Abstract
In this paper, we study a system of multidimensional coupled reflected backward stochastic differential equations (RBSDEs) with interconnected generators and barriers and mixed reflections, i.e. oblique and normal reflections. This system of equations is arising in the context of optimal switching problem when both sides of the balance sheet are considered. This problem incorporates both the action of switching between investment modes and the action of abandoning the investment project before its maturity once it becomes unprofitable. Pricing such real options (switch option and abandon option) is equivalent to solve the system of coupled RBSDEs considered in the paper, for which we show the existence of a continuous adapted minimal solution via a Picard iteration method.
Keywords: Real options, Optimal switching, Balance sheet, Trade-off strategies, Merger and acquisition, Backward SDEs, Mixed reflections
Introduction
Optimal switching problem (OSP) has attracted a lot of interest in the recent years (see among others [1, 2, 5–7, 11, 12]), since it can be related to many practical applications, for example the problem of valuation investment opportunities. OSP consists in finding an optimal management strategy for a production company that can run in m, 
, different modes. A management strategy 
is a combination of a nondecreasing sequence of stopping times 
, and a sequence of random variables 
taking values in the set of possible production modes 
. At time 
, in order to maximize the profit of the company, the manager decides to switch the production from the current mode 
to 
. When the production of the company is working under a strategy 
, it generates a gain equal to 
. The OSP amounts to finding an optimal management strategy 
such that 
. The OSP is connected with multidimensional RBSDEs with oblique reflections and interconnected barriers.
One dimensional BSDEs with normal reflections were first introduced by [10]. The multidimensional case was studied by Gegout-Petit and Pardoux [9], and then further investigated in many other works see e.g. [8, 13]. Multidimensional BSDEs with oblique reflections occurring in the context of OSPs were first introduced by [12]. They consider RBSDEs with generator taking the form 
and barrier 
where 
are constant switching costs and 
. Later, Hamadène and Zhang [11] generalized the preceding work by considering general generators and barriers of the following types 
and 
. Xu [17] dealt with the same kind of RBSDEs but when the generator, which is discontinuous w.r.t. 
, and the barrier take respectively the following forms 
and 
. Then, Aazizi et al. [1] extended the results of [17] to the case of generators and barriers of the form 
and 
.
In this paper, we are interested by Balance sheet OSP (BSOSP) which is a combination between the classical OSP described above and optimal stopping involving the balance sheet. BSOSP incorporates both the action of switching between modes and the action of abandoning a project once it becomes unprofitable. There are only few papers dealing with BSOSPs. Djehiche and Hamdi [4] considered the 2-modes case, i.e. 
. Their generators are of the form 
, 
and their barriers of type 
and 
, where 
and 
are switching costs and 
. Recently, the BSOSP multi-modes case was solved by Eddahbi et al. [5] when the barriers are of the form 
and 
(see Eddahbi et al. [6] for the mean–field case).
Now, let us describe precisely the problem studied in this paper by introducing some notations. Let 
be a given real number, and 
is a fixed probability space endowed with a d–dimensional Brownian motion 
. 
is the natural filtration of the Brownian motion augmented by the 
–null sets of 
. All the measurability notion will refer to this filtration. The euclidean norm of a vector 
is denoted |z|. Furthermore, we introduce the following spaces of processes. 
is the space of 
–valued processes 
, such that 
. 
(resp. 
) is the set of 
–valued adapted and continuous (resp. càdlàg ) processes 
such that 
is the set of 
–valued, progressively measurable processes 
such that 
(resp. 
) is the set of non-decreasing processes K, satisfying 
and that belong to 
.
Next, to illustrate the BSOSP studied in this paper, let us deal with a concrete example. Consider a company that has m modes of production (if 
, minimal, average and maximal production modes). The manager of the company has two options. A switch option, i.e. in order to maximize its global profit, she switches the production between the modes depending on their random performances but this switching incorporates a cost called switching cost. The manager has also an abandon option i.e. stop the production once it becomes unprofitable.
More precisely, being in mode 
, one have to switch at time t to another mode 
, once we have that the expected profit 
in this mode falls below the following barrier
 |
1 |
where 
is nonlinear random function (a special case is when 
, where 
is a switching cost from mode i to mode j), 
is the expected cost in mode i, and 
is the cost incurred when exiting/terminating the production while in mode i. Since we consider both sides of the balance sheet, the manager has to switch at time t to another mode 
, as soon as the expected cost in mode i, 
rises above the following barrier
 |
2 |
where 
is a cost of default (i.e. in this case the project is no longer profitable and thus leads to the abandon of this latter even before its maturity), and 
is the benefit incurred when exiting/terminating the production while in mode i. It is well known that the BSOSP can be formulated using the following system of Snell envelopes
 |
3 |
 |
4 |
where 
are 
–stopping times which represent the exit times from the production in mode i, 
and 
denote respectively the running profit and cost per unit time dt and 
and 
are respectively the values at time T of the profit and the cost yields.
The BSOSP consists in showing existence and uniqueness of the processes
and also proving that the following stopping times are optimal
 |
Since the Snell envelope is strongly connected to RBSDEs, solving the BSOSP is equivalent to showing existence of continuous solution to the following general (since we take 
where 
, 
) system of BSDEs with mixed reflections: for 
where T is called the time horizon, 
and 
are called the terminal conditions, the random functions 
and 
are respectively 
–progressively measurable for each 
, called the generators. 
is a real nonlinear random function, and 
, 
, and 
are previously given 
–adapted processes with some suitable regularity. The unknowns are the processes 
which are required to be 
–adapted. Moreover, 
and 
are non-decreasing processes. The second condition in (5) (resp. (6)) says that the first component 
(resp. 
) of the solution of RBSDE (5) (resp. (6)) is forced to stay above (resp. below) the barrier 
(resp. 
). The role of 
(resp. 
) is to push 
(resp. 
) upwards (resp. downwards) in order to keep it above (resp. below) the respective barrier in a minimal way in the sense of the third condition of RBSDE (5) (resp. (6)) which is called the minimal boundary condition i.e. 
(resp. 
) increases only when 
(resp. 
) touches the respective barrier.
Let us make precise the notion of a solution of the system of RBSDEs (S).
Definition 1
A 6-uplet of processes 
is called solution of the system of RBSDEs (S) if the two triples 
and 
belong to 
and satisfy the system (S).
The main contribution of our paper is to establish the existence of a continuous minimal adapted solution to system of RBSDEs (S). To this end we use a Picard iteration method (see El Karoui et al. [10] for more details). Uniqueness of the solution does not hold, since it is not verified even for the two-modes case and for a less general form of RBSDE (S) (see the counter-example in [4, subsection 3.1]).
Clearly, our results generalize the related works in the literature, since our RBSDE (S) is more general in many features. Actually, the expected profits and cost yields 
and 
are respectively interconnected in the generators 
and 
. This dependence can be interpreted as a nonzero-sum game problem, where the players’ utilities affect each other. Furthermore, the solutions 
and 
are also interconnected in the barriers. Note that, the general barrier 
which is random and nonlinear, makes the dependence on the unknown process implicit. This, allows one to consider more general switching cost, for instance in the case of risk sensitive switching problem.
The remainder of the paper is organized as follows. Section 2 is devoted to the assumptions. In Sect. 3, we state and prove the main result of the paper.
Assumptions
Let us introduce the following assumptions:
[H1]: For each 
, 
and 
satisfies:
-
(i)
. -
(ii)
The mappings
and 
are Lipschitz continuous in 
uniformly in t, and are continuous in 
for 
. -
(iii)
and 
are increasing in 
for 
. This assumption means that the m-players are partners.
[H2]: For each 
, 
satisfies:
-
(i)
is continuous in (t, y); -
(ii)
is increasing in y; -
(iii)
. -
(iv)
There is no sequence
, 
, and 
such that 
, 
, 
. This means that there is no free loop of instantaneous switchings.
[H3]: For 
and 
, 
, 
and 
belong to 
.
[H4]: For any 
the random variables 
and 
are 
–measurable and belong to 
. Moreover we assume that
 |
[H5]: For every 
, the processes 
and 
are semimartingales of the form 
and
where 
and 
are respectively 
and 
-valued 
–progressively measurable processes which are 
–square integrable.
Main Result
Next we state and prove the main result of this paper.
Theorem 1
Assume that [H1]–[H5] hold. Then for all 
, the system of RBSDEs (S) admits a continuous minimal solution 
.
Proof
The whole proof is performed in six steps.
Step 1: Construction of Picard’s sequence of solutions
Consider the following sequence of RBSDEs defined recursively, for 
and 
, as follows: For 
we start with the following BSDE:
 |
7 |
and RBSDE: 
where 
Now, for 
consider the following system of RBSDEs:
Note that by [H1](i) and (ii) we have that 
and 
are uniformly Lipschitz continuous in 
and satisfy the following integrability condition
 |
Thus, from [14] it follows that for each 
BSDE (7) admits a unique solution 
. Thus, there exists a constant 
such that
 |
10 |
and thus in view of [10, Proposition 2.3] we deduce that RBSDE (8) has a unique solution 
.
As a by product, under the assumptions [H1]–[H4], in view of [10, Proposition 2.3] the solution 
exists and is unique. This in turn, in view of the following estimate, which holds due to assumption [H2](iii), [H3] and the fact that 
,
 |
combined with [10, Proposition 2.3], leads to the existence of the unique solution
.
Next, for 
, consider the following system
Based on the arguments used previously, we can show by using an induction argument that for any 
, the system of RBSDEs (11) has a unique solution
 |
Step 2: Convergence of the sequences
Let us set,
 |
Note that by [H1](i) and (ii) we have that 
and 
are uniformly Lipschitz continuous in 
and satisfy the following integrability condition
 |
12 |
Consider the following BSDE
 |
It follows from [14] that this BSDE admits a unique solution 
. Next, let 
be solutions of the following system of reflected BSDEs, for any 
and 
, as follows
By using previous arguments, and thanks to the fact that 
and assumption [H3], applying [1, Theorem 3.1] yields that this RBSDE admits a solution 
. Moreover, the following holds Next, by using an induction argument, plus a repeated use of the comparison theorem, we can easily show that for any 
, 
, 
:
 |
14 |
Consequently, we deduce the following
 |
15 |
 |
16 |
Next, from (14) combined with (15) and (16) we deduce that the sequences 
and 
admit limits. Therefore, let 
and 
, 
be two optional processes which are respectively the limits of 
and 
.
Applying Fatou’s Lemma and the dominated convergence theorem, we obtain
 |
17 |
Step 3: Uniform estimates for the sequences.
, 
By [H2](iii), we obtain in view of the facts 
, 
and (14) combined with [H3] the following estimate of the barriers of RBSDE (11): for all 
and all 
 |
18 |
 |
19 |
Finally, with the estimates (15), (16), (18) and (19) at hand, applying the results in [10] we obtain that
 |
20 |
Step 4: Continuity of the limit processes.
and
, 
To this end, let us first establish the absolute continuity of the increasing process 
w.r.t t for every 
.
We will first show that the claim holds true for 
. Let
 |
Applying Itô–Tanaka formula to 
, and in view of assumption 
we obtain
 |
21 |
where 
is the local time at 0 of the continuous semimartingale 
,
 |
and
 |
Note that 
and 
are uniformly Lipschitz continuous in 
. Thus, using the fact that 
and 
belong respectively to 
, yields that there is a constant 
such that
 |
22 |
Moreover, in view of the above and assumption 
, there exists a constant 
such that
 |
23 |
Then, applying [10, Proposition 4.2] yields for all 
 |
24 |
which means that, 
is absolutely continuous w.r.t. t. Next, in the same spirit, we can show, thanks to [10, Proposition 4.2], for all 
that the process 
is absolutely continuous w.r.t. t. Furthermore, we can obtain that: there exists a constant 
such that for all 
and 
,
 |
25 |
Notice that, by combining [H1](ii) together with (15), (16), (20) we obtain that there is a constant 
such that
 |
26 |
Next, in view of estimates (20), (25) and (26), we deduce that there exists a subsequence along which all 
, 
and
converge weakly in their respective spaces 
, 
and 
to the processes 
, 
and 
.
Next, for any 
and any stopping time 
we have
 |
Taking the weak limits in each side and along this subsequence yields
 |
Since the processes appearing in each side are optional, using the Optional Section Theorem (see e.g. [3], Chapter IV pp.220), it follows that
 |
27 |
Therefore, the process 
is continuous. Relying both on Dini’s Theorem and on Lebesgue’s dominated convergence one (17), we also get that
 |
28 |
We will now focus on the continuity of the sequence of processes 
, 
. Actually, applying Peng’s Monotone Limit Theorem (see [15]) yields that for every 
, the limit process 
is càdlàg . Based on what has been already shown in previous steps, by mimicking the arguments of [15] we can easily show that there exist two processes 
and 
such that 
satisfies the first equation of RBSDE (S). Moreover, passing to the limit in the fifth inequality of RBSDE (11), implies that 
Thus, for 
, 
satisfies
 |
29 |
It remains to prove the minimal boundary condition. Next, consider the following RBSDE whose solution exists thanks to [16]:
Note that RBSDEs (29) and (30) have the same lower barrier. In fact, since 
is the smallest 
–supermartingale with lower barrier 
, we have that for any 
, 
(see [16, Theorem 2.1]). On the other hand since for any 
and 
, 
and 
, applying the comparison theorem in view of [H2](ii) yields that 
, and then passing to the limit implies that 
. Summing up we have that for any 
. From the uniqueness of the Doob-Meyer decomposition, it follows that 
, 
–a.s., and 
for any 
, 
–a.s. Then, for 
, 
satisfies RBSDE (5) but with the following minimal boundary condition
 |
31 |
From the first equation of (29) and since the process 
is increasing, it follows that 
Assume that 
, for some 
. Thus 
. From the minimality condition (31), we have
 |
Let 
be the optimal index for which the maximum is attained. Thus,
 |
32 |
This obviously yields that 
and thus 
. Repeating the above procedure we obtain for 
 |
Since each 
can take only values in 
which is a finite set, then there must be a loop in 
. we may assume w.l.o.g. that 
for some 
noting again that the 
’s are mutually different i.e. for each k, 
. Therefore, we have 
and 
This contradicts assumption [H2](iv). Consequently, 
. Hence, the processes 
and 
, 
are continuous.
Step 5: Identification of the limit
Next, we show that
 |
Actually, since 
and 
is continuous then relying both on Dini’s Theorem and on Lebesgue’s dominated convergence one (17), we get that
 |
33 |
Further, we can easily show by applying Itô’s formula to 
and using standard arguments (see e.g. [10]) that
 |
From this, and [H1](ii) combined with (27), (28) it holds that
 |
Next, passing to the limit in the second inequality of RBSDE (11), yields that 
Furthermore, thanks to the weak convergence of 
to the process 
and the strong convergences (28) and (33), we deduce that
 |
In fact, this implies that 
is a solution to the second part of RBSDE (S). Summing up 
is a solution of RBSDE (S). Finally, it remains to show that this solution is the minimal one.
Step 6: Minimality of the solution of RBSDE (S)
Let 
be another solution of RBSDE (S). Since 
and 
, for all 
, and thanks to the monotonicity of 
applying the comparison theorem yields that for each 
: 
and 
, for all 
. Passing to the limit when 
implies that for each 
: 
and 
, which is the desired result. This ends the proof of Theorem 1.
Conclusion and Perspectives
In this paper we have proved the existence of a continuous minimal solution to RBSDE (S) which is arising from BSOSPs. Let us comment on a possible generalization of the results obtained in this paper. Actually, the full balance sheet case is still an open problem and constitutes a challenge. By the full balance sheet case we mean that we consider the two sides of the balance sheet. Indeed, in this case the expected cost in mode i, 
should rise above the following barrier
 |
34 |
instead of 
where 
is a real nonlinear random function satisfying [H2], except for [H2]–(iii) which is replaced by 
. We want to stress out that, the new assumption [H2] is satisfied when 
takes the particular form 
where 
is the switching cost from mode i to mode j.
A full BSOSP amounts to establishing existence of a continuous solution to the system of RBSDEs (S), but with the upper barrier (34) for 
. Note that, as in the proof of Theorem 1 (Step 4) the absolute continuity of the process 
w.r.t. t will play a primordial role to derive convergence of the corresponding approximating sequence. To do so we need to use the Itô–Tanaka formula (see Step 4), which makes it difficult to solve the system of RBSDEs (S) for the full balance sheet case. Note that even in the case when the functions 
and 
take the particular forms respectively 
and 
let alone the general case, the question of existence of solutions to the corresponding system of RBSDEs (S) for the full balance sheet case, is still open. This issue was discussed in [5] and in [6] in the mean field case.
Acknowledgment
The second named author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no (RG-1441-339).
Contributor Information
Valeria V. Krzhizhanovskaya, Email: V.Krzhizhanovskaya@uva.nl
Gábor Závodszky, Email: G.Zavodszky@uva.nl.
Michael H. Lees, Email: m.h.lees@uva.nl
Jack J. Dongarra, Email: dongarra@icl.utk.edu
Peter M. A. Sloot, Email: p.m.a.sloot@uva.nl
Sérgio Brissos, Email: sergio.brissos@intellegibilis.com.
João Teixeira, Email: joao.teixeira@intellegibilis.com.
Rachid Belfadli, Email: r.belfadli@uca.ma.
M’hamed Eddahbi, Email: meddahbi@ksu.edu.sa.
Imade Fakhouri, Email: imade.fakhouri@um6p.ma.
Youssef Ouknine, Email: ouknine@uca.ac.ma, Email: youssef.ouknine@um6p.ma.
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