optimal stopping of expected profit and cost yields in an investment under uncertainty

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Optimal stopping of expected profitand cost yields in an investment underuncertaintyBoualem Djehiche a , Said Hamadène b & Marie-Amélie Morlais ba Department of Mathematics, The Royal Institute of Technology,SE-100 44, Stockholm, Swedenb Département de Mathématiques, Equipe Statistique et Processus,Université du Maine, Avenue Olivier Messiaen, 72085, Le Mans,Cedex 9, FrancePublished online: 08 Jun 2011.

To cite this article: Boualem Djehiche , Said Hamadne & Marie-Amlie Morlais (2011) Optimalstopping of expected profit and cost yields in an investment under uncertainty, Stochastics AnInternational Journal of Probability and Stochastic Processes: formerly Stochastics and StochasticsReports, 83:4-6, 431-448, DOI: 10.1080/17442508.2010.516828

To link to this article: http://dx.doi.org/10.1080/17442508.2010.516828

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Optimal stopping of expected profit and cost yields in an investmentunder uncertainty

Boualem Djehichea*, Said Hamadeneb1 and Marie-Amelie Morlaisb2

aDepartment of Mathematics, The Royal Institute of Technology, SE-100 44 Stockholm, Sweden;bDepartement de Mathematiques, Equipe Statistique et Processus, Universite du Maine,

Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France

(Received 29 October 2009; final version received 16 August 2010)

We consider a finite horizon optimal stopping problem related to trade-off strategiesbetween expected profit and cost cash flows of an investment under uncertainty.The optimal problem is first formulated in terms of a system of Snell envelopes for theprofit and cost yields which act as obstacles to each other. We then construct both aminimal solution and a maximal solution using an approximation scheme of theassociated system of reflected backward stochastic differential equations (SDEs).We also address the question of uniqueness of solutions of this system of SDEs. Whenthe dependence of the cash flows on the sources of uncertainty, such as fluctuationmarket prices, assumed to evolve according to a diffusion process, is made explicit, weobtain a connection between these solutions and viscosity solutions of a system ofvariational inequalities with interconnected obstacles.

Keywords: optimal stopping; Snell envelop; backward stochastic differentialequations; merger and acquisition

AMS Subject Classification: 60G40; 93E20; 62P20; 91B99

1. Introduction

The trade-off between the expected profit and cost yields is a central theme in the cash flow

analysis of any investment project or any industry which produces a commodity or provides

services that are subject to uncertainties such as fluctuating market prices or demand and

supply flows (see [1,3,9] and the references therein). The project is profitable when the

expected profit yield is larger than the expected cost yield, a relationship that cannot always

be sustained, due to many sources of uncertainty. Timing of exit from the project based on

optimal trade-off between expected profit and cost yields is thus a crucial decision.

An approach to this problem, which is widely used in portfolio choice with transaction

costs (see [6] and the references therein), is to impose a predetermined form of the cost

yield, to formulate an optimal switching or impulse control problem for the expected profit

yield and to determine exit and re-entry strategies. But in many investment projects subject

to uncertain demand and supply flows, such as merger and acquisition operations, it is

often impossible to fully capture the expected cost yield with a given predetermined

model. It is this situation that we focus on in this paper. Our hope is to give some insight

into the foundations of the complex structure of the type of uncertain cash flows related to,

ISSN 1744-2508 print/ISSN 1744-2516 online

q 2011 Taylor & Francis

http://dx.doi.org/10.1080/17442508.2010.516828

http://www.tandfonline.com

*Corresponding author. Email: [email protected]

Stochastics: An International Journal of Probability and Stochastic Processes

Vol. 83, Nos. 4–6, August–December 2011, 431–448

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e.g. merger and acquisition operations, where there is almost nothing written on this

topical issue.

In this work, we do not assume any predetermined model for the cost yield. We rather

approach the problem by formulating a finite horizon optimal stopping problem that

involves both the expected profit and cost yields which will act as obstacles to each other.

More precisely, given the profit (resp. cost) c1ðtÞdt (resp. c2ðtÞdt) per unit time dt and the

cost a(t) (resp. profit b(t)) incurred when exiting/abandoning the project, if we let Y 1 and

Y 2 denote the expected profit and the cost yields, respectively, the decision to exit the

project at time t depends on whether Y1t $ Y2

t 2 aðtÞ or Y2t # Y1

t þ bðtÞ. If Ft denotes the

history of the project up to time t, the expected profit yield at time t is expressed in terms of

a Snell envelope as follows:

Y1t ¼ ess supt$tE

ðtt

c1ðsÞ ds þ ðY2t 2 aðtÞÞ1½t,T� þ j11½t¼T�jF t

� �; ð1:1Þ

where the supremum is taken over all exit times t from the project. Moreover, for any

t # T , the random time

t*t ¼ inf s $ t; Y1

s ¼ Y2s 2 aðsÞ

� �^ T ; ð1:2Þ

related to the cost Y 2 2 a incurred when exiting the project should be an optimal time to

abandon the project after t, in which case, we should also get

Y1t ¼ E

ðt*t

t

c1ðsÞ ds þ ðY2t*

t2 aðt*

t ÞÞ1½t*t ,T� þ j11½t*

t ¼T�jF t

" #: ð1:3Þ

In a similar fashion, the expected cost yield at time t reads

Y2t ¼ ess infs$tE

ðst

c2ðsÞ ds þ ðY1s þ bðs ÞÞ1½s,T� þ j21½s¼T�jF t

� �; ð1:4Þ

where the infimum is taken over all exit times s from the project. The random time

s *t ¼ inf s $ t; Y2

s ¼ Y1s þ bðsÞ

� �^ T ð1:5Þ

related to the profit Y 1 þ b incurred when exiting the project should be optimal after t as

well. In this case, we should get

Y2t ¼ E

ðs *t

t

c2ðsÞ ds þ Y1s *

tþ bðs *

t Þ� �

1½s *t ,T� þ j21½s *

t ¼T�jF t

" #: ð1:6Þ

In other words, the cost Y 2 2 a and the profit Y 1 þ b act as obstacles that define the

exit strategy.

The main result of this paper is to show existence of the pair ðY 1; Y 2Þ that solves the

system of Equations (1.1) and (1.4) and also to prove that t* and s * given, respectively,

by (1.2) and (1.5) are optimal strategies for our problem. Using the relation between Snell

envelopes, reflected backward stochastic differential equations (RBSDEs) and variational

inequalities (VI, see [2] for more details), it then follows that solving the system of

Equations (1.1) and (1.4) is equivalent to finding a solution to the following RBSDEs with

B. Djehiche et al.432

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interconnected obstacles: for all t # T ,

ðSÞ

Y1t ¼ j1 þ

Ð T

tc1ðsÞ ds þ ðK1

T 2 K1t Þ2

Ð T

tZ1

s dBs;

Y2t ¼ j2 þ

Ð T

tc2ðsÞ ds 2 ðK2

T 2 K2t Þ2

Ð T

tZ2

s dBs;

Y1t $ Y2

t 2 aðtÞ and Y2t # Y1

t þ bðtÞ;Ð T

0Y1

s 2 ðY2s 2 aðsÞÞ

� �dK1

s ¼ 0 andÐ T

0ðY1

s þ bðsÞ2 Y2s Þ dK2

s ¼ 0:

8>>>>>><>>>>>>:Using an approximation scheme for systems of RBSDEs, we establish existence of both a

maximal solution and a minimal solution of (S). When the dependence of the cash flows

ðY 1; Y 2Þ on the sources of uncertainty, such as fluctuation market prices, which are

assumed to evolve according to a diffusion process X is made explicit, we also obtain a

connection between the solutions of system (S) and the viscosity solutions of the following

system of VI with interconnected obstacles:

ðVIÞ

min{u1ðt; xÞ2 u2ðt; xÞ þ aðt; xÞ;2›tu1ðt; xÞ2 Lu1ðt; xÞ2 c1ðt; xÞ} ¼ 0;

max{u2ðt; xÞ2 bðt; xÞ2 u1ðt; xÞ;2›tu2ðt; xÞ2 Lu2ðt; xÞ2 c2ðt; xÞ} ¼ 0;

u1ðT ; xÞ ¼ g1ðxÞ; u2ðT ; xÞ ¼ g2ðxÞ:

8>><>>:This paper is organized as follows: Section 2 is devoted to the formulation of the

optimal stopping problem under consideration. In Section 3, we construct a minimal

solution and a maximal solution of (S), using an approximation scheme, where the

minimal solution is obtained as a limit of an increasing sequence of solutions of a system

of RBSDEs, while the maximal one is obtained as a limit of a decreasing sequence of

solutions of another system of RBSDEs. Next, we address the question of uniqueness of

the solution of (S). In general, uniqueness does not hold as it is shown through two

counter-examples. However, we give some sufficient conditions on c1;c2; a and b, for

which a uniqueness result is derived. Finally, in Section 4, we establish a connection

between the solutions of system (S) and viscosity solutions of the system of VI with

interconnected obstacles (VI). We actually show that (VI) admit a solution. Uniqueness

and finer regularity properties of the solutions of (VI) require heavy PDE techniques which

we prefer not to include in this paper but will appear elsewhere.

2. Preliminaries and the main result

In this section, we introduce some basic notions and results concerning RBDSEs, which

will be needed in the subsequent sections.

Throughout this paper, T . 0 denotes an arbitrarily fixed time horizon, and ðV;F ;PÞis a given probability space on which is defined a d-dimensional Brownian motion

B ¼ ðBtÞ0#t#T . We also denote by F ¼ ðF tÞ0#t#T the filtration generated by B and

completed by the P-null sets of F. Throughout the sequel, we always denote the process

restricted to ½0; T� by B and assume that all processes are defined on ½0; T�.

We shall also introduce the following spaces of processes which will be frequently

used in the sequel:

. S2 is the set of all continuous F-adapted processes Y ¼ ðYtÞ such that E½supt[½0;T�

jYtj2� , 1;

. A2 is the subset of S2 of increasing processes ðKtÞt#T with K0 ¼ 0;

. Md;2 denotes the set of F-adapted and d-dimensional processes Z such that

EðÐ T

0jZsj

2dsÞ , 1.

Stochastics: An International Journal of Probability and Stochastic Processes 433

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The following results on RBSDEs are by now well known. For a proof, the reader is

referred to [5]. A solution for the RBSDE associated with a triple (f ; j; S), where f :

ðt;v; y; zÞ 7! f ðt;v; y; zÞ (R-valued) is the generator, j is the terminal condition and

S U ðStÞt#T is the lower barrier, is a triple ðYt; Zt;KtÞ0#t#T of F-adapted stochastic

processes that satisfies

Y [ S2; K [ A2 and Z [ Md;2;

Yt ¼ jþÐ T

tf ðs;v; Ys; ZsÞds þ ðKT 2 KtÞ2

Ð T

tZs dBs;

Yt $ St; 0 # t # T ;Ð T

0ðSt 2 YtÞ dKt ¼ 0:

8>>>>><>>>>>:ð2:7Þ

The RBSDE(f ; j; S) is said standard if the following conditions are satisfied:

(A1): The generator f is Lipschitz with respect to ðy; zÞ uniformly in ðt;vÞ.(A2): The process ðf ðt;v; 0; 0; 0ÞÞ0#t#T is F-progressively measurable and dt ^ dP-

square integrable.

(A3): The random variable j is in L2 V;F T ;P� �

.

(A4): The barrier S is continuous F-adapted and satisfies E½sup0#s#T jS

þs j

2� , 1 and

ST # j, P-a.s.

Theorem 2.1. (See [5]) Let the coefficients ðf ; j; SÞ satisfy Assumptions (A1)–(A4). Then

the RBSDE (2.7) associated with ðf ; j; SÞ has a unique F-progressively measurable

solution (Y ; Z;K) which belongs to S2 £Md;2 £A2. Moreover, process Y enjoys the

following representation property as a Snell envelope: for all t # T ,

Y1t ¼ ess supt$tE

ðtt

f ðs; Y1s ; Z1

s Þds þ St1½t,T� þ j11½t¼T�jF t

� �: ð2:8Þ

The proof of Theorem 2.1 is related to the following, by now standard, estimates and

comparison results for RBSDEs. For the proof, see Proposition 3.5 and Theorem 4.1 in [5].

Lemma 2.1. Let ðY ; Z;KÞ be a solution of the RBSDE ðf ; j; SÞ. Then there exists a constant

C depending only on the time horizon T and on the Lipschitz constant of f such that

E sup0#t#T

jYtj2þ

ðT

0

jZsj2

ds þ jKT j2

# CE

ðT

0

j f ðs; 0; 0Þj2

ds þ jjj2þ sup

0#t#T

jSþt j

2

:

ð2:9Þ

Lemma 2.2. (Comparison of solutions) Assume that ðY ; Z;KÞ and ðY 0; Z 0;K 0Þ are solutions

of the RBSDEs associated with (f ; j; S) and ðf 0; j0; S0Þ, respectively, where only one of the

two generators f or f 0 is assumed to be Lipschitz continuous. If

. j # j 0, P-a.s.,

. f ðt; y; zÞ # f 0ðt; y; zÞ; dP^ dt-a.s. and for all (y; z),

. P-a.s., for all t # T , St # S0t,

then

P-a:s: ;t # T ; Yt # Y 0t: ð2:10Þ

The previous results are also valid for RBSDE with upper barriers. Indeed, if ðY; Z;KÞ

solves the RBSDE associated with ðf ; j;UÞ with upper barrier equal to U, then


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(2Y ;2Z;K) solves the RBSDE associated with ð~f;2j; SÞ with parameters given by

S ¼ 2U and ~fðs; y; zÞ ¼ 2f ðs;2y;2zÞ.

The first objective in this paper is to study existence of solutions of the coupled system

of RBSDEs (S). Let us introduce the following assumptions:

(B1): For each i ¼ 1; 2, the mappings ðt;v; y; zÞ 7! ciðt;v; y; zÞ are Lipschitz in ðy; zÞ

uniformly in ðt;vÞ meaning that there exists C . 0 such that

jciðt;v; y; zÞ2 ciðt;v; y0; z0Þj # Cðjy 2 y0j þ jz 2 z0jÞ; for all t; y; z; y0; z0:

Moreover, the processes ðc iðt; 0; 0; 0ÞÞ0#t#T are F-progressively measurable and

dt^ dP -square integrable.

(B2): The obstacles ðaðt;vÞÞ0#t#T and ðbðt;vÞÞ0#t#T belong to S2.

(B3): The random variables j 1 and j 2 are FT -measurable and square integrable.

Moreover, we assume that P-a:s:; j1 2 j2 $ max{ 2 aðTÞ;2bðTÞ}.

From now on, we will also make use of either one of the two following assumptions:

(B4): The process ðbðtÞÞ0#t#T is of Ito type, i.e. for any t # T .

bðtÞ ¼ bð0Þ þ

ðt

0

U2s ds þ

ðt

0

V2s dBs; ð2:11Þ

for some F-progressively measurable processes U 2 and V 2 which are dt^ dP-

integrable and square integrable, respectively.

(B40): The process ðaðtÞÞ0#t#T is of Ito type, i.e. for any t # T ,

aðtÞ ¼ að0Þ þ

ðt

0

U1s ds þ

ðt

0

V1s dBs; ð2:12Þ

for some F-progressively measurable processes U 1 and V 1 which are dt^ dP-

integrable and square integrable, respectively

Remark 2.1. Assumption (B4) is required to prove the continuity of the minimal solution,

which is obtained by using an increasing approximation scheme, whereas Assumption

(B40) is required to get the continuity of the maximal solution.

Let us now make precise: on the one hand, the notion of a solution and, on the other

hand, the notions of minimal and maximal solutions of system (S).

Definition 2.1. A 6-tuple of processes ðY 1; Z 1; K 1; Y 2; Z 2; K 2Þ is a solution of system

(S) if the two triples ðY 1; Z 1; K 1Þ and ðY 2; Z 2; K 2Þ belong to S2 £Md;2 £A2 and if it

satisfies (S).

The process ðY 1; Z 1; K 1; Y 2; Z 2; K 2Þ is a minimal solution of the system (S) if it is a

solution of (S) and if whenever another 6-tuple of processes ð ~Y1; ~Z1; ~K1; ~Y2; ~Z2; ~K2Þ is a

solution of (S), then

P-a:s: ;t # T ; ~Y1

t $ Y1t and Y2

t $ Y2t ;

whereas it is a maximal solution (S) if

P-a:s: ;t # T; ~Y1

t # Y1t and ~Y

2

t # Y2t :

The following theorems, related to existence of minimal and maximal solutions of S,

are the main results of this paper.


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Theorem 2.2. Assume that the data ðc1;c2; j1; j2; a; bÞ satisfy Assumptions (B1)–(B4).

Then system (S) of RBSDEs associated with ðc1;c2; j1; j2; a; bÞ admits a minimal

solution ðY 1; Z 1;K 1; Y 2; Z 2;K 2Þ.

Theorem 2.3. Suppose that the data ðc1;c2; j1; j2; a; bÞ satisfy Assumptions (B1)–(B3)

and, in addition, Assumption (B40) on the process ðaðtÞÞt#T . Then, system (S) of RBSDEs

associated with ðc1;c2; j1; j2; a; bÞ admits a maximal solution ðY 1; Z 1;K 1; Y 2; Z 2;K 2Þ.

The proof of Theorem 2.3 can be obtained from Theorem 2.2 by considering the

minimal solution of the system associated with ð2c1ðt;v;2y;2zÞ;2c2ðt;v;2y;2zÞ;2j1;2j2;2a;2bÞ.

Section 3 is devoted to the proof of Theorem 2.2.

3. Proof of Theorem 2.2

Step 1: Construction of the sequences and properties.

We first introduce two increasing approximation schemes ðY 1;n; Z 1;n;K 1;nÞ and

ðY 2;n; Z 2;n;K 2;nÞ that converge to the minimal solution of (S).

Consider the following BSDEs defined recursively, for any n $ 1, by

ðY 1;0; Z 1;0Þ [ S2 £Md;2;

Y1;0t ¼ j1 þ

Ð T

tc1ðs; Y1;0

s ; Z1;0s Þds 2

Ð T

tZ1;0

s dBs; t # T;

8<: ð3:13Þ

and for n $ 0 and any t # T ,

ðSnÞ

Y2;nþ1t ¼ j2 þ

Ð T

tc2ðs;Y

2;nþ1s ;Z2;nþ1

s Þds 2 ðK2;nþ1T 2 K

2;nþ1t Þ2

Ð T

tZ2;nþ1

s dBs;

Y2;nþ1t # Y

1;nt þ bðtÞ;

Y1;nþ1t ¼ j1 þ

Ð T

tc1ðs;Y

1;nþ1s ;Z1;nþ1

s Þds þ ðK1;nþ1T 2 K

1;nþ1t Þ2

Ð T

tZ1;nþ1

s dBs;

Y1;nþ1t $ Y

2;nþ1t 2 aðtÞ;Ð T

0ðY1;nþ1

s 2 ðY2;nþ1s 2 aðsÞÞdK1;nþ1

s ¼ 0 andÐ T

0ðY1;n

s þ bðsÞ2 Y2;nþ1s ÞdK2;nþ1

s ¼ 0:

8>>>>>>>>><>>>>>>>>>:In view of Assumptions (B1)–(B4), it is easily shown by induction that for any n $ 1,

the triples ðY 1;n; Z 1;n;K 1;nÞ and ðY 2;n; Z 2;n;K 2;nÞ are well defined and belong to the space

S2 £Md;2 £A2, because the pair of processes ðY 1;0; Z 1;0Þ solution of (3.13) exists.

Additionally, by the Comparison Lemma 2.2, we have P-a.s., for all t # T , Y1;0t # Y

1;1t

because the process K 1;1 is increasing and then K1;1T 2 K

1;1t $ 0;;t # T . Next, using

once more the comparison result, we obtain Y 2;1 # Y 2;2: Finally, an induction argument

leads to

;n $ 0; P-a:s:; for all t # T ; Y1;nt # Y1;nþ1

t and Y2;nþ1t # Y2;nþ2

t :

Next, let us consider the following standard BSDE:

�Y2 [ S2 and �Z [ Md;2;

�Y2t ¼ j2 þ

Ð T

tc2ðs; �Y

2s ; �Z

2s Þds 2

Ð T

t�Z

2s dBs; t # T :

8<:


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The solution of this equation exists (see, e.g. [7]). Furthermore, since the process K 2;n

is non-decreasing then using standard comparison theorem for BSDEs (see, e.g. [7])

we obtain

P-a:s:; ;t # T; ;n Y2;nt # �Yt: ð3:14Þ

Finally, let ð ~Y; ~Z; ~KÞ be the solution of the following RBSDE associated with ðc1; j1; �YÞ,i.e. for any t # T ,

ð ~Y; ~Z; ~KÞ [ S2 £Md;2 £ S2;

~Yt ¼ j1 þÐ T

tc1ðs; ~Ys; ~ZsÞds þ ð ~KT 2 ~KtÞ2

Ð T

t~ZsdBs;

~Yt $ �Yt 2 aðtÞ;Ð T

0ð ~Ys 2 ð �Ys 2 aðsÞÞd ~Ks ¼ 0:

8>>>>><>>>>>:Again using the Comparison Lemma 2.2 and relying on (3.14), we have

P-a:s:; ;t # T ; ;n; Y1;nt # ~Yt: ð3:15Þ

Therefore, from (3.14) and (3.15), it follows that

E supn$0

sup0#t#T

ðjY1;nt j þ jY2;n

t jÞ2� �

, 1: ð3:16Þ

Moreover, using the estimates given in Lemma 2.1 for standard RBSDEs, there exists a

real constant C $ 0 such that for all n $ 0,

E

ðT

0

jZ1;ns j þ jZ2;n

s j� �2

ds

� �þ E ðK

1;nT Þ2 þ ðK

2;nT Þ2

h i# C: ð3:17Þ

Let Y 1 and Y 2 be two optional processes defined, for all t # T , by

Y1t ¼ lim

n!1Y1;n

t and Y2t ¼ lim

n!1Y2;n

t :

Step 2: Existence of a solution for (S).

Since the processes b and Y 1;n are of Ito type, then thanks to a result by El Karoui et al.

([5], Proposition 4.2, p. 713), the process K 2;n is absolutely continuous w.r.t. t. Moreover,

we have, for all t # T ,

dK2;nt # 1

½Y2;nþ1t ¼Y

1;nt þbt�

c2ðt; Y1;nt þ bt; Z1;n

t þ V2t Þ þ U2

t þ c1ðt; Y1;nt ; Z1;n

t Þ� �þ

dt:

Hence, by (B1) and (B4), there exists a constant C $ 0 such that, for all n $ 1,

E

ðT

0

dK2;nt

dt

!2

dt

24 35 # C:

In view of this estimate together with (3.17), there exists a subsequence along which

ððdK2;nt =dtÞ0#t#T Þn$1, ððc2ðt; Y

2;nþ1t ; Z

2;nþ1t ÞÞ0#t#T Þn$1 and ððZ

2;nþ1t Þ0#t#T Þn$1 converge

weakly in their respective spaces to the processes ðk2t Þt#T , ðw2ðtÞÞt#T and ðZ2

t Þt#T which

also belong to M1;2, M1;2 and Md;2, respectively.

Next, for any n $ 0 and any stopping time t, we have

Y2;nþ1t ¼ Y

2;nþ10 2

ðt0

c2ðs; Y2;nþ1s ; Z2;nþ1

s Þds þ K2;nþ1t þ

ðt0

Z2;nþ1s dBs:


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Taking the weak limits on each side and along this subsequence yields

Y2t ¼ Y2

0 2

ðt0

w2ðsÞds þ

ðt0

k2s ds þ

ðt0

Z2s dBs; P-a:s:

Since the processes appearing in each side are optional, using the optional section theorem

(see, e.g. [2], Chapter IV, p. 220), it follows that

P-a:s:; ;t # T ; Y2t ¼ Y2

0 2

ðt

0

w2ðsÞds þ

ðt

0

k2s ds þ

ðt

0

Z2s dBs: ð3:18Þ

Therefore, the process Y 2 is continuous. Relying on both Dini’s Theorem and Lebesgue’s

dominated convergence one, we also get that

limn!1

E supt#T

jY2;nt 2 Y2

t j2

� �¼ 0:

We will now focus on the convergence of ðY 1;nÞn$0. Using estimates (3.16) and (3.17) and

then applying Peng’s monotone limit theorem (see [8]) to the sequence (Y 1;n;Z 1;n;K 1;n),

we get that Y 1 is cadlag. Moreover, there exist an F-adapted cadlag non-decreasing

process K1 and a process Z1 of Md;2 such that ðZ 1;nÞn$0 converges to Z 1 in Lpðdt^ dPÞ for

any p [ ½1; 2Þ. Moreover, for any stopping time t, the sequence ðK1;nt Þn$1 converges

weakly to K1t in L2ðV;F t; dPÞ. Relying now on the Snell envelope representation (see [5],

Proposition 2.3, p. 705), we have, for any n $ 1 and t # T ,

Y1;nþ1t ¼ ess supt$tE

ðtt


s Þds þ ðY2;nþ1t 2 aðtÞÞ1½t,T� þ j11½t¼T�jF t

� �:

ð3:19Þ

But for any n $ 0 and t # T ,

E ess supt$tE

ðtt


s Þds þ ðY2;nþ1t 2 aðtÞÞ1½t,T� þ j11½t¼T�jF t

� ��2ess supt$tE

ðtt

c1ðs; Y1s ; Z1

s Þds þ ðY2t 2 aðtÞÞ1½t,T� þ j11½t¼T�jF t

� �� # E ess supt$t E

ðtt

ðc1ðs; Y1;nþ1s ; Z1;nþ1

s Þ2 c1ðs; Y1s ; Z1

s ÞÞds

��þðY2;nþ1

t 2 Y2tÞÞ1½t,T�jF t

��# E E

ðT

0


s Þ2 c1ðs; Y1s ; Z1

s Þ�� ds þ supt#T jY

2;nþ1t 2 Y2

t kF t

� �� :

As ðY 2;nÞn$1 converges uniformly (with respect to t) in L2ðdPÞ to Y 2 (actually the

convergence holds in S2), ðY 1;nÞn$0 converges to Y 1 in L2ðdt^ dPÞ, thanks to (3.16) and

ðZ 1;nÞn$0 converges to Z 1 in Lpðdt^ dPÞ ð1 # p , 2Þ, then the right-hand side of the

previous inequality converges to 0 as n !1. But for any t # T , once more thanks to

(3.16), ðY1;nt Þn$0 converges to Y1

t in L2ðdPÞ, therefore, from equality (3.19), we deduce

that for any t # T:

Y1t ¼ ess supt$tE

ðtt

c1ðs;Y1s ;Z

1s Þds þ ðY2

t 2 aðtÞÞ1½t,T� þ j11½t¼T�jF t

� �; P-a:s: ð3:20Þ


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As Y 1 and the right-hand side of the latter equality are cadlag, then those processes are

indistinguishable, i.e. P-a.s., for all t # T , equality (3.20) holds.

Next, the process ðY2t 2 aðtÞÞt#T is continuous and Y2

T 2 aðTÞ ¼ j2 2 aðTÞ # j1

through assumption (B3); therefore, using Theorem 2.1, we get that the right-hand side of

(3.20) is continuous.

Henceforth, Y 1 is continuous and, using Dini’s theorem, the convergence of ðY 1;nÞn$1

to Y 1 holds in S2. Relying next on the Doob–Meyer decomposition of the supermartingale

ðY1t þ

Ð t

0c1ðs;Y

1s ;Z

1s ÞdsÞt#T (see also Theorem 2.1), there exist Z 1 and K 1 such that, for all

t # T ,

Y1t ¼ j1 þ

Ð T

tc1ðs; Y1

s ; Z1s Þds þ ðK1

T 2 K1t Þ2

Ð T

tZ1

s dBs;

Y1t $ Y2

t 2 aðtÞ andÐ T

0Y1

s 2 ðY2s 2 aðsÞÞ

� �dK1

s ¼ 0:

8<:Since the convergence of ðY 1;nÞn$1 to Y 1 holds in S2, we can now rely on standard

arguments and, in particular, on Ito’s formula applied to ðY 2;n 2 Y 2;mÞ2 (m; n $ 0) to

claim that ðZ 2;nÞn$1 is a Cauchy sequence, and, therefore, that converges to Z 2 in Md;2.

Using this and taking into account the decomposition obtained in (3.18), we finally get, for

any t # T ,

Y2t ¼ j2 þ

Ð T

tc2ðs; Y2

s ; Z2s Þds 2

Ð T

tk2

s ds 2Ð T

tZ2

s dBs;

Y2t # Y1

t þ bðtÞ:

8<:Due to the weak convergence of ððdK

2;nt =dtÞt#T Þn$1 to the process k 2 and the strong

convergence of (Y 1;n) and ðY 2;nÞ in S2, it follows that

0 ¼

ðT

0

ðY1;ns þ bðsÞ2 Y2;nþ1

s ÞdK2;nþ1s !

ðT

0

ðY1s þ bðsÞ2 Y2

s Þk2s ds ¼ 0;

which implies that ðY 2; Z 2;K 2 UÐ :

0k2

s dsÞ is solution for the second part of ðSÞ, and

henceforth, the 6-tuple ðY 1; Z 1;K 1; Y 2; Z 2;K 2Þ is a solution of (S).

This solution is actually a minimal one. Indeed, if there is another one

ðY1; Z1;K1; Y2; Z2;K2Þ then, by comparison, we obviously get Y1 $ Y 1;0 and then

Y2 $ Y 2;1. Finally, by induction we have, for any n $ 1, Y1 $ Y 1;n and then Y2 $ Y 2;n,

which implies the desired result after taking the limit as n goes to 1.

3.1 On the uniqueness of the solution of system (S)As shown in Section 3.2, in general, we do not have uniqueness of the solution of (S).

However, in some specific cases, such as in the following result, uniqueness holds.

Theorem 3.1. Assume that

(i) the mappings c1 and c2 do not depend on ðy; zÞ, i.e. ci U ðciðt;vÞÞ, i ¼ 1; 2

(ii) the barriers a and b satisfy

P-a:s:

ðT

0

1½aðsÞ¼bðsÞ�ds ¼ 0: ð3:21Þ

Then, the solution of (S) is unique.


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Proof. The proof relies on the uniqueness of the solution of a RBSDE with one lower

barrier. Indeed, let ðY 1; Z 1;K 1; Y 2; Z 2;K 2Þ be a solution of (S) and, for t # T , let us set

Yt ¼ Y1t 2 Y2

t , Zt ¼ Z1t 2 Z2

t and Kt ¼ K1t þ K2

t . Therefore, the triple ðY ; Z;KÞ belongs to

S2 £Md;2 £A2. Moreover, for any t # T , it satisfies

Yt ¼ j1 2 j2 þÐ T

t{c1ðu;vÞ2 c2ðu;vÞ}du þ ðKT 2 KtÞ2

Ð T

tZudBu;

Yt $ max{ 2 aðtÞ;2bðtÞ} andÐ T

0Ys þ min{aðsÞ; bðsÞ}� �

dKs ¼ 0:

8<: ð3:22Þ

Indeed, the two first relations being obvious, it only remains to show the third one. ButðT

0

Ys þ min{aðsÞ; bðsÞ}� �

dKs ¼

ðT

0

Y1s 2 Y2

s þ aðsÞ� �

1½aðsÞ#bðsÞ�dðK1s þ K2

s Þ

þ

ðT

0

Y1s 2 Y2

s þ bðsÞ� �

1½aðsÞ.bðsÞ�dðK1s þ K2

s Þ: ð3:23Þ

However, ðT

0

Y1s 2 Y2

s þ aðsÞ� �

1½aðsÞ#bðsÞ�dK1s ¼ 0;

because, for any t # T , dK1t ¼ 1½Y1

t 2Y2t þaðtÞ¼0�dK1

t . On the other hand,

0 #

ðT

0

Y1s 2 Y2

s þ aðsÞ� �

1½aðsÞ#bðsÞ�dK2s #

ðT

0

Y1s 2 Y2

s þ bðsÞ� �

1½aðsÞ#bðsÞ�dK2s ¼ 0;

because for any t # T , it holds that dK2t ¼ 1½Y2

t 2Y1t þbðtÞ¼0�dK2

t : In the same way, one can

show that the second term in (3.23) is null, and therefore, the third relation in (3.22) holds

true. It follows that ðY; Z;KÞ is a solution for the one lower barrier RBSDE associated with

ðc1 2 c2; j1 2 j2;max{ 2 aðtÞ;2bðtÞ}Þ. As the solution of the latter equation is unique by

Theorem 2.1, then for any solution ðY 1; Z 1;K 1; Y 2; Z 2;K 2Þ of (S), the differences Y 1 2

Y 2 and Z 1 2 Z 2 and the increasing process K 1 þ K 2 are unique.

Next, let us express K 1 and K 2 in terms of K. For any t # T , we claim that

K1t ¼

ðt

0

1½Y1s2Y2

sþaðsÞ¼0�dK1s

¼

ðt

0

1½Y1s2Y2

sþaðsÞ¼0�1½aðsÞ,bðsÞ� dK1s þ

ðt

0

1½Y1s2Y2

sþaðsÞ¼0�1½aðsÞ.bðsÞ� dK1s ; ð3:24Þ

where to get this second equality, we make use of the both absolute continuity of dK 1, the

increasing property of K 1, and condition (ii) on the barriers to argue thatðT

0

1½Y1s2Y2

sþaðsÞ�1½aðsÞ¼bðsÞ� dK1s ¼

ðT

0

ds1½Y1s2Y2

sþaðsÞ�1½aðsÞ¼bðsÞ�

dK1s

ds

¼ 0:

On the other hand,ðT

0

1½Y1s2Y2

sþaðsÞ¼0�1½aðsÞ,bðsÞ� dK2s ¼

ðT

0

1½Y1s2Y2

sþaðsÞ¼0�1½aðsÞ,bðsÞ�1½Y1s2Y2

sþbðsÞ¼0� dK2s ¼ 0:

In a similar fashion, we haveðT

0

1½Y1s2Y2

sþaðsÞ¼0�1½aðsÞ.bðsÞ� dK2s ¼ 0:


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Therefore, going back to (3.24), we obtain, for all t # T ,

K1t ¼

ðt

0

1½Y1s2Y2

sþaðsÞ¼0�dKs;

which implies that K 1 is unique and then so is K 2. Next, writing the equations satisfied by

Y 1 and Y 2 and taking the conditional expectation (w.r.t. F t), we obtain their uniqueness.

From standard arguments, uniqueness of Z 1 and Z 2 follows immediately. A

Remark 3.1. This uniqueness result can be slightly generalized to generators of the

following forms:

c1ðt;v; y; zÞ ¼ ~c1ðt;vÞ þ aty þ btz and c2ðt;v; y; zÞ ¼ ~c2ðt;vÞ þ aty þ btz;

where a and b are P-progressively measurable bounded processes taking their values in R

and Rd, respectively. The proof is the same as the previous one, noting that the process

Y ¼ Y 1 2 Y 2 solves a linear RBSDE, which is explicitly solvable.

For the sake of completeness, we now consider the more general case, i.e. when

condition (3.21) on the barriers is no more satisfied.

Lemma 3.1. Under (B1)–(B3) together with (B4) or (B40), if condition (3.21) on the

barriers is no more satisfied, then on any interval ½a;b� of ½0; T�, where a ; b, uniqueness

of a solution for system (S) holds only in the trivial cases where neither the first component

Y 1 nor the second component Y 2 is reflected processes.

Proof. To prove this, let us consider a non-trivial interval ½a;b� with 0 , a , b , T and

where aðtÞ ¼ bðtÞ; ;t [ ½a;b� and assume that uniqueness of the solution of (S) holds. So

let ðY1; Z1;K1; Y2; Z2;K2Þ be the minimal solution of (S) which is then equal to the

maximal one by uniqueness. Next, let us consider the following system of RBSDEs on the

time interval ½a;b�: for any s [ ½a;b�,

ðSminÞ

dY1s ¼ 2c1ðs; Y1

s ; Z1s Þds þ Z1

s dBs; Y1b ¼ Y1

b;

dY2s ¼ 2c2ðs; Y2

s ; Z2s Þds þ Z2

s dBs þ dK2s ; Y2

b ¼ Y2b;

Y2s # Y1

s þ bðsÞ;Ð ba

Y1s þ bðsÞ2 Y2

s

� �dK2

s ¼ 0:

8>>><>>>:Note also that both existence and uniqueness for ðSminÞ on ½a;b� result from standard

results for BSDEs (or RBSDEs with one barrier).

On the other hand, let us consider the following system of RBSDEs which is similar to

(S) but on the time interval ½0;a� and which actually has a solution. For any t [ ½0;a�,

g1t ¼ Y1

a þÐ a

tc1ðs; g

1s ; u

1s Þds þ ðz1

a 2 z1t Þ2

Ð atu1

s dBs;

g2t ¼ Y2

a þÐ a

tc2ðs; g

2s ; u

2s Þds þ ðz2

a 2 z2t Þ2

Ð atu2

s dBs;

g1t $ g2

t 2 aðtÞ and g2t # g1

t þ bðtÞ;Ð a0g1

s 2 ðg2s 2 aðsÞÞ

� �dz1

s ¼ 0 andÐ a

0ðg1

s þ bðsÞ2 g2s Þdz

2s ¼ 0;

8>>>>><>>>>>:where both Y1

a and Y2a stand for the value at time t ¼ a of the first component Y 1 (and the

second component Y 2) of the unique solution to system (Smin) defined on ½a;b�. Therefore,


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the following process

ðg1t ; u

1t ; z

1t ; g

2t ; u

2t ; z

2t Þ1½t#a� þ ðY1

t ; Z1t ; z

1a; Y2

t ; Z2t ;K2

t 2 K2a þ z2

aÞ1½a,t#b�

�þðY1

t ; Z1t ;K1

t 2 K1b þ z1

a;Y2t ;Z

2t ;K2

t 2 K2b þ K2

b 2 K2a þ z2

aÞ1½b,t#T�

ot#T

;

obtained by concatenation, is also a solution for (S). Using once more uniqueness for (S)

yields for any t [ ½a;b�

Y1t ¼ Y1

t and Z1t ¼ Z1

t :

It implies that process Y1 is not reflected on the time interval ½a;b�. Now going back to

system ðSminÞ and making the reflection on Y 2 and not on Y 1, we define a new system

denoted by (Smax), and, therefore, proceeding as above, we obtain that the solution

ð �Y1; �Y2Þ on ½a;b� is such that the second component is not reflected. By uniqueness, we

can claim that ðY1; Y2Þ ; ð �Y1; �Y2Þ on ½a;b�, and hence, Y2 is not reflected on ½a;b�. A

Remark 3.2. On any time interval where the solution of (Smin) provides a solution of the

system (S), it is straightforward to check, by using standard comparison result for BSDEs

or RBSDEs (see Theorem 2.2), that solution (Y1; Y2) is the minimal solution of (S).

Similarly, whenever the solution ( �Y1; �Y2) associated with system (Smax) is a solution of

(S), it can be proved that it is the maximal one.

3.2 Non-uniqueness: two counter-examples

In this section, we provide two explicit counter-examples of system (S) where uniqueness

does not hold.

First example. In this first example, we study the case when the two penalties a and b are

equal to zero and the terminal conditions j1 and j2 are equal. In addition, the two generators

ðs;vÞ! ciðs;v; y; zÞ, for i ¼ 1; 2, are assumed to be independent of ðy; zÞ and satisfy

P-a:s:; ;t # T ; c1ðt;vÞ2 c2ðt;vÞ , 0: ð3:25Þ

Therefore, this leads to the following system of RBSDEs:

ðS1Þ

Y1t ¼ j1 þ

Ð T

tc1ðsÞds 2

Ð T

tZ1

s dBs þ ðK1T 2 K1

t Þ;

Y2t ¼ j1 þ

Ð T

tc2ðsÞds 2

Ð T

tZ2

s dBs 2 ðK2T 2 K2

t Þ;

Y1t $ Y2

t ; t # T ;Ð T

0Y1

s 2 Y2s

� �dðK1

s þ K2s Þ ¼ 0:

8>>>>>><>>>>>>:Introduce ðY 1;min; Y 2;minÞ (resp. ðY 1;max; Y 2;maxÞ) as being the minimal (resp. the

maximal) solution of the system (S1) as constructed in Section 2.

Our objective is to establish that, even in this simple example, uniqueness for system

(S1) fails to hold. To do this, we prove that the minimal and maximal solutions for that

system do not coincide. So let us consider an arbitrary solution ðY 1; Y 2Þ of the system ðS1Þ.

If we set �Y ¼ Y 1 2 Y 2, then we have, for all t # T ,

�Yt ¼Ð T

tðc1ðsÞ2 c2ðsÞÞds þ ð �KT 2 �KtÞ þ

Ð T

t�Zs dBs;

�Yt $ 0; t # T ;Ð T

0�Yt d �Kt ¼ 0:

8>>><>>>: ð3:26Þ


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Hence, standard results imply that the solution of this RBSDE is unique. Then taking

conditional expectation w.r.t. F t in (3.26), it yields

Y1t 2 Y2

t ¼ E

ðT

t

ðc1ðsÞ2 c2ðsÞÞds þ ð �KT 2 �KtÞjF t

:

Therefore, thanks to both (3.25) and the constraint condition on Y 1 2 Y 2, we obtain the

strict increasing property of �K. Next, for i ¼ 1; 2 denoting by K i;min and by K i;max, the pair

of increasing processes associated with each component of both the minimal solution and

the maximal solution, we show below

dK1;mint ; 0 and dK2;max

t ; 0; for any t # T :

To this end, let us consider the following system: for any s # T ,

dY1s ¼ 2c1ðsÞds þ Z1

s dBs; Y1T ¼ j1;

dY2s ¼ 2c2ðsÞds þ Z2

s dBs þ dK2s ; Y2

T ¼ j1;

Y2s # Y1

s ;Ð T

0Y1

s 2 Y2s

� �dK2

s ¼ 0:

8>>><>>>: ð3:27Þ

Then, for any t # T , we have

Y1t 2 Y1;min

t ¼ 2

ðT

t

ðZ1s 2 Z1;min

s ÞdBs 2 K1;minT 2 K1;min

t

� �:

As Y1t 2 Y

1;mint $ 0, it follows from standard arguments that K

1;mint ¼ K

1;minT for any t # T

and then K 1;min ; 0. Using now uniqueness of BSDEs, reflected or not, we obtain that

Y1 ; Y 1;min and then Y2 ; Y 2;min. In the same way, by considering the reflection on the

other equation in (3.27), we obtain that K 2;max ; 0.

Next, using again the uniqueness of the triple ð �Y ¼ Y 1 2 Y 2; �Z ¼ Z 1 2 Z 2; �K ¼

K 1 þ K 2Þ and solving the one lower barrier RBSDE (3.26), it follows that

�K ¼ K 1;min þ K 2;min ¼ K 2;min ¼ K 1;max þ K 2;max ¼ K 1;max:

The process �K uniquely defined by (3.26) being strictly increasing then, in view of the

second line of the previous equality, K 1;max and K 2;min are also strictly increasing.

Consequently, both Y 1;min and Y 1;max solve BSDEs with the same generator c1 and the

same terminal condition j1. However, Y 1;min solves a standard BSDE without any

reflection, whereas Y 1;max solves a RBSDE with the associated process K 1;max which is

strictly increasing, which yields that Y 1;min – Y 1;max and achieves the proof. A

Second example. Let us assume the following structure of the generators:

c1ðt;v; yÞ ¼ y and c2ðt;v; yÞ ¼ 2y:

We also assume that for any t # T , at ¼ bt ¼ 0. Then, we are led to consider the following

system, for all t [ ½0; T�,

ðS2Þ

Y1t ¼ 1 þ

Ð T

tY1

s ds 2Ð T

tZ1

s dBs þ ðK1T 2 K1

t Þ;

Y1t $ Y2

t andÐ T

tðY1

s 2 Y2s ÞdK1

s ¼ 0;

Y2t ¼ 1 þ

Ð T

t2Y2

s ds 2Ð T

tZ2

s dBs 2 ðK2T 2 K2

t Þ;

Y1t $ Y2

t andÐ T

tðY2

s 2 Y1s ÞdK2

s ¼ 0:

8>>>>>>><>>>>>>>:


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In this second example, we will show that the minimal and maximal solutions are not

equal, and, therefore, uniqueness does not hold. We note that, considering an arbitrary

solution of system (S2), the difference Y 1 2 Y 2 does not solve any more RBSDEs, which

was the crucial fact, therefore we rely on in the previous example.

To prove that uniqueness does not hold for (S2), let us consider the minimal

(resp. maximal) solution (Y 1;min; Y 2;min) (resp. (Y 1;max; Y 2;max)) of (S2) which is given,

for all t [ ½0; T�, by

dY1;mint ¼ 2Y

1;mint dt 2 Z

1;mint dBt and Y

1;minT ¼ 1;

dY2;mint ¼ 22Y

2;mint dt 2 Z

2;mint dBt þ dK

2;mint and Y

2;minT ¼ 1;

Y1;mint $ Y

2;mint and

Ð T

0ðY

1;mint 2 Y

2;mint ÞdK

2;mint ¼ 0;

8>>><>>>:and

dY1;maxt ¼ 2Y

1;maxt dt 2 Z

1;maxt dBt 2 dK

1;maxt and Y

1;maxT ¼ 1;

dY2;maxt ¼ 22Y

2;maxt dt 2 Z

2;maxt dBt and Y

2;maxT ¼ 1;

Y1;maxt $ Y

2;maxt and

Ð T

0ðY

1;maxt 2 Y

2;maxt ÞdK

1;maxt ¼ 0;

8>>><>>>:meaning that K 1;min ; 0 and K 2;max ; 0. But, the solution of the first system is given, for

all t # T , by

Y1;mint ¼ Y2;min

t ¼ eT2t; Z1;mint ¼ Z2;min

t ¼ 0 and K2;mint ¼ eT ð1 2 e2tÞ:

On the other hand, one of the second systems is given, for all t # T , by

Y1;maxt ¼ Y2;max

t ¼ e2ðT2tÞ; Z1;maxt ¼ Z2;max

t ¼ 0 and K1;maxt ¼

1

2e2T ð1 2 e22tÞ:

Therefore, as we can see, uniqueness for (S2) does not hold in this case. Finally, let us

point out that in order to exhibit the solutions of the previous systems, we have kept in

mind two facts: (1) the solutions of those systems are deterministic and (2) we have used

properties of the Snell envelope of processes. A

Remark 3.3. As a slight generalization of the previous counter-example, let us consider

here the more general case when the generators take the following form:

c1ðs; yÞ ¼ a1s y and c2ðs; yÞ ¼ a2

s y;

where a1 and a2 are deterministic functions, integrable on ½0; T� and satisfy

a1t , a2

t ;;t # T .

We are led to study the following system given, for all t [ ½0; T�, by

ðS02Þ

dY1t ¼ 2a1

t Y1t dt þ Z1

t dBt 2 dK1t ; Y1

T ¼ 1;

dY2t ¼ 2a2

t Y2t dt þ Z2

t dBt þ dK2t ; Y2

T ¼ 1;

Y1t $ Y2

t andÐ T

0ðY2

t 2 Y1t ÞdK1

t ¼Ð T

0ðY2

t 2 Y1t ÞdK2

t ¼ 0:

8>>><>>>:Similarly as in the previous proof, we consider here the minimal solution (resp. the

maximal solution) of the previous system which is denoted by (Y 1;min; Y 2;min) (resp. by


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(Y 1;max; Y 2;max)). In addition and as already mentioned in Remark 3.2, the increasing

processes associated with these solutions are such that

K 1;min ; 0 and K 2;max ; 0:

Thus, Y 1;min and Y 2;max solve a standard BSDE which, by uniqueness, implies that, for all

t [ ½0; T�,

Y1;mint ¼ exp

ðT

t

a1s ds

; Y2;max

t ¼ exp

ðT

t

a2s ds

and Z1;min

t ¼ Z2;maxt ¼ 0:

Next, since the data of the system are deterministic and using the characterization of the

Snell envelope of a process as the smallest supermartingale which dominates it (see, e.g.

[4]), we get that

;t [ ½0;T�; Y2;mint ¼ Y1;min

t ; Z2;mint ¼ 0 and K2;min

t ¼

ðt

0

a2s 2a1

s

� �Y1;min

s ds: ð3:28Þ

Thus, we have obtained the minimal solution of ðS02Þ.

Concerning the maximal solution, we proceed in the same way and then we can check

that Y 1;max, Z 1;max and K 1;max are given by

;t [ ½0; T�; Y1;maxt ¼ Y2;max

t ; Z1;maxt ¼ 0 and K1;max

t ¼

ðt

0

a2s 2 a1

s

� �Y2;max

s ds:

Therefore, since a1 , a2, we obtain, for i ¼ 1; 2, Y i;min , Y i;max, which entails that, in

this case, uniqueness does not hold for ðS02Þ as well.

4. Study of the related system of VI

4.1 The system of VI

In this section, we briefly describe the connection between solutions of (S) and existence

viscosity solutions of the following system of VI with interconnected obstacles:

ðVIÞ

min u1ðt; xÞ2 u2ðt; xÞ þ aðt; xÞ;2›tu1ðt; xÞ2 Lu1ðt; xÞ2 c1ðt; xÞ

� �¼ 0;

max u2ðt; xÞ2 u1ðt; xÞ2 bðt; xÞ;2›tu2ðt; xÞ2 Lu2ðt; xÞ2 c2ðt; xÞ

� �¼ 0;

u1ðT ; xÞ ¼ g1ðxÞ and u2ðT ; xÞ ¼ g2ðxÞ;

8>><>>:when the dependence of dynamics of the cash flows Y 1; Y 2 of, e.g. the fluctuations of the

market prices, X, assumed to be a diffusion process, is made explicit. Note that the

obstacles may depend on the diffusion process X. In (VI), L denotes the infinitesimal

generator of the diffusion X.

For ðt; xÞ [ ½0; T� £ Rk, let X t;x U ðXt;xs Þs#T be the solution of the following standard

differential equation:

dXt;xs ¼ mðs;Xt;x

s Þds þ s ðs;Xt;xs ÞdBs; t # s # T; Xt;x

s ¼ x; s # t; ð4:29Þ

where the two functions m U ðmðt; xÞÞ and s U ðs ðt; xÞÞ defined on ½0; T� £ Rk and taking

their respective values in Rk and Rk£d are uniformly Lipschitz w.r.t. x and have linear

growth. This means that, for all t, x, y,

ðC1Þjmðt; xÞ2 mðt; yÞj þ js ðt; xÞ2 s ðt; yÞj # Cjx 2 yj;

jmðt; xÞj þ js ðt; xÞj # Cð1 þ jxjÞ:

(


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These properties ensure both existence and uniqueness of a solution for (4.29).

Additionally and for any u, u $ 2, there exists a constant C such that for any x in Rk

E sup0#s#T

Xt;xs

�� u� �# Cð1 þ jxj

uÞ: ð4:30Þ

Next, in this setting, the infinitesimal generator L of the diffusion X :¼bX t;x is defined,

for any function F in C1;2ð½0; T� £ RÞ, as follows:

LðFÞðt; xÞ ¼ kmðt; xÞ;Fðt; xÞlþ1

2Trace ss TD2Fðt; xÞ

� �;

where T stands for the transpose operation. Let us now make the following assumption on

the functions ci, i ¼ 1; 2:

ðC2Þ

The functions c1 and c2 do not depend on ðy; zÞ; i:e: c1 ¼ c1ðt; xÞ and

c2 ¼ c2ðt; xÞ; they are jointly continuous in ðt; xÞ and have polynomial

growth; i:e: there exist two positive constants q and C such that;

for all ðt; xÞ; jciðt; xÞj # Cð1 þ jxjqÞ; i ¼ 1; 2:

8>>>>><>>>>>:The assumption that each generator ci does not depend on ðy; zÞ but only on x is quite

natural especially for applications in economics. It means that the payoffs are not of

recursive type, and the utilities c1 and c2 depend only on the process X t;x which stands,

e.g. for the price of a commodity such as electricity or oil price in the market.

Next, let gi U ðgiðxÞÞ, i ¼ 1; 2, a U ðaðt; xÞÞ and b U ðbðt; xÞÞ be the given functions

defined, respectively, on Rk and ½0; T� £ Rk, with values in R and satisfy

ðC3Þ

ðiÞ a and b are of polynomial growth; belong to C1;2ð½0;T�£RdÞ and satisfy

;ðt;xÞ[ ½0;T�£Rk;Ð T

01½aðs;Xt;x

s Þ¼bðs;Xt;xs Þ�ds ¼ 0;

ðiiÞ g1 and g2 are continuous and of polynomial growth;

ðiiiÞ ;x [ Rk; g1ðxÞ2 g2ðxÞ$ max{2 aðT ;xÞ;2bðT ;xÞ}:

8>>>>><>>>>>:With the help of the solutions of the system of RBSDEs (S), we show below that (VI)

has a solution in viscosity sense whose definition is the following.

Definition 1. Let ðu1; u2Þ be a pair of continuous functions on ½0; T� £ Rk. It is called

(i) a viscosity supersolution (resp. subsolution) of the system (VI) if for any ðt0; x0Þ [½0; T� £ Rk and any pair of functions ðw1;w2Þ [ ðC 1;2ð½0; T� £ RkÞÞ2 such that

ðw1;w2Þðt0; x0Þ ¼ ðu1; u2Þðt0; x0Þ and for any i ¼ 1; 2, ðt0; x0Þ is a maximum

(resp. minimum) of wi 2 ui, then we have

min u1ðt0; x0Þ2 u2ðt0; x0Þ þ aðt0; x0Þ;2›tw1ðt0; x0Þ2 Lw1ðt0; x0Þ2 c1ðt0; x0Þf g $ 0

ðresp: # 0Þ;

and

max u2ðt0; x0Þ2 u1ðt0; x0Þ2 bðt0; x0Þ;2›tw2ðt0; x0Þ2 Lw2ðt0; x0Þf

2c2ðt0; x0Þg $ 0 ðresp: # 0Þ:

(ii) a viscosity solution of system ðVIÞ if it is both a viscosity supersolution and

subsolution.


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4.2 Construction and regularity of viscosity solution of system

Our objective is to construct and identify a continuous viscosity solution for the system (VI)

by relying on both standard results for the representation of viscosity solutions by BSDEs

(see, for example, [5], Theorem 8.5) and on the results obtained in the previous sections.

For ðt; xÞ [ ½0; T� £ Rk, let ðY 1;ðt;xÞ; Y 2;ðt;xÞ; Z 1;ðt;xÞ; Z 2;ðt;xÞ;K 1;ðt;xÞ;K 2;ðt;xÞÞ be a solution

of the following system of RBSDEs: for any s [ ½t;T�,

ð ~SÞ

Y1;ðt;xÞs ¼ g1ðX

ðt;xÞT Þ þ

Ð T

sc1ðu;Xt;x

u Þ du þ ðK1;ðt;xÞT 2 K1;ðt;xÞ

s Þ2Ð T

sZ1;ðt;xÞ

u dBu;

Y2;ðt;xÞs ¼ g2ðX

ðt;xÞT Þ þ

Ð T

sc2ðu;Xt;x

u Þ du 2 ðK2;ðt;xÞT 2 K2;ðt;xÞ

s Þ2Ð T

sZ2;ðt;xÞ

u dBu;

Y1;ðt;xÞs $ Y2;ðt;xÞ

s 2 aðs;Xt;xs Þ and Y2;ðt;xÞ

s # Y1;ðt;xÞs þ bðs;Xt;x

s Þ;Ð T

tY1;ðt;xÞ

s 2 ðY2;ðt;xÞs 2 aðs;Xt;x

s ÞÞ� �

dK1;ðt;xÞs ¼ 0;Ð T

tðY1;ðt;xÞ

s þ bðs;Xt;xs Þ2 Y2;ðt;xÞ

s ÞdK2;ðt;xÞs ¼ 0:

8>>>>>>>>><>>>>>>>>>:Note that, thanks to Theorems 2.2 and 3.1 and Assumptions (C1)–(C3), especially the fact

that the processes ðaðs;Xt;xs ÞÞs[½t;T� and ðbðs;Xt;x

s ÞÞs[½t;T� verify (B4) and (B40), respectively,

both the minimal and maximal solutions exist and, furthermore, these two solutions

coincide. Indeed, uniqueness holds because the functions ci, i ¼ 1; 2, do not depend on

ðy; zÞ and the barriers satisfy the condition (ii) of Theorem 3.1. Moreover, this unique

solution is obtained as a limit of the increasing and decreasing schemes because the

functions a U ðaðt; xÞÞ and b U ðbðt; xÞÞ belong to C1;2ð½0; T� £ RkÞ and are of polynomial

growth.

According to the construction of the minimal solution of ð ~SÞ we have, for any

s [ ½t; T�,

Y1;ðt;xÞs ; Y2;ðt;xÞ

s

� �¼ lim

n!1Y1;ðt;xÞ;n

s ; Y2;ðt;xÞ;ns

� �;

where ðY 1;ðt;xÞ;n; Y 2;ðt;xÞ;nÞ are defined in the same way as in (Sn) but with the specific data

above. Thanks to Theorem 8.5 in [5], and by an induction argument there exist

deterministic functions uin U ðui

nðt; xÞÞ, i ¼ 1; 2, continuous on ½0; T� £ Rk such that, for

any s [ ½t; T�,

Y1;ðt;xÞ;ns ; Y2;ðt;xÞ;n

s

� �¼ u1

nðs;Xt;xs Þ; u2

nðs;Xt;xs Þ

� �:

Moreover, there exist two positive constants a1 and a2 such that, for any

ðt; xÞ [ ½0; T� £ Rk,

juinðt; xÞj þ jui

nðt; xÞj # a1ð1 þ jxja2Þ:

Finally, the sequences ðuinÞn$0, i ¼ 1; 2, are increasing because Y i;ðt;xÞ;n # Y i;ðt;xÞ;nþ1,

i ¼ 1; 2. Therefore, there exist two deterministic lower semi-continuous functions ui,

i ¼ 1; 2, with polynomial growth such that, for i ¼ 1; 2 and for any ðt; xÞ [ ½0; T� £ Rk,

Y1;ðt;xÞs ; Y2;ðt;xÞ

s

� �¼ u1ðs;Xt;x

s Þ; u2ðs;Xt;xs Þ

� �; ;s [ ½t; T�: ð4:31Þ

Now, and in the same fashion, when considering the decreasing approximating

scheme, it follows from the same result in [5] and from the uniqueness that u1 and u2 are

also upper semi-continuous. Therefore, u1 and u2 are continuous with polynomial growth.

Finally, relying on Theorem 8.5 in [5], we directly obtain the following result.


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Theorem 4.1. The pair ðu1; u2Þ defined in (4.31) is a continuous viscosity solution for the

system (VI).

Acknowledgements

The authors are grateful to Paavo Salminen and the two anonymous reviewers for their insightfulremarks and suggestions that improved the content of the work. This work was completed while thefirst author was visiting the Department of Mathematics of Universite du Maine. Financial supportfrom MATPYL (RPL) is gratefully acknowledged.

Notes

1. Email: [email protected]. Email: [email protected]

References

[1] M.J. Brennan and E.S. Schwartz, Evaluating natural resource investments, J. Bus. 58 (1985),pp. 135–137.

[2] C. Dellacherie and P.A. Meyer, Probabilites et potentiels, Chapter I–IV Hermann, Paris, 1975.[3] A. Dixit and R.S. Pindyck, Investment Under Uncertainty, Princeton University Press,

Princeton, NJ, 1994.[4] N. El Karoui, Les aspects probabilistes du controle stochastique, in Ecole d’ete de probabilites

de Saint-Flour, Lecture Notes in Mathematics No 876, Springer Verlag, Berlin, 1980.[5] N. El Karoui, C. Kapoudjan, E. Pardoux, S. Peng, and M.-C. Quenez, Reflected solutions of

backward SDE’s and related problems for PDE’s, Ann. Prob. 25(2) (1997), pp. 702–737.[6] R. Korn, Some applications of impulse control in mathematical finance, Math. Methods Oper.

Res. 50 (1999), pp. 493–518.[7] E. Pardoux and S. Peng, Adapted solution of BSDE, Syst. Control Lett. 14 (1990), pp. 55–61.[8] S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of

Doob-Meyer’s type, Prob. Theory Relat. Fields 113 (1999), pp. 473–499.[9] L. Trigeorgis, Real Options: Managerial Flexibility and Strategy in Resource Allocation,

MIT Press, Cambridge, MA, 1996.


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optimal stopping of expected profit and cost yields in an investment under uncertainty

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