So the main result can be proved: the existence of processes
Assume that
Proof: Theorem 4.8 is applied with
First one remarks that
as conditional expectation of non-negative random variables.Second
- (3)
Third one has
: indeed, using the facts that and are positive,
The facts that
- (4)
One now turns to the checking of (7) and (8). Theorem 4.34 [23, p. 189], applied to the semi martingale
Using the third inequality of system (S), one has
As a result, the quadruplet
Then Equality (29) in Proposition 4.3 is applied with
Similarly, using the third inequality of system (S), one has
6. Numerical resolution
Recall that the optimal strategy
By using the classical Euler scheme for sample path trajectories of the process X where
Let us now focus on our problem: namely, how to simulate the process Y, and therefore the optimal strategy. Recall that
First of all, when t tends to infinity,
To approximate the backward component
- (1)
One notes that with
: Assumption is satisfied since so thus
Interpretation: Recall once again that the optimal strategy
In the case of reasonable costs, as in Figures 5 and 6, the firm can switch the technology more often: actually at times
Figures
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
if
if
then
Lemma 7.2. Assume that f and L satisfy respectively
Proof. Ito's formula and
Using the Lipschitz property of g, we obtain
It follows that
Moreover, for any
Since ϕ is decreasing, we get
Similarly we get
Lemma 7.3. The solutions of the reflected BSDE (Theorem 4.2) and of the double reflected BSDE (Theorem 4.8) are regular.
Proof: Let T be a finite stopping time and
If the process Y is a solution to reflected BSDE, we get
So a sufficient condition is: for any
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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.