getdoc8dfc. 272KB Jun 04 2011 12:04:39 AM
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Vol. 11 (2006), Paper no. 5, pages 121–145.
Journal URL
http://www.math.washington.edu/∼ejpecp/
BSDEs with two reflecting barriers driven by a Brownian and a
Poisson noise and related Dynkin game
S. Hamadène & M. Hassani
Laboratoire de Statistique et Processus,
Université du Maine, 72085 Le Mans Cedex 9, France.
e-mails: hamadene@univ-lemans.fr & mohammed.hassani@univ-lemans.fr
Abstract
In this paper we study BSDEs with two reflecting barriers driven by a Brownian motion
and an independent Poisson process. We show the existence and uniqueness of local and
global solutions. As an application we solve the related zero-sum Dynkin game.
Key words: Backward stochastic differential equation ; Poisson measure ; Dynkin game ;
Mokobodzki’s condition.
.
AMS 2000 Subject Classification: Primary 91A15, 60G40, 91A60; Secondary 60G99.
Submitted to EJP on December 23, 2003. Final version accepted on January 25, 2006.
121
1
Introduction
Let (Ft )t≤T be a filtration generated by a Brownian motion (Bt )t≤T and an independent Poisson measure µ(t, ω, de) defined on a probability space (Ω, F, P ). A solution for the backward
stochastic differential equation (BSDE for short) with two reflecting barriers associated with a
coefficient f (t, ω, y, z), a terminal value ξ and a lower (resp. an upper) barrier (Lt )t≤T (resp.
(Ut )t≤T ) is a quintuple of Ft -predictable processes (Yt , Zt , Vt , Kt+ , Kt− )t≤T which satisfies:
Z
+
−
Vt (e)µ̃(dt, de), t ≤ T ; YT = ξ
−dYt = f (t, Yt , Zt , Vt )dt + dKt − dKt − Zt dBt −
E
Lt ≤ Yt ≤ Ut , ∀t ≤ T
±
K
(1)
are continuous non-decreasing and (Yt − Lt )dKt+ = (Yt − Ut )dKt− = 0 (K0± = 0),
where µ̃ is the compensated measure of µ.
Nonlinear BSDEs have been first introduced by Pardoux and Peng [19], who proved the existence
and uniqueness of a solution under suitable hypotheses on the coefficient and the terminal value
of the BSDE. Since, these equations have gradually become an important mathematical tool
which is encountered in many fields such as finance ([6], [4], [8], [9],...), stochastic games and
optimal control ([10], [11], ...), partial differential equations ([1], [18], [20],...).
In the case when the filtration is generated only by a Brownian motion and when we consider
just one lower barrier (set U ≡ +∞ and K − ≡ 0 in (1)), the problem of existence and uniqueness
of a solution for (1) is considered and solved by El-Karoui et al. in [6]. Their work has been
generalized by Cvitanic & Karatzas in [4] where they deal with BSDEs with two reflecting
barriers.
BSDEs without reflection (in (1) one should take L = −∞ and U = +∞, thereby K ± = 0)
driven by a Brownian motion and an independent Poisson measure have been considered first
by Tang & Li [17] then by Barles et al. in connection with partial-integral differential equations
in [1]. In both papers the authors showed the existence and uniqueness of a solution.
The extension to the case of BSDEs with one reflecting barrier has been established by Hamadène
& Ouknine in [13]. The authors showed the existence and uniqueness of the solution when the
coefficient f is Lipschitz. Two proofs have been given, the first one is based on the penalization
scheme as for the second, it is obtained in using the Snell envelope notion. However both
methods make use of a contraction argument since the usual comparison theorem fails to work
in the general framework.
So in this work we study BSDEs with two reflecting barriers driven by a Brownian motion and
an independent Poisson measure. This is the natural extension of Hamadène & Ouknine’s work.
However there are at least four motivations for considering this problem. The first one is related
to Dynkin zero-sum game. The second is in connection with the real option area since the
stopping and starting problem leads to a BSDE with two reflecting barriers (see e.g. [12]). The
third one is that our problem can provide solutions for variational inequality problems with two
obstacles when the generator is of partial-integral type. Finally we provide a condition easy to
check in practice under which the well known Mokobodski’s hypothesis is satisfied.
122
In this paper we begin to show the existence of an adapted and rcll (right continuous and left
limited) process Y := (Yt )t≤T which in a way is a local solution for (1). Actually we prove that
for any stopping time τ there exist another stopping time θτ ≥ τ and a quadruple of processes
(Z, V, K + , K − ) which with Y verify (1) on the interval [τ, θτ ]. In addition the process Y reaches
the barriers U and L between τ and θτ . In the proof of our theorem, the key point is that the
predictable projection of a process π whose jumping times are inaccessible is equal to π− , the
process of left limits associated with π.
This result is then applied to deal with the zero-sum Dynkin game associated with L, U , ξ and
a process (gs )s≤T which stands for the instantaneous payoff. We show that this game is closely
related to the notion of local solution for (1). Besides we obtain the existence of a saddle-point
for the game under conditions out of the scope of the known results on this subject. Finally we
give some feature of the value function of the game (see Remark 4.3). Our result can be applied
in mathematical finance to deal with American game (or recallable) options whose underlying
derivatives contain a Poisson part (see [15] for this type of option).
Further we consider the problem of existence and uniqueness of a global solution for (1). When
the weak Mokobodski’s assumption [WM] is satisfied, which roughly speaking turns into the
existence of a difference of non-negative supermartingales between L and U , we show existence
and uniqueness of the solution. Then we address the issue of the verification of the condition
[WM]. Actually we prove that under the fully separation of the barriers, i.e., Lt < Ut for any
t ≤ T , the condition [WM] holds true.
This paper is organized as follows :
In Section 2, we deal with local solutions for (1) while Section 3 is devoted to zero-sum Dynkin
games. At the end, in Section 4, we address the problem of existence of a global solution for
(1). ✷
2
Reflected BSDEs driven by a Brownian motion and an independent Poisson point process
Let (Ω, F, (Ft )t≤T ) be a stochastic basis such that F0 contains all P -null sets of F, Ft+ =
ǫ>0 Ft+ǫ = Ft , ∀t < T , and suppose that the filtration is generated by the following two
mutually independent processes :
T
- a d-dimensional Brownian motion (Bt )t≤T
- a Poisson random measure µ on IR+ × E, where E := IRl \ {0} is equipped with its Borel fields
E, with compensator ν(dt, de) = dtλ(de), such that {µ̃([0, t] × A) = (µ − ν)([0, t] × A)}t≤T is a
martingale for every
A ∈ E satisfying λ(A) < ∞. The measure λ is assumed to be σ-finite on
R
(E, E) and verifies E (1 ∧ |e|2 )λ(de) < ∞.
Now let:
- D be the set of Ft -adapted right continuous with left limits processes (Yt )t≤T with values in
IR and D2 := {Y ∈ D, IE[supt≤T |Yt |2 ] < ∞}
- P̃ (resp. P) be the Ft -progressive (resp. predictable) tribe on Ω × [0, T ]
- H2,k (resp. Hk ) be the set of P̃-measurable processes Z := (Zt )t≤T with values in IRk and
dP ⊗ dt-square integrable (resp. P -a.s. Z(ω) := (Zt (ω))t≤T is dt-square integrable)
123
- L2 (resp. L) be the set of mappings V : Ω × [0, T ] × E → IR which are P ⊗ E-measurable and
R
R
R
1
IE[ 0T kVs k2 ds] < ∞ (resp. 0T kVs k2 ds < ∞, P-a.s.) where kvk := ( E |v(e)|2 λ(de)) 2 for v : E →
IR
2 (resp. C ) the space of continuous F -adapted and non-decreasing processes (k )
- Cci
ci
t
t t≤T such
2
that k0 = 0 and IE[kT ] < ∞ (resp. kT < ∞, P-a.s.)
- for a stopping time τ , Tτ denotes the set of stopping times θ such that θ ≥ τ
- for a given rcll process (wt )t≤T , wt− = limsրt ws , t ≤ T (w0− = w0 ) ; w− := (wt− )t≤T and
△w := w − w− ✷
We are now given four objects:
- a terminal value ξ ∈ L2 (Ω, FT , P )
- a map f : Ω × [0, T ] × IR1+d × L2 (E, E, λ; IR) → IR which with (ω, t, y, z, v) associates
f (ω, t, y, z, v), P̃ ⊗ B(IR1+d ) ⊗ B(L2 (E, E, λ; IR))-measurable and satisfying :
(i) the process (f (t, 0, 0, 0))t≤T belongs to H2,1
(ii) f is uniformly Lipschitz with respect to (y, z, v), i.e., there exists a constant C ≥ 0 such
that for any y, y ′ , z, z ′ ∈ IR and v, v ′ ∈ L2 (E, E, λ; IR),
P − a.s., |f (ω, t, y, z, v) − f (ω, t, y ′ , z ′ , v ′ )| ≤ C(|y − y ′ | + |z − z ′ | + kv − v ′ k)
- two obstacles L := (Lt )t≤T and U := (Ut )t≤T which are Ft −progressively measurable rcll, real
valued processes satisfying Lt ≤ Ut , ∀t ≤ T and LT ≤ ξ ≤ UT , P-a.s.. In addition they belong
to D2 , i.e.,
IE[ sup {|Lt | + |Ut |}2 ] < ∞.
0≤t≤T
Besides we assume that their jumping times occur only at inaccessible stopping times which
roughly speaking means that they are not predictable (see e.g. [2], pp.215 for the accurate
definition). If this latter condition is not satisfied and especially if the upper (resp. lower)
barrier U (resp. L) has positive (resp. negative) jumps then Y could have predictable jumps
and the processes K ± would be no longer continuous. Therefore the setting of the problem is
not the same as in (1).
Let us now introduce our two barrier reflected BSDE with jumps associated with (f, ξ, L, U ).
A solution is a 5-uple (Y, Z, V, K + , K − ) := (Yt , Zt , Vt , Kt+ , Kt− )t≤T of processes with values in
IR1+d × L2 (E, E, λ; IR) × IR+ × IR+ such that:
(i)
(ii)
Y ∈ D2 , Z ∈ Hd , V ∈ L and K ± ∈ Cci
Yt = ξ +
Z
t
T
f (s, Ys , Zs , Vs )ds + (KT+ − Kt+ ) − (KT− − Kt− )
Z
(iii) ∀t ≤ T, Lt ≤ Yt ≤ Ut and
0
−
T
Z
t
T
Zs dBs −
(Yt − Lt )dKt+ =
Z
0
T
Z TZ
t
E
Vs (e)µ̃(ds, de) , ∀t ≤ T
(Yt − Ut )dKt− = 0.
(2)
Note that equation (2) has not a solution in general. Actually one can take L = U with L not
being a semimartingale, then obviously we cannot find a 5-uple which satisfies the relation (ii).
✷
124
2.1
BSDEs with one reflecting barrier
To begin with we recall the following result by Hamadène & Ouknine [13] related to reflected
BSDEs with one upper barrier (in (2), L ≡ −∞ and K + = 0) driven by a Brownian motion and
an independent Poisson process.
Theorem 2.1 : There exits a quadruple (Y, Z, K, V ) := (Yt , Zt , Kt , Vt )t≤T of processes with
values in IR1+d × IR+ × L2 which satisfies :
2 and V ∈ L2
Y ∈ D2 , Z ∈ H2,d , K ∈ Cci
Z T
Yt = ξ +
f (s, Ys , Zs , Vs )ds − (KT − Kt ) −
t
Z
∀t ≤ T, Yt ≤ Ut and
T
Z
t
T
Zs dBs −
Z TZ
t
U
Vs (e)µ̃(ds, de), t ≤ T
(Ut − Yt )dKt = 0. ✷
0
(3)
In general we do not have a comparison result for solutions of BSDEs driven by a Brownian
motion and an independent Poisson process, reflected or not (see e.g. [1] for a counter-example).
However in some specific cases, when the coefficients have some features and especially when
they do not depend on the variable v, we actually have comparison.
So let us give another pair (f ′ , ξ ′ ) where f ′ : (ω, t, y, z, v) 7−→ f ′ (ω, t, y, z, v) ∈ IR and ξ ′ ∈
L2 (Ω, FT , P ; IR). On the other hand, assume there exists a quadruple of processes (Y ′ , Z ′ , V ′ , K ′ )
2 and solution for the BSDE with one reflecting upper barrier
which belongs to D2 ×H2,d ×L2 ×Cci
associated with (f ′ (ω, t, y, z, v), ξ ′ , U ). Then we have:
Lemma 2.1 : Assume that :
(i) f is independent of v
(ii) P-a.s. for any t ≤ T , f (t, Yt′ , Zt′ ) ≤ f ′ (s, Yt′ , Zt′ , Vt′ ) and ξ ≤ ξ ′ .
Then P-a.s., ∀t ≤ T , Yt ≤ Yt′ .
Proof : Let X = (Xt )t≤T be a rcll semi-martingale, then using Tanaka’s formula with the
function (x+ )2 = (max{x, 0})2 reads:
(Xt+ )2
=
(XT+ )2 −2
Z
t
T
Z
+
Xs−
dXs −
T
t
1[Xs >0] d < X c , X c >s −
X
+ 2
+
{(Xs+ )2 −(Xs−
) −2Xs−
∆Xs }.
t0] d < X c , X c >s ≤ (XT+ )2 − 2
T
t
+
Xs−
dXs .
Now using this formula with Y − Y ′ yields:
((Yt − Yt′ )+ )2 +
Z
t
T
1[Ys −Ys′ >0] |Zs − Zs′ |2 ds ≤ −2
Z
t
T
′ +
(Ys− − Ys−
) d(Ys − Ys′ ).
′ )+ d(K − K ′ ) = T (Y − Y ′ )+ d(K − K ′ ) and as usual this last term is nonBut tT (Ys− − Ys−
s
s
s
s
s
s
t
negative since (Ys − Ys′ )+ dKs′ = 0 (indeed when Yt > Yt′ we cannot have Yt′ = Ut ). Besides
R
R
125
f (s, Ys′ , Zs′ ) ≤ f ′ (s, Ys′ , Zs′ , Vs′ ) and f is Lipschitz continuous then there exist bounded and Ft adapted processes (as )s≤T and (bs )s≤T such that:
f (s, Ys , Zs ) = f (s, Ys′ , Zs′ ) + as (Ys − Ys′ ) + bs (Zs − Zs′ ).
Therefore we have:
′ )+ {a (Y − Y ′ ) + b (Z − Z ′ )}ds
((Yt − Yt′ )+ )2 + tT 1[Ys −Ys′ >0] |Zs − Zs′ |2 ds ≤ 2 tT (Ys− − Ys−
s s
s
s
s
s
R
RT ′
+
′
−2 t (Ys− − Ys− ) {(Zs − Zs )dBs + E Vs (e)µ̃(ds, de)}.
R
R
Taking now expectation, using in an appropriate way the inequality |a.b| ≤ ǫ|a|2 +ǫ−1 |b|2 (ǫ > 0),
and Gronwall’s one we obtain IE[((Yt − Yt′ )+ )2 ] = 0 for any t ≤ T . The result follows thoroughly
since Y and Y ′ are rcll. ✷
2.2
Local solutions of BSDEs with two reflecting barriers
Throughout this part we assume that the function f does not depend on v. The main reason
is, as pointed out in Lemma 2.1, in that case we can use comparison in order to deduce results
for the two reflecting barrier BSDE associated with (f, ξ, L, U ). Actually we have the following
result related to the existence of local solutions for (2):
Theorem 2.2 : There exists a unique Ft -optional process Y := (Yt )t≤T such that:
(i) YT = ξ and P-a.s. for any t ≤ T , Lt ≤ Yt ≤ Ut
(ii) for any Ft -stopping time τ there exists a quintuple (θτ , Z τ , V τ , K τ,+ , K τ,− ) which belongs
2 × C 2 such that: P-a.s.,
to Tτ × H2,d × L2 × Cci
ci
E(f, ξ, L, U ) :
Z θτ
Z θτ
τ
d(Ksτ,+ − Ksτ,− )
f (s, Ys , Zs )ds +
∀t ∈ [τ, θτ ], Yt = Yθτ +
t
t
Z
Z
Z
θτ
θτ
−
Z
τ
θτ
(Us − Ys )dKsτ,− =
Z
τ
θτ
t
Zsτ dBs −
(Ys −
t
τ,+
Ls )dKs
E
Vsτ (e)µ̃(ds, de),
(4)
=0
(iii) if we set ντ := inf {s ≥ τ, Ys = Us } ∧ T and στ := inf {s ≥ τ, Ys = Ls } ∧ T then ντ ∨ στ ≤
θτ , Yντ = Uντ on [ντ < T ] and Yστ = Lστ on [στ < T ].
Hereafter we call the process Y := (Yt )t≤T a local solution for the BSDE with two reflecting
barriers associated with (f, ξ, L, U ) which we denote E(f, ξ, L, U ).
P roof : Let us show uniqueness. Let Y and Y ′ be two Ft -optional processes which satisfy
(i) − (iii). Then for any stopping time τ we have :
Yστ ∧ντ′
= Yστ 1[στ ≤ντ′ ] + Yντ′ 1[ντ′
i
on
Electr
o
u
a
rn l
o
f
P
c
r
ob
abil
ity
Vol. 11 (2006), Paper no. 5, pages 121–145.
Journal URL
http://www.math.washington.edu/∼ejpecp/
BSDEs with two reflecting barriers driven by a Brownian and a
Poisson noise and related Dynkin game
S. Hamadène & M. Hassani
Laboratoire de Statistique et Processus,
Université du Maine, 72085 Le Mans Cedex 9, France.
e-mails: hamadene@univ-lemans.fr & mohammed.hassani@univ-lemans.fr
Abstract
In this paper we study BSDEs with two reflecting barriers driven by a Brownian motion
and an independent Poisson process. We show the existence and uniqueness of local and
global solutions. As an application we solve the related zero-sum Dynkin game.
Key words: Backward stochastic differential equation ; Poisson measure ; Dynkin game ;
Mokobodzki’s condition.
.
AMS 2000 Subject Classification: Primary 91A15, 60G40, 91A60; Secondary 60G99.
Submitted to EJP on December 23, 2003. Final version accepted on January 25, 2006.
121
1
Introduction
Let (Ft )t≤T be a filtration generated by a Brownian motion (Bt )t≤T and an independent Poisson measure µ(t, ω, de) defined on a probability space (Ω, F, P ). A solution for the backward
stochastic differential equation (BSDE for short) with two reflecting barriers associated with a
coefficient f (t, ω, y, z), a terminal value ξ and a lower (resp. an upper) barrier (Lt )t≤T (resp.
(Ut )t≤T ) is a quintuple of Ft -predictable processes (Yt , Zt , Vt , Kt+ , Kt− )t≤T which satisfies:
Z
+
−
Vt (e)µ̃(dt, de), t ≤ T ; YT = ξ
−dYt = f (t, Yt , Zt , Vt )dt + dKt − dKt − Zt dBt −
E
Lt ≤ Yt ≤ Ut , ∀t ≤ T
±
K
(1)
are continuous non-decreasing and (Yt − Lt )dKt+ = (Yt − Ut )dKt− = 0 (K0± = 0),
where µ̃ is the compensated measure of µ.
Nonlinear BSDEs have been first introduced by Pardoux and Peng [19], who proved the existence
and uniqueness of a solution under suitable hypotheses on the coefficient and the terminal value
of the BSDE. Since, these equations have gradually become an important mathematical tool
which is encountered in many fields such as finance ([6], [4], [8], [9],...), stochastic games and
optimal control ([10], [11], ...), partial differential equations ([1], [18], [20],...).
In the case when the filtration is generated only by a Brownian motion and when we consider
just one lower barrier (set U ≡ +∞ and K − ≡ 0 in (1)), the problem of existence and uniqueness
of a solution for (1) is considered and solved by El-Karoui et al. in [6]. Their work has been
generalized by Cvitanic & Karatzas in [4] where they deal with BSDEs with two reflecting
barriers.
BSDEs without reflection (in (1) one should take L = −∞ and U = +∞, thereby K ± = 0)
driven by a Brownian motion and an independent Poisson measure have been considered first
by Tang & Li [17] then by Barles et al. in connection with partial-integral differential equations
in [1]. In both papers the authors showed the existence and uniqueness of a solution.
The extension to the case of BSDEs with one reflecting barrier has been established by Hamadène
& Ouknine in [13]. The authors showed the existence and uniqueness of the solution when the
coefficient f is Lipschitz. Two proofs have been given, the first one is based on the penalization
scheme as for the second, it is obtained in using the Snell envelope notion. However both
methods make use of a contraction argument since the usual comparison theorem fails to work
in the general framework.
So in this work we study BSDEs with two reflecting barriers driven by a Brownian motion and
an independent Poisson measure. This is the natural extension of Hamadène & Ouknine’s work.
However there are at least four motivations for considering this problem. The first one is related
to Dynkin zero-sum game. The second is in connection with the real option area since the
stopping and starting problem leads to a BSDE with two reflecting barriers (see e.g. [12]). The
third one is that our problem can provide solutions for variational inequality problems with two
obstacles when the generator is of partial-integral type. Finally we provide a condition easy to
check in practice under which the well known Mokobodski’s hypothesis is satisfied.
122
In this paper we begin to show the existence of an adapted and rcll (right continuous and left
limited) process Y := (Yt )t≤T which in a way is a local solution for (1). Actually we prove that
for any stopping time τ there exist another stopping time θτ ≥ τ and a quadruple of processes
(Z, V, K + , K − ) which with Y verify (1) on the interval [τ, θτ ]. In addition the process Y reaches
the barriers U and L between τ and θτ . In the proof of our theorem, the key point is that the
predictable projection of a process π whose jumping times are inaccessible is equal to π− , the
process of left limits associated with π.
This result is then applied to deal with the zero-sum Dynkin game associated with L, U , ξ and
a process (gs )s≤T which stands for the instantaneous payoff. We show that this game is closely
related to the notion of local solution for (1). Besides we obtain the existence of a saddle-point
for the game under conditions out of the scope of the known results on this subject. Finally we
give some feature of the value function of the game (see Remark 4.3). Our result can be applied
in mathematical finance to deal with American game (or recallable) options whose underlying
derivatives contain a Poisson part (see [15] for this type of option).
Further we consider the problem of existence and uniqueness of a global solution for (1). When
the weak Mokobodski’s assumption [WM] is satisfied, which roughly speaking turns into the
existence of a difference of non-negative supermartingales between L and U , we show existence
and uniqueness of the solution. Then we address the issue of the verification of the condition
[WM]. Actually we prove that under the fully separation of the barriers, i.e., Lt < Ut for any
t ≤ T , the condition [WM] holds true.
This paper is organized as follows :
In Section 2, we deal with local solutions for (1) while Section 3 is devoted to zero-sum Dynkin
games. At the end, in Section 4, we address the problem of existence of a global solution for
(1). ✷
2
Reflected BSDEs driven by a Brownian motion and an independent Poisson point process
Let (Ω, F, (Ft )t≤T ) be a stochastic basis such that F0 contains all P -null sets of F, Ft+ =
ǫ>0 Ft+ǫ = Ft , ∀t < T , and suppose that the filtration is generated by the following two
mutually independent processes :
T
- a d-dimensional Brownian motion (Bt )t≤T
- a Poisson random measure µ on IR+ × E, where E := IRl \ {0} is equipped with its Borel fields
E, with compensator ν(dt, de) = dtλ(de), such that {µ̃([0, t] × A) = (µ − ν)([0, t] × A)}t≤T is a
martingale for every
A ∈ E satisfying λ(A) < ∞. The measure λ is assumed to be σ-finite on
R
(E, E) and verifies E (1 ∧ |e|2 )λ(de) < ∞.
Now let:
- D be the set of Ft -adapted right continuous with left limits processes (Yt )t≤T with values in
IR and D2 := {Y ∈ D, IE[supt≤T |Yt |2 ] < ∞}
- P̃ (resp. P) be the Ft -progressive (resp. predictable) tribe on Ω × [0, T ]
- H2,k (resp. Hk ) be the set of P̃-measurable processes Z := (Zt )t≤T with values in IRk and
dP ⊗ dt-square integrable (resp. P -a.s. Z(ω) := (Zt (ω))t≤T is dt-square integrable)
123
- L2 (resp. L) be the set of mappings V : Ω × [0, T ] × E → IR which are P ⊗ E-measurable and
R
R
R
1
IE[ 0T kVs k2 ds] < ∞ (resp. 0T kVs k2 ds < ∞, P-a.s.) where kvk := ( E |v(e)|2 λ(de)) 2 for v : E →
IR
2 (resp. C ) the space of continuous F -adapted and non-decreasing processes (k )
- Cci
ci
t
t t≤T such
2
that k0 = 0 and IE[kT ] < ∞ (resp. kT < ∞, P-a.s.)
- for a stopping time τ , Tτ denotes the set of stopping times θ such that θ ≥ τ
- for a given rcll process (wt )t≤T , wt− = limsրt ws , t ≤ T (w0− = w0 ) ; w− := (wt− )t≤T and
△w := w − w− ✷
We are now given four objects:
- a terminal value ξ ∈ L2 (Ω, FT , P )
- a map f : Ω × [0, T ] × IR1+d × L2 (E, E, λ; IR) → IR which with (ω, t, y, z, v) associates
f (ω, t, y, z, v), P̃ ⊗ B(IR1+d ) ⊗ B(L2 (E, E, λ; IR))-measurable and satisfying :
(i) the process (f (t, 0, 0, 0))t≤T belongs to H2,1
(ii) f is uniformly Lipschitz with respect to (y, z, v), i.e., there exists a constant C ≥ 0 such
that for any y, y ′ , z, z ′ ∈ IR and v, v ′ ∈ L2 (E, E, λ; IR),
P − a.s., |f (ω, t, y, z, v) − f (ω, t, y ′ , z ′ , v ′ )| ≤ C(|y − y ′ | + |z − z ′ | + kv − v ′ k)
- two obstacles L := (Lt )t≤T and U := (Ut )t≤T which are Ft −progressively measurable rcll, real
valued processes satisfying Lt ≤ Ut , ∀t ≤ T and LT ≤ ξ ≤ UT , P-a.s.. In addition they belong
to D2 , i.e.,
IE[ sup {|Lt | + |Ut |}2 ] < ∞.
0≤t≤T
Besides we assume that their jumping times occur only at inaccessible stopping times which
roughly speaking means that they are not predictable (see e.g. [2], pp.215 for the accurate
definition). If this latter condition is not satisfied and especially if the upper (resp. lower)
barrier U (resp. L) has positive (resp. negative) jumps then Y could have predictable jumps
and the processes K ± would be no longer continuous. Therefore the setting of the problem is
not the same as in (1).
Let us now introduce our two barrier reflected BSDE with jumps associated with (f, ξ, L, U ).
A solution is a 5-uple (Y, Z, V, K + , K − ) := (Yt , Zt , Vt , Kt+ , Kt− )t≤T of processes with values in
IR1+d × L2 (E, E, λ; IR) × IR+ × IR+ such that:
(i)
(ii)
Y ∈ D2 , Z ∈ Hd , V ∈ L and K ± ∈ Cci
Yt = ξ +
Z
t
T
f (s, Ys , Zs , Vs )ds + (KT+ − Kt+ ) − (KT− − Kt− )
Z
(iii) ∀t ≤ T, Lt ≤ Yt ≤ Ut and
0
−
T
Z
t
T
Zs dBs −
(Yt − Lt )dKt+ =
Z
0
T
Z TZ
t
E
Vs (e)µ̃(ds, de) , ∀t ≤ T
(Yt − Ut )dKt− = 0.
(2)
Note that equation (2) has not a solution in general. Actually one can take L = U with L not
being a semimartingale, then obviously we cannot find a 5-uple which satisfies the relation (ii).
✷
124
2.1
BSDEs with one reflecting barrier
To begin with we recall the following result by Hamadène & Ouknine [13] related to reflected
BSDEs with one upper barrier (in (2), L ≡ −∞ and K + = 0) driven by a Brownian motion and
an independent Poisson process.
Theorem 2.1 : There exits a quadruple (Y, Z, K, V ) := (Yt , Zt , Kt , Vt )t≤T of processes with
values in IR1+d × IR+ × L2 which satisfies :
2 and V ∈ L2
Y ∈ D2 , Z ∈ H2,d , K ∈ Cci
Z T
Yt = ξ +
f (s, Ys , Zs , Vs )ds − (KT − Kt ) −
t
Z
∀t ≤ T, Yt ≤ Ut and
T
Z
t
T
Zs dBs −
Z TZ
t
U
Vs (e)µ̃(ds, de), t ≤ T
(Ut − Yt )dKt = 0. ✷
0
(3)
In general we do not have a comparison result for solutions of BSDEs driven by a Brownian
motion and an independent Poisson process, reflected or not (see e.g. [1] for a counter-example).
However in some specific cases, when the coefficients have some features and especially when
they do not depend on the variable v, we actually have comparison.
So let us give another pair (f ′ , ξ ′ ) where f ′ : (ω, t, y, z, v) 7−→ f ′ (ω, t, y, z, v) ∈ IR and ξ ′ ∈
L2 (Ω, FT , P ; IR). On the other hand, assume there exists a quadruple of processes (Y ′ , Z ′ , V ′ , K ′ )
2 and solution for the BSDE with one reflecting upper barrier
which belongs to D2 ×H2,d ×L2 ×Cci
associated with (f ′ (ω, t, y, z, v), ξ ′ , U ). Then we have:
Lemma 2.1 : Assume that :
(i) f is independent of v
(ii) P-a.s. for any t ≤ T , f (t, Yt′ , Zt′ ) ≤ f ′ (s, Yt′ , Zt′ , Vt′ ) and ξ ≤ ξ ′ .
Then P-a.s., ∀t ≤ T , Yt ≤ Yt′ .
Proof : Let X = (Xt )t≤T be a rcll semi-martingale, then using Tanaka’s formula with the
function (x+ )2 = (max{x, 0})2 reads:
(Xt+ )2
=
(XT+ )2 −2
Z
t
T
Z
+
Xs−
dXs −
T
t
1[Xs >0] d < X c , X c >s −
X
+ 2
+
{(Xs+ )2 −(Xs−
) −2Xs−
∆Xs }.
t0] d < X c , X c >s ≤ (XT+ )2 − 2
T
t
+
Xs−
dXs .
Now using this formula with Y − Y ′ yields:
((Yt − Yt′ )+ )2 +
Z
t
T
1[Ys −Ys′ >0] |Zs − Zs′ |2 ds ≤ −2
Z
t
T
′ +
(Ys− − Ys−
) d(Ys − Ys′ ).
′ )+ d(K − K ′ ) = T (Y − Y ′ )+ d(K − K ′ ) and as usual this last term is nonBut tT (Ys− − Ys−
s
s
s
s
s
s
t
negative since (Ys − Ys′ )+ dKs′ = 0 (indeed when Yt > Yt′ we cannot have Yt′ = Ut ). Besides
R
R
125
f (s, Ys′ , Zs′ ) ≤ f ′ (s, Ys′ , Zs′ , Vs′ ) and f is Lipschitz continuous then there exist bounded and Ft adapted processes (as )s≤T and (bs )s≤T such that:
f (s, Ys , Zs ) = f (s, Ys′ , Zs′ ) + as (Ys − Ys′ ) + bs (Zs − Zs′ ).
Therefore we have:
′ )+ {a (Y − Y ′ ) + b (Z − Z ′ )}ds
((Yt − Yt′ )+ )2 + tT 1[Ys −Ys′ >0] |Zs − Zs′ |2 ds ≤ 2 tT (Ys− − Ys−
s s
s
s
s
s
R
RT ′
+
′
−2 t (Ys− − Ys− ) {(Zs − Zs )dBs + E Vs (e)µ̃(ds, de)}.
R
R
Taking now expectation, using in an appropriate way the inequality |a.b| ≤ ǫ|a|2 +ǫ−1 |b|2 (ǫ > 0),
and Gronwall’s one we obtain IE[((Yt − Yt′ )+ )2 ] = 0 for any t ≤ T . The result follows thoroughly
since Y and Y ′ are rcll. ✷
2.2
Local solutions of BSDEs with two reflecting barriers
Throughout this part we assume that the function f does not depend on v. The main reason
is, as pointed out in Lemma 2.1, in that case we can use comparison in order to deduce results
for the two reflecting barrier BSDE associated with (f, ξ, L, U ). Actually we have the following
result related to the existence of local solutions for (2):
Theorem 2.2 : There exists a unique Ft -optional process Y := (Yt )t≤T such that:
(i) YT = ξ and P-a.s. for any t ≤ T , Lt ≤ Yt ≤ Ut
(ii) for any Ft -stopping time τ there exists a quintuple (θτ , Z τ , V τ , K τ,+ , K τ,− ) which belongs
2 × C 2 such that: P-a.s.,
to Tτ × H2,d × L2 × Cci
ci
E(f, ξ, L, U ) :
Z θτ
Z θτ
τ
d(Ksτ,+ − Ksτ,− )
f (s, Ys , Zs )ds +
∀t ∈ [τ, θτ ], Yt = Yθτ +
t
t
Z
Z
Z
θτ
θτ
−
Z
τ
θτ
(Us − Ys )dKsτ,− =
Z
τ
θτ
t
Zsτ dBs −
(Ys −
t
τ,+
Ls )dKs
E
Vsτ (e)µ̃(ds, de),
(4)
=0
(iii) if we set ντ := inf {s ≥ τ, Ys = Us } ∧ T and στ := inf {s ≥ τ, Ys = Ls } ∧ T then ντ ∨ στ ≤
θτ , Yντ = Uντ on [ντ < T ] and Yστ = Lστ on [στ < T ].
Hereafter we call the process Y := (Yt )t≤T a local solution for the BSDE with two reflecting
barriers associated with (f, ξ, L, U ) which we denote E(f, ξ, L, U ).
P roof : Let us show uniqueness. Let Y and Y ′ be two Ft -optional processes which satisfy
(i) − (iii). Then for any stopping time τ we have :
Yστ ∧ντ′
= Yστ 1[στ ≤ντ′ ] + Yντ′ 1[ντ′