Reected and Doubly Reected BSDEs with Jumps: A
Priori Estimates and Comparison
Stéphane Crépey
∗
Département de Mathématiques
Université d'Évry Val d'Essonne
91025 Évry Cedex, France
Anis Matoussi
Département de Mathématiques
Université du Maine
F-72085 Le Mans Cedex 9, France
September 18, 2008
Abstract.
It is now established that under quite general circumstances, including in models with
jumps, existence of a solution to a reected BSDE is guaranteed under mild conditions, whereas existence of a solution to a doubly reected BSDE is essentially equivalent to the so-called Mokobodski
condition. As for uniqueness of solutions, it holds under mild integrability conditions. However, for
practical purposes, existence and uniqueness is not enough. In order to give further developments
to these results in Markovian set-ups, one also need a (simply or doubly) reected BSDE to be wellposed, in the sense that the solution satises suitable bound and error estimates, and one further
needs a suitable comparison theorem. In this paper we derive such estimates and comparison results.
In the last section applicability of the results is illustrated on a pricing problem in nance.
Key words:
Reected BSDEs, Jumps, A priori Estimates, Comparison Theorem, Markovian BS-
DEs, Finance, Convertible Bonds.
1
Introduction
It is now established that under quite general circumstances, including in models with jumps, existence of a solution to a (simply) reected BSDE (RBSDE for short in the sequel) is guaranteed
under mild conditions, whereas existence of a solution to a doubly reected BSDE (R2BSDE) is
equivalent to the so-called Mokobodski condition. This condition essentially postulates the existence
of a quasimartingale between the barriers (see in particular HamadèneHassani [22, Theorem 4.1]
and previous works in this direction [12, 23, 27, 28, 20, 18]). As for uniqueness of solutions, it is
guaranteed under mild integrability conditions (see e.g. HamadèneHassani [22, Remark 4.1]).
However, for practical purposes, existence and uniqueness is not enough. Let us for instance consider
the application of R2BSDEs to convertible bonds in nance (see Section 6 and [5, 6, 8]). In this case
∗ The
research of S. Crépey beneted from the support of the Europlace Institute of Finance and of Ito33.
2
Reflected BSDEs with Jumps
the state-process (rst component)
Y
of a solution to a related R2BSDE may be interpreted in terms
of an arbitrage price process for the bond. As demonstrated in [7], the mere existence of a solution
to the related R2BSDE is a result with important theoretical consequences in terms of pricing and
hedging the bond. Yet, in order to give further developments to these results in Markovian set-ups,
we also need the R2BSDE to be well-posed, in the sense that the solution satises suitable bound
and error estimates, and we also need a suitable comparison theorem.
Now, as opposed to the situation prevailing for RBSDEs (see, e.g., El Karoui et al. [16]), universal
a priori estimates cannot be obtained for R2BSDEs. In order to get estimates for R2BSDEs, one
needs to specialize the problem a little bit. Likewise, universal comparison theorems do not hold in
models with jumps (see [2] for a counter-example in the simple case of a BSDE, without barriers).
Section 2 presents an abstract set-up in which our results are derived, as well as the BSDEs under consideration (Subsection 2.1). In Sections 3 and 4 we establish the a priori bound and error
estimates (Theorem 3.2) and our comparison theorem (Theorem 4.2). The a priori error estimates
immediately imply uniqueness of a solution to our problems (Subsection 5.1).
Assuming an ad-
ditional martingale representation property and the quasi-left continuity of the barriers, we then
give existence results (Subsection 5.2).
In Section 6 we show that all the required assumptions
are satised in the case of the convertible bonds related R2BSDEs, in a rather generic Markovian
specication of our abstract set-up. These R2BSDEs thus admit (unique) solutions.
These results can be used to develop a related variational inequality approach in the Markovian case
(see [10, 11]).
2
Set-Up
In all the paper we work with a nite time horizon
F = (Ft )t∈[0,T ]
with
FT = F,
By default we declare that a random variable is
time interval
[0, T ]
and
T > 0, a probability space (Ω, F, P) and a ltration
satisfying the usual conditions of right-continuity and completeness.
F-adapted.
F -measurable,
and that a process is dened on the
We may and do assume that all semimartingales are càdlàg,
without loss of generality.
B = (Bt )t∈[0,T ] be a d-dimensional standard Brownian motion. Given an auxiliary measured
(E, BE , ρ), where ρ is a non-negative σ -nite measure on (E, BE ), let µ = (µ(dt, de))t∈[0,T ],e∈E
e = P ⊗ BE where
be an integer valued random measure on [0, T ] × E, B([0, T ]) ⊗ BE . Denoting P
P is the predictable sigma eld on Ω × [0, T ], recall that an integer valued random measure µ on
e sigma nite, N ∪ {+∞} valued random measure
[0, T ] × E, B([0, T ]) ⊗ BE is an optional and P
such that µ(ω, {t} × E) ≤ 1, identically (JacodShiryaev [25, Denition II.1.13 page 68]; see also
Let
space
[1, 30]).
e-measurable non-negative
µ is dened by ζt (ω, e)ρ(de)dt, for a P
uniformly bounded (random) function ζ. The motivation for the introduction of the random density
ζ is to account for dependence between factors in applications, for instance in the context of nancial
We assume that the compensator of
modeling (see section 6.2 and [10, 11, 3, 9]). We refer the reader to the literature [1, 25, 30] regarding
e-measurable integrands with
P
µ(dt, de), its compensator dt ⊗ ζdρ := ζt (ω, e)ρ(de)dt, or its
measure) µ
e(dt, de) = µ(dt, de) − ζt (ω, e)ρ(de)dt.
the denition of the integral process of
respect to random measures
such as
compensatrix (compensated
By default in the sequel, all (in)equalities between random quantities are to be understood
almost surely,
dP ⊗ dt
almost everywhere or
dP ⊗ dt ⊗ ζdρ
dP
almost everywhere, as suitable in the
situation at hand. For simplicity we omit all dependences in
ω
of any process or random function
in the notation.
We denote by:
• |X|, the (d-dimensional) Euclidean norm of a vector or row vector X in Rd or R1⊗d ;
• Mρ = M(E, BE , ρ; R), the set of measurable functions from (E, BE , ρ) to R endowed
topology of convergence in measure;
with the
3
S. Crépey and A. Matoussi
•
v ∈ Mρ
for
t ∈ [0, T ] :
and
|v|t =
Z
1
v(e)2 ζt (e)ρ(de) 2 ∈ R+ ∪ {+∞} ;
(1)
E
• B(O),
the Borel sigma eld on
O,
for any topological space
Let us now introduce some Banach (or Hilbert, in case of
O.
L2 , Hd2
or
Hµ2 )
spaces of processes or
random functions:
• L2 ,
the space of square integrable real-valued (FT -measurable) random variables
ξ
such that
h i 21
< +∞ ;
kξk2 := E ξ 2
• Sdp ,
for any real
p≥2
(or
S p,
in case
d = 1),
the space of
Rd -valued
càdlàg processes
X
such that
h
i p1
kXkSdp := E sup |Xt |p
< +∞ ;
t∈[0,T ]
• Hd2
(or
H2 ,
in case
d = 1),
the space of
R1⊗d -valued
T
hZ
kZkH2d := E
predictable processes
Z
such that
i 12
|Zt |2 dt
< +∞ ;
0
• Hµ2 ,
kV kH2µ
• A2 ,
e-measurable
P
the space of
T
hZ
:= E
|Vt |2t dt
i 21
V : Ω × [0, T ] × E → R
hZ
= E
0
0
T
Z
K± ∈ S2
i 21
< +∞ ;
E
K
with (continuous and non decreasing)
null at time 0;
A2 .
the space of non-decreasing processes in
Remark 2.1
such that (cf. (1))
Vt (e)2 ζt (e)ρ(de)dt
the space of nite variation continuous processes
Jordan components
• A2i ,
functions
By a slight abuse of notation, we shall also write
kXkH2
for
hZ
E
T
i 21
|Xt |2 dt
0
the case of a progressively measurable (not necessarily predictable) real-valued process
in
X.
Observe that in particular:
Z
•
·
Z ·Z
Zt dBt
0
Vt (e)e
µ(dt, de)
and
0
E
K − , and
• K = K+ −
• K = K + , for
any
K ± dene
K ∈ A2i .
are (true) martingales, for any
mutually singular measures on
Z ∈ Hd2
R+ ,
and
for any
V ∈ Hµ2 ;
K ∈ A2 ;
It is worth noting that our results admit a straightforward extension to the case where the Brownian
motion
B
is replaced by a more general continuous local martingale. In this case, the space
dened as the space of
R
1⊗d
-valued predictable processes
kZkH2d
(kXkH2 being still dened as
T
hZ
:= E
|Zt |2 dhBit
X ).
is
such that
i 21
< +∞
0
hZ
kXkH2 = E
0
real-valued process
Z
Hd2
T
i 21
Xt2 dt
, in the case of a progressively measurable
4
Reflected BSDEs with Jumps
2.1
Reected and Doubly Reected BSDEs
2.1.1 Basic Problems
ξ , and a P ⊗ B(R) ⊗ B(R1⊗d ) ⊗
× Mρ → R. In all the paper, we work
Let us be given a real-valued random variable (terminal condition)
B(Mρ )-measurable
driver coecient
g : Ω × [0, T ] × R × R
1⊗d
under the following Standing Assumptions:
(H.0) ξ ∈ L2 ;
(H.1.i) g· (y, z, v) is a progressively measurable process, for any y ∈ R, z ∈ R1⊗d , v ∈ Mρ ;
(H.1.ii) kg· (0, 0, 0)kH2 < +∞ ;
(H.1.iii) g is uniformly Λ Lipschitz continuous with respect to (y, z, v), in the sense that Λ
0 0 0
1⊗d
constant such that for any t ∈ [0, T ] and (y, z, v), (y , z , v ) ∈ R × R
× Mρ , identically:
is a
|gt (y, z, v) − gt (y 0 , z 0 , v 0 )| ≤ Λ(|y − y 0 | + |z − z 0 | + |v − v 0 |t ) .
We also introduce the barriers (or obstacles) L and U ,
(H.2.i) L and U are càdlàg processes in S 2 ;
(H.2.ii) Lt ≤ Ut , t ∈ [0, T ) and LT ≤ ξ ≤ UT , P-a.s.
Denition 2.2
A solution to the R2BSDE with data
such that:
(g, ξ, L, U )
is a quadruple
(Y, Z, V, K)
such
that:
(i) Y ∈ S 2 , Z ∈ Hd2 , V ∈ Hµ2 , K ∈ A2
Z T
(ii) Yt = ξ +
gs (Ys , Zs , Vs )ds + KT − Kt
Zt T
Z TZ
−
Zs dBs −
Vs (e)e
µ(ds, de)
t
t
for any
t ∈ [0, T ], P-a.s.
E
(E)











(iii) Lt ≤ Yt ≤ Ut for any t ∈ [0, T ], P-a.s.,
Z T
Z T
+
and
(Yt − Lt )dKt =
(Ut − Yt )dKt− = 0, P-a.s.
0












0
The inequalities and the integral conditions in (E )(iii) are called the barrier constraints and the
minimality conditions, respectively.
Let us now consider the case when there is only one barrier, say, for instance, a lower barrier
solution to the RBSDE with data
(g, ξ, L)
is a quadruple
(Y, Z, V, K)
(i) Y ∈ S 2 , Z ∈ Hd2 , V ∈ Hµ2 , K ∈ A2i
Z T
(ii) Yt = ξ +
gs (Ys , Zs , Vs )ds + KT − Kt
Zt T
Z TZ
−
Zs dBs −
Vs (e)e
µ(ds, de) for any t ∈ [0, T ], P-a.s.
t
t
E
Z T
(iii) Lt ≤ Yt for any t ∈ [0, T ], P-a.s. and
(Yt − Lt )dKt = 0, P-a.s.
0
When there is no barrier, we dene likewise solutions to the BSDE with data
Remark 2.3
L.
A
such that:










(E 0 )









(g, ξ).
(i) All these denitions (as well as the ones introduced in section 2.1.2 below) admit
obvious extensions to problems in which the driving term contains a further nite variation process
A
(not necessarily absolutely continuous).
(ii) Since the integrands are càdlàg and the integrators lie in
A2
in the minimality conditions, these
are equivalent to
Z
0
T
(Yt− − Lt− )dKt+ = 0 ,
Z
0
T
(Ut− − Yt− )dKt− = 0 .
5
S. Crépey and A. Matoussi
2.1.2 Extensions with stopping time
Motivated by applications (see [5, 7, 8]), we now consider two generalizations of the above problems
involving a further stopping time
τ ∈T.
Reected BSDE with random terminal time
R2BSDE with random terminal time
that
T
is replaced by
τ
τ
A
solution to a BSDE, resp. RBSDE, resp.
is dened as in Denition 2.2, with the only dierence being
therein (including in the denition of the involved spaces of random variables,
processes and random functions; so in particular we assume here that
to a BSDE with random terminal
τ
ξ is Fτ -measurable). A solution
[0, τ ] ⊆ [0, T ].
is thus dened over the random time interval
0
In particular we denote in the sequel by (Ē ) the RBSDE with random terminal time
(g, ξ, L) on [0, τ ] (assuming in this case that ξ is Fτ -measurable). Note that
τ = T, (Ē 0 ) reduces to (E 0 ). So (Ē 0 ) is a rst possible generalization of (E 0 ).
Remark 2.4
τ
and data
in the special case
(Y, Z, V, K) to (Ē 0 ) on [0, τ ], let us prolongate (Y, Z, V, K) to
the whole interval [0, T ] so that on (τ, T ] the prolongated processes and random functions Y, K ,
Z and V satisfy Y = Yτ , K = Kτ , Z = V = 0. One thus gets a solution to the RBSDE (E 0 ) with
data (1·≤τ g, ξ, L·∧τ ). Note that the data (1·≤τ g, ξ, L·∧τ ) satisfy (H.0), (H.1) and (the Assumptions
regarding L in) (H.2) on [0, T ], provided (g, ξ, L) satisfy (H.0), (H.1) and (H.2) with τ instead of
T therein. Given these observations, the estimates and comparison results derived in this paper for
0
solutions to RBSDEs (on [0, T ]) will thus in eect be applicable to solutions to (Ē ).
(i) Given a solution
(ii) BSDEs with random terminal time were introduced in Darling and Pardoux [13] (without barriers
and in a context of Brownian ltrations). In [13], the random terminal time is a priori unbounded,
whereas in this paper
0 ≤ τ ≤ T.
In this respect, the situation that we consider here is rather
elementary.
Upper barrier with delayed activation
We shall also consider τ -R2BSDEs, namely the gen[0, T ] in which the upper barrier U is inactive before τ. Formally,
we replace U by Ūt := 1{t<τ } ∞ + 1{t≥τ } Ut in (E )(iii), with the convention that 0 × ±∞ = 0. The
resulting problem is denoted by (Ē ). Note that in the special case τ = 0, resp. τ = T, (Ē ) reduces
0
0
to (E ), resp. (E ). Thus (Ē ) is a generalization of both (E ) and (E ).
eralization of the R2BSDE (E ) on
3
A Priori Bound and Error Estimates
A (càdlàg) quasimartingale
X
can be dened as a dierence of two non-negative supermartingales
(see sections VI.38 to VI.42 and Appendix 2 of Dellacherie and Meyer [14]; see also Protter [31,
X = X 1 − X 2 of a quasimartingale X as
2
a dierence of two non-negative supermartingales X and X , there exists a (unique) decomposition
1
2
X = X̄ − X̄ , referred to as the Rao decomposition of X in the sequel, which is minimal in the
1
1
2
2
1
2
sense that X ≥ X̄ , X ≥ X̄ , for any such decomposition X = X − X ([14, section VI.40]).
2
Also note that any quasimartingale X belonging to S is a special semimartingale with canonical
decomposition X = X0 + M + A such that M is a uniformly integrable martingale and A is a
Chapter III, section 4]). Among the various decompositions
1
predictable nite variation process of integrable variation ([14, Appendix 2.4]).
L (resp. U ) is a quasimartingale in S 2 , we have an explicit representation
K − ) of a solution to (E ) (Lemma 3.1). This will enable us to derive related
We shall now see that when
for the process
K
+
(resp.
a priori bound and error estimates in Theorem 3.2.
The results of this section thus extend to R2BSDEs with jumps the results of El Karoui et al. [16]
(see also [15] for a survey) regarding RBSDEs in a continuous set-up: representation of
K+
(cf. [16,
Proposition 4.2]) and a priori bound and error estimates (cf. [16, Propositions 3.5 and 3.6]).
Note that in El Karoui et al.
[16], the representation of
K+
is incidental and the estimates are
6
Reflected BSDEs with Jumps
universal, whereas in our case, the representation of
K+
or
K−
is actually used in the derivation of
the estimates, assuming that one of the barriers is a quasimartingale in
S2
(or a suitable limit in
S2
of quasimartingales).
We only state and prove the results regarding
with data
L. The results for U
follow by considering the problem
(−g, −ξ, −L, −U ).
Lemma 3.1
(i) Let
be a solution to (E ), in case when
(Y, Z, V, K)
L
is a quasimartingale in
S2
with canonical decomposition
Lt = L0 + Mt + At , t ∈ [0, T ]
for a uniformly integrable martingale
dKt+
where
A = A+ − A−
M
and a predictable process of integrable variation
1{Yt =Lt } gt− (Yt , Zt , Vt )dt
≤
(2)
is the Jordan decomposition of
+
dA−
t
,
A.
Then
(3)
A.
(ii) If, in addition,
dA−
t ≤ αt dt
for a progressively measurable time-integrable process
k + such that
(4)
α,
then
K+
is an Lebesgue-absolutely contin-
uous process with density
kt+ ≤ 1{Yt =Lt } gt− (Yt , Zt , Vt ) + αt , t ∈ [0, T ] .
Proof
Note that (3) immediately implies (5), under condition (4).
(5)
Therefore it only remains to
prove (i). By (E ), we have:
d(Yt − Lt ) = −gt (Yt , Zt , Vt )dt − d(Kt+ − Kt− ) − dAt
Z
+Zt dBt +
Vt (e)e
µ(dt, de) − dMt .
(6)
E
Besides, we have by application of the MeyerTanaka formula to the semimartingale
by
Θ
the local time of
Y −L
at
0
Y − L, denoting
(see e.g. [31, page 214]):
d(Yt − Lt )+ = −1{Yt >Lt } gt (Yt , Zt , Vt )dt
− 1{Yt >Lt } dKt+ + 1{Yt >Lt } dKt− − 1{Yt >Lt } dAt
Z
+ 1{Yt >Lt } Zt dBt +
1{Yt− >Lt− } Vt (e) µ
e(dt, de) − 1{Yt− >Lt− } dMt
(7)
E
By the lower
1
+ 1{Yt− >Lt− } (Yt − Lt )− + 1{Yt− ≤Lt− } (Yt − Lt )+ + dΘt .
2
barrier constraint on Y, we have that
(Y − L)− = 0 , (Y − L)+ = Y − L , 1{Yt− =Lt− } dKt+ = dKt+ .
Whence by identication of (6) and (7):
Z
1{Yt− =Lt− } Zt dBt +
Vt (e)e
µ(dt, de) − dMt =
E
1
+ dA+
t + dΘt + 1{Yt− =Lt− } ∆(Y − L)t
2
−
gt− (Yt , Zt , Vt )dt + dA−
.
t + dKt
1{Yt =Lt } gt+ (Yt , Zt , Vt )dt
+ dKt+ − 1{Yt =Lt }
Since
M
(8)
is integrable, the second line of (8) denes a non-decreasing integrable process. Denoting
its compensator by
R
and its compensatrix by
Z
1{Yt− =Lt− } Zt dBt +
e,
R
it comes:
et =
Vt (e)e
µ(dt, de) − dMt − dR
E
−
dRt − 1{Yt =Lt } gt− (Yt , Zt , Vt )dt + dA−
+ dKt+ .
t + dKt
(9)
7
S. Crépey and A. Matoussi
Note that
A−
is predictable, like
A (see Dellacherie and Meyer [14,
page 129]). Since
K+
is continu-
ous, all terms are predictable in the second line of (9), whence equality to zero in (9). In particular:
−
dKt+ + dRt = 1{Yt =Lt } gt− (Yt , Zt , Vt )dt + dA−
,
t + dKt
(10)
−
dKt+ ≤ 1{Yt =Lt } gt− (Yt , Zt , Vt )dt + dA−
.
t + dKt
(11)
whence
Inequality (3) follows by mutual singularity of
K+
and
2
K −.
The proof of the following Theorem (a priori bound and error estimates) is deferred to Appendix A.
Theorem 3.2
We consider a sequence of R2BSDEs of the form considered in Lemma 3.1(i), with
n, the data being bounded in the sense that the driver coecients g n
data and solutions indexed by
are
Λ
equi-Lipschitz continuous, and for some constant
c1 :
kξ n k22 + kg·n (0, 0, 0)k2H2 + kLn k2S 2 + kU n k2S 2 + kAn,− k2S 2 ≤ c1 .
Then we have for some constant
(12)
c(Λ) :
kY n k2S 2 + kZ n k2H2 + kV n k2H2 + kK n,+ k2S 2 + kK n,− k2S 2 ≤ c(Λ)c1 .
Indexing by
n,p
the dierences
d
µ
·n − ·p ,
we also have:
(13)
kY n,p k2S 2 + kZ n,p k2H2 + kV n,p k2H2µ + kK n,p k2S 2 ≤
d
n,p 2
c(Λ)c1 kξ k2 + kg·n,p (Y·n , Z·n , V·n )k2H2 + kLn,p kS 2 + kU n,p kS 2 .
(14)
n,−
satisfy the assumptions of Lemma 3.1(ii), so dA
≤ αtn dt
n
n
for some progressively measurable processes α with kα kH2 nite for every n ∈ N. Then we may
n 2
n,p
replace kL kS 2 and kL
kS 2 by kLn k2H2 and kLn,p kH2 in (12) and (14).
Assume further that the barriers
Ln
n
Suppose additionally that kα kH2 is bounded over N and that when n → ∞ :
• g·n (Y· , Z· , V· ) H2 -converges to g· (Y· , Z· , V· ) locally uniformly w.r.t. (Y, Z, V ) ∈ S 2 × Hd2 × Hµ2 , and
• (ξ n , Ln , U n ) L2 × H2 × S 2 -converges to (ξ, L, U ).
n
n
n
n
2
2
2
2
Then (Y , Z , V , K ) S × Hd × Hµ × S -converges to a solution (Y, Z, V, K) of (E ). Moreover,
(Y, Z, V, K) also satises (13)(14) (with n = ∞ therein).
Remark 3.1
(i) By symmetry, analog results are valid when the
n,+
(with dA
≤ αtn dt for some progressively measurable processes
over n ∈ N, for the last part of the theorem).
α
Un
n
are quasimartingales in
such that
n
kα kH2
S2
is bounded
(ii) The reader can check by inspection of the proofs in Appendix A that Theorem 3.2 is in fact valid
for more general sequences of
(the same for every
τ -R2BSDEs
(see section 2.1.2), given a further stopping time
τ ∈T
n).
0
In the case of RBSDEs like (E ), the following results can be proven along the same lines as Theorem
3.2.
Theorem 3.3
Let us consider a sequence of RBSDEs, the data being bounded in the sense that the
g n are Λ equi-Lipschitz continuous, and for some constant c1 :
driver coecients
kξ n k22 + kg·n (0, 0, 0)k2H2 + kLn k2S 2 ≤ c1 .
Then we have for some constant
(15)
c(Λ) :
kY n k2S 2 + kZ n k2H2 + kV n k2H2 + kK n k2S 2 ≤ c(Λ)c1 .
d
µ
(16)
8
Reflected BSDEs with Jumps
Indexing by
n,p
the dierences
·n − ·p ,
we also have:
kY n,p k2S 2 + kZ n,p k2H2 + kV n,p k2H2µ + kK n,p k2S 2 ≤
d
n,p 2
c(Λ)c1 kξ k2 + kg·n,p (Y·n , Z·n , V·n )k2H2 + kLn,p kS 2 .
n
If, moreover, the barriers L satisfy the assumptions of Lemma 3.1(ii), then we may replace
n 2
n,p
2
and kL
kS by kL kH2 and kLn,p kH2 in (15) and (17).
(17)
kLn k2S 2
Suppose that when n → ∞ :
• g·n (Y· , Z· , V· ) H2 -converges to g· (Y· , Z· , V· ) locally uniformly w.r.t. (Y, Z, V ) ∈ S 2 × Hd2 × Hµ2 , and
• (ξ n , Ln ) L2 × S 2 -converges to (ξ, L) (or merely (ξ n , Ln ) L2 × H2 -converges to (ξ, L), in case when
n
the barriers L are as in Lemma 3.1(ii)).
n
n
n
n
2
2
2
2
0
Then (Y , Z , V , K ) S × Hd × Hµ × S -converges to a solution (Y, Z, V, K) of (E ). Moreover,
(Y, Z, V, K) also satises (16)(17) (with n = ∞ therein).
2
4
Comparison
In this section we specialize (H.1) to the case where
Z
gt (y, z, v) = get y, z,
v(e)ηt (e)ζt (e) ρ(de) ,
(18)
E
e-measurable non-negative function ηt (e) with |ηt |t uniformly bounded, and a P ⊗ B(R) ⊗
P
B(R ) ⊗ B(R)-measurable function ge : Ω × [0, T ] × R × R1⊗d × R → R such that:
(H.1.i)' ge· (y, z, r) is a progressively measurable process, for any y ∈ R, z ∈ R1⊗d , r ∈ R;
(H.1.ii)' keg· (0, 0, 0)kH2 < +∞;
(H.1.iii)' |egt (y, z, r)−egt (y0 , z 0 , r0 )| ≤ Λ |y −y0 |+|z −z 0 |+|r −r0 | , for any t ∈ [0, T ], y, y0 ∈ R, z, z 0 ∈
R1⊗d and r, r0 ∈ R ;
(H.1.iv)' r 7→ get (y, z, r) is non-decreasing, for any (t, y, z) ∈ [0, T ] × R × R1⊗d .
for a
1⊗d
Using in particular the fact that
Z
(v(e) − v 0 (e))ηt (e)ζt (e) ρ(de) ≤ |v − v 0 |t |ηt |
E
with
|ηt |t
uniformly bounded, so
g
dened by (18) satises (H.1).
0
Our next goal is to prove a comparison result for (E ) (or (E ), see Remark 4.1(ii)) in this case, thus
extending to RBSDEs and R2BSDEs the comparison theorem of Barles et al. [2, Proposition 2.6
page 63] (see also Royer [32]) for classic BSDEs (without barriers). We refer the reader to Barles et
al. [2, Remark 2.7 page 64] for a counter-example in the general case, not assuming (H.1.iv)'.
To this end we shall rst prove the following Lemma relative to a linear BSDE (without barriers).
This BSDE is slightly non-standard inasmuch as its driving term contains a nite variation non
absolutely continuous process. This poses no special problem, however (see Remark 2.3(i)).
Lemma 4.1 (Linear BSDE)
Let us be given
ξ ∈ L2 ,
a process
A ∈ A2
and
get (y, z, r) = βt y + zπtT + κt r
R1⊗d -valued, processes β and κ, resp. π, with
terminal condition ξ at T and driving term dened by,
for uniformly bounded predictable real-valued, resp.
κη > −1. Let (Y, Z, V )
t ∈ [0, T ] :
solve the BSDE with
for
Z
At +
0
t
ges y, z,
Z
E
v(e)ηs (e)ζs (e)ρ(de) ds .
9
S. Crépey and A. Matoussi
Then, for any
τ ∈T :
τ
Z
h
Γ0 Y0 = E Γτ Yτ +
0
Γ
where the càdlàg adjoint process
i
Γs dAs F0 , P-a.s.
(19)
is the solution of the following linear (forward) SDE:
Z
ηt (e)e
µ(dt, de) , t ∈ [0, T ]
dΓt = Γt− βt dt + πt dBt + κt
(20)
E
with initial condition
Γ0 = 1.
In particular,
Γ>0
on
[0, T ].
Proof. Using (20), the integration by parts formula gives, for
Z
τ
Z
Z
i
+ κs
Γ0 Y0 = Γτ Yτ +
Γs− dAs + βs Ys +
Vs (e)ηs (e)ζs (e)ρ(de) ds
0
E
Z τ
Z τZ
−
Γs− Zs dBs −
Γs− Vs (e)e
µ(ds, de)
0
0
E
Z τ
Z
−
Ys− Γs− βs ds + πs dBs + κs
ηs (e)e
µ(ds, de)
E
Z0 τ
Z τZ
−
Γs Zs πsT ds −
Γs− Vs (e)κs ηs (e)µ(ds, de)
0
0
Z τ
Z τE
= Γτ Yτ +
Γs dAs −
Γs Zs + Ys πs ) dBs
0
0
Z τZ
−
Γs− (1 + κs ηs (e))Vs (e) + κs ηs (e)Ys− µ
e(ds, de) .
0
h
τ ∈T :
Zs πsT
E
·
Γs dAs is a local martingale. Moreover, sup[0,T ] |Y | belongs to L2 , and so does
0
(by Burkholder's inequality) sup[0,T ] |Γ|, hence their product is integrable. Thus the local martingale
Z ·
In particular
ΓY +
Γs dAs is a uniformly integrable martingale, whose value at time 0 is the F0 -conditional
0
expectation of its value at the stopping time τ ∈ T . This yields (19). Finally, we recognize in Γ the
ΓY +
stochastic exponential of
Z
·
·
Z
Θ :=
βs ds +
Z ·Z
πs dBs +
0
0
which is explicitely given in terms of
1
c
Γt = eΘt − 2 hΘ
it
Θ
κs ηs (e)e
µ(ds, de) ,
0
E
by
Y
1 + ∆Θs e−∆Θs , t ∈ [0, T ] .
(21)
0<s≤t
Therefore
Γ > 0,
Theorem 4.2
since
2
κη > −1.
(Y, Z, V, K) and (Y 0 , Z 0 , V 0 , K 0 ) be solutions to the R2BSDEs with data (g, ξ, L, U )
and (g , ξ , L , U ) satisfying assumptions (H.0)(H.1)(H.2). We assume further that g satises
0
(H.1)'. Then Y ≤ Y , dP ⊗ dt almost everywhere, whenever:
0
(i) ξ ≤ ξ , P almost surely,
0
0
0
0
0
0
0
(ii) g· (Y· , Z· , V· ) ≤ g· (Y· , Z· , V· ), dP ⊗ dt almost everywhere,
0
0
(iii) L ≤ L and U ≤ U , dP ⊗ dt almost everywhere.
0
0
0
Let
0
Proof. We write the proof in case
d = 1,
for notational simplicity. Let us denote
ξ = ξ − ξ0,
and for
10
Reflected BSDEs with Jumps
t ∈ [0, T ] :
δt = gt (Yt0 , Zt0 , Vt0 ) − gt0 (Yt0 , Zt0 , Vt0 )
−1
Yt − Yt0
gt (Yt , Zt , Vt ) − gt (Yt0 , Zt , Vt )
if
Yt 6= Yt0
βt =
0
if
Yt = Yt0 ,
−1
Zt − Zt0
gt (Yt0 , Zt , Vt ) − gt (Yt0 , Zt0 , Vt )
if
Zt 6= Zt0
πt =
0
if
Zt = Zt0

R
g (Y 0 , Z 0 , V ) − gt (Yt0 , Zt0 , Vt0 )

R t t t t0
if
Vt (e) − Vt0 (e) ηt (e)ζt (e)ρ(de) 6= 0
E
κt =
(Vt (e) − Vt (e))ηt (e)ζt (e)ρ(de)
E
R

0
if
Vt (e) − Vt0 (e) ηt (e)ζt (e)ρ(de) = 0 .
E
By assumption (H.1)' on
g,
we have:
gt (Yt0 , Zt0 , Vt ) − gt (Yt0 , Zt0 , Vt0 ) =
Z
Z
0
0
0
0
get Yt , Zt ,
Vt (e)ηt (e)ζt (e) ρ(de) − get Yt , Zt ,
Vt0 (e)ηt (e)ζt (e) ρ(de) .
E
E
ge
The Lipschitz continuity property of
with respect to
(y, z, r)
implies that
uniformly bounded progressively measurable processes. Moreover
on
[0, T ],
by assumption (H.1.iv)' on
Now, by linearity,
Y , Z, V
terminal condition
ξ¯ = ξ − ξ 0
T,
β, π, κ
are real-valued
is nite. Furthermore
κ≥0
g.
:= Y − Y 0 , Z − Z 0 , V − V 0
at
kδkH2
in which
solves the following linear BSDE with
At := Kt − Kt0 +
Z
t
δs ds
(see Remark 2.3(i)):
0
T
Z
T
Z
Y t = ξ¯ + AT − At +
t
T
Z
E
V s (e)e
µ(ds, de) , t ∈ [0, T ] .
t
t
Lemma 4.1 then yields, for any
E
τ ∈T :
τ
Z
h
Γ0 Y 0 = E Γτ Y τ +
V s (e)ηs (e)ζs (e)ρ(de) ds
Z
Z s dBs −
−
Z
Y s βs + Z s πs + κs
Z
τ
Γs δs ds +
0
Γs d(Ks+
+
Ks0− )
Z
−
0
0
τ
i
Γs d(Ks0+ + Ks− )F0 .
(22)
Now:
• κ ≥ 0, hence Γ > 0, by Lemma
• δ ≤ 0 and dK 0+ , dK − ≥ 0.
4.1;
Therefore choosing
τ = inf
then
0
Yτ ≤ 0
and
n
o
n
o
s ∈ [0, T ] ; Ys = Ls ∧ inf s ∈ [0, T ] ; Ys0 = Us0 ∧ T
K + = K 0− = 0
on
[0, τ ],
Y 0 ≤ 0, P almost surely, by (22). Since time
Yt ≤ Yt0 , P almost surely, for any t ∈ [0, T ].
0
that Yt ≤ Yt for any t ∈ [0, T ], P almost surely. 2
yielding
plays no special role in the problem, we have in fact
As
Y
and
Y0
are càdlàg processes, we conclude
Remark 4.1
(i) By inspection of the above proof, it appears that one may relax assumptions
(H.1.ii) and (H.1.iii) on
g0
into
kg.0 (Y.0 , Z.0 , V.0 )kH2 < ∞
in Theorem 4.2.
(ii) This comparison theorem admits obvious specications to RBSDEs and BSDEs. We thus recover
Barles et al. [2, Proposition 2.6 page 63] (see also Royer [32]).
5
Existence and Uniqueness Results
0
0
Recall that (Ē ) is more general than (E ), whereas (Ē ) can be considered as a generalization of either
0
(E ) or (E ) (see section 2.1.2). So some of the statements are in a sense redundant in Propositions
5.1 and 5.2 below. However we nd it convenient to state them explicitly, for more clarity.
11
S. Crépey and A. Matoussi
5.1
Uniqueness
Proposition 5.1
Under assumptions (H.0)(H.1)(H.2):
0
(i) Uniqueness holds for (E ) and (E );
τ ∈ T , uniqueness holds for the
ξ Fτ -measurable) and for the τ -R2BSDE (Ē ).
(ii) Given a further stopping time
0
time (Ē ) (assuming
Proof.
RBSDE with random terminal
0
(i) Uniqueness for (E ) results directly from the error estimate (17).
As for (E ), careful
examination of the proof of estimate (14) in section A.2 shows that in the special case
U
n,p
= 0,
Ln,p =
estimate (14) can be strengthened under weaker Assumptions, namely we have
kY n,p k2S 2 + kZ n,p k2H2 + kV n,p k2H2µ + kK n,p k2S 2 ≤
d
n,p 2
c(Λ)c1 kξ kL2 + kg·n,p (Y·n , Z·n , V·n )k2H2
for any sequence of R2BSDEs with common barriers
L
and
U
(23)
and such that
kξ n k22 + kg·n (0, 0, 0)k2H2 ≤ c1
(without any of the Assumptions specic to Lemma 3.1). Uniqueness for (E ) then directly follows
from (23).
0
(ii) Given Remark 2.4(i), uniqueness for (Ē ) follows from the uniqueness, by part (i), for the RBSDE
with data
(1·≤τ g, ξ, L·∧τ ). Finally, uniqueness for (Ē ) can be established as that for (E ) above, given
2
Remark 3.1(ii).
5.2
Existence
In this section we work under the following square integrable martingale predictable representation
assumption:
(H)
Every square integrable martingale
Z
Mt = M 0 +
M
t
admits a representation
Z tZ
0
for some
Z ∈ Hd2
and
Vs (e)e
µ(ds, de) , t ∈ [0, T ]
Zs dBs +
0
(24)
E
V ∈ Hµ2 .
We also strengthen Assumption (H.2.i) into:
(H.2.i)0 L
and
U
are càdlàg quasi-left continuous processes in
Recall that for a càdlàg process
X,
S 2.
quasi-left continuity is equivalent to the existence of sequence
of totally inaccessible stopping times which exhausts the jumps of
X,
whence
p
X = X·−
(Jacod
Shiryaev [25, Propositions I.2.26 page 22 and I.2.35 page 25]). We thus work in this section under
assumptions (H)(H.0)(H.1)(H.2)', where (H.2)' denotes (H.2) with (H.2.i) replaced by (H.2.i)'.
The proof of the following proposition, which is essentially contained in earlier results by Hamadène
and Ouknine [21] and Hamadène [22], is given in Appendix B. By the Mokobodski condition in
this proposition, we mean the existence of a quasimartingale
X
with Rao components in
S2
and
L ≤ X ≤ U over [0, T ]. This is of course tantamount to the existence of non-negative
1
2
2
1
2
supermartingales X , X belonging to S and such that L ≤ X − X ≤ U over [0, T ] (cf. rst
2
paragraph of section 3). X is then obviously a quasimartingale in S . Note that the question whether
2
2
any quasimartingale in S has Rao components in S is unsolved, to the best of our knowledge.
such that
Proposition 5.2
Assuming (H)(H.0)(H.1)(H.2)':
0
ξ is Fτ -measurable, here) (Ē 0 );
(ii) Existence of a solution to (E ) is equivalent to the Mokobodski condition, which also implies
(i) Existence holds for (E ) and (assuming that
existence of a solution to (Ē ). In particular, existence holds for (E ), whence (Ē ), when L or U is a
2
quasimartingale with Rao components in S (in which case, L or U is obviously a quasimartingale
2
in S as postulated in Lemma 3.1(i)).
12
Reflected BSDEs with Jumps
The complete characterization of existence for (Ē ) depends of course on the specication of the
stopping time
τ. Recall
to (Ē ) in this case), whereas
equivalence between existence
6
τ = T, (Ē ) reduces to (E 0 ) (whence always a solution
in the special case τ = 0 (Ē ) reduces to (E ) (whence in this case
of a solution to (Ē ) and the Mokobodski condition).
that in the special case
An Application in Finance
In the case of the convertible bonds related R2BSDEs in nance (see section 1), the lower barrier
L
is given by a call payo functional of the underlying stock price process
S,
the latter being typi-
cally modeled as a jump-diusion (with possibly random coecients). This motivates the following
developments.
6.1
Abstract Set-Up
Proposition 6.1
Let
S
be given as an Itô-Lévy process with square integrable special semimartingale
decomposition components, so
Z
t
St = S0 +
t
Z
Z tZ
as ds +
0
vs (e)e
µ(ds, de) , t ∈ [0, T ]
zs dBs +
0
0
(25)
E
z ∈ Hd2 , v ∈ Hµ2 , and a progressively measurable time-integrable process a such that kakH2 <
+∞. Let in turn L be given as L = S ∨ c, for some constant c ∈ R ∪ {−∞}.
2
Then L is a (càdlàg) quasi-left continuous quasimartingale with Rao components in S . Moreover L
satises all the conditions in Lemma 3.1 (including the hypotheses on L in (H.2)), with in particular
a− , the negative part of a in (25), for α in (4)(5).
for some
Proof. We have by the MeyerTanaka (or simply Itô-Lévy, in case
c = −∞)
formula, much like in
the proof of Lemma 3.1:
Z
dLt = 1{St >c} zt dBt +
E
1{St− >c} vt (e) µ
e(dt, de) − 1{St >c} a−
t dt
1
+ 1{St− >c} (St − c) + 1{St− ≤c} (St − c)+ + dΘt + 1{St >c} a+
t dt
2
(26)
−
where
Θ
S at c (or 0, in case c = −∞). We
Z T
1
Lt = E LT −
1{Su >c} au du − (ΘT − Θt )
2
t
is the local time of
thus have for
t ∈ [0, T ] :
"
−
X
1{Su− >c} (Su − c)− + 1{Su−

+
1
2

≤c} (Su − c) Ft = Lt − Lt
(27)
t<u≤T
t ∈ [0, T ] :
where we set, for
T
L1t = E L+
T +
Z
L2t = E L−
T +
X
Z
1{Su >c} a−
u du Ft
t
t
T
1
1{Su >c} a+
u du + (ΘT − Θt )+
2
1{Su− >c} (Su − c)− + 1{Su− ≤c} (Su − c)+ Ft
t<u≤T
Here
L1
and
L2
are non-negative supermartingales, as optional projections of non-increasing pro-
cesses. Moreover,
L
and
with Rao components in
L1 ,
S 2.
and thus, in turn,
L2 ,
belong to
S 2. L
is therefore a quasimartingale
13
S. Crépey and A. Matoussi
Observe further that the second line of (26) denes a non-decreasing integrable process. Denoting
by
R
e
R
and
its compensator and its compensatrix, we get:
Z
et
1{St− >c} vt (e) µ
e(dt, de) − dR
dLt = 1{St >c} zt dBt +
E
+ dRt −
1{St >c} a−
t dt
(28)
.
Z
So the predictable nite variation component
Z
R
and
A
of
L
is given by
A = R−
·
1{St >c} a−
t dt,
where
0
·
1{St >c} a−
t dt
are non-decreasing processes, thus the Jordan component
A−
of
A
satises
0
−
dA−
t ≤ 1{St >c} at dt.
6.2
2
JumpDiusion Setting with Regimes
Motivated by applications (see [9, 7, 10, 11, 4]), we now present a rather generic specication for a
Markovian model
F
(which in the context of nancial applications will correspond to a Markovian
factor process underlying a nancial derivative), and we show how it ts into the abstract set-up of
the present paper.
6.2.1 Specication of the Model
d and k, we dene the following linear operator G
(t, x, i) ∈ [0, T ] × Rd × I, where I = {1, · · · , k} :
Given integers
ui (t, x)
for
Gui (t, x) = ∂t ui (t, x) +
1
2
d
X
acting on regular functions
ail,q (t, x)∂x2l xq ui (t, x)
u=
(29)
l,q=1
+
d X
bil (t, x) −
Z
l=1
Z
+
Rd
δli (t, x, y)f i (t, x, y)m(dy) ∂xl ui (t, x)
ui (t, x + δ i (t, x, y)) − ui (t, x) f i (t, x, y)m(dy)
Rd
+
X
λi,j (t, x)(uj (t, x) − ui (t, x)) .
j∈I
In this equation,
m(dy) is a nite
jump measure on
Rd ,
and all the coecients are Borel-measurable
functions such that:
•
the
ai (t, x)
are
d-dimensional
dimensional dispersion matrices
covariance matrices, with
i
ai (t, x) = σ i (t, x)σ i (t, x)T
for some
d-
σ (t, x) :
• the bi (t, x) are d-dimensional drift vector coecients;
• the intensity functions f i (t, x, y) are bounded, and the jump size functions δ i (t, x, y) are absolutely
integrable with respect to m(dy);
• the [λi,j (t, x)]i,j∈I are P
intensity matrices such that the λi,j (t, x) are non-negative and bounded for
i 6= j, and λi,i (t, x) = − j∈I\{i} λi,j (t, x).
i
We shall often nd convenient to denote v(t, x, i, · · · ) rather than v (t, x, · · · ) for a function v of
(t, x, i, · · · ), and λ(t, x, i, j), for λi,j (t, x). For instance, the notation f (t, Xt , Nt , y) (or even f (t, Ft , y),
N
with Ft = (Xt , Nt ) below) will typically be used rather than f t (t, Xt , y). Also note that a function
d
u on [0, T ] × R × I may equivalently be referred to as a system u = (ui )i∈I of functions ui = ui (t, x)
d
on [0, T ] × R .
The construction of a model corresponding to the previous data is a non-trivial issue treated in
detail in [10] (see also [11], or see Theorems 4.1 and 5.4 in Chapter 4 of Ethier and Kurtz [19] for
14
Reflected BSDEs with Jumps
abstract conditions regarding the existence and uniqueness of a solution to the martingale problem
with generator
G ).
We will thus be rather formal at this point of the present paper, referring the
reader to [10, 11] for the complete statement of suitable conditions below.
(Ω, F, P) on [0, T ] endowed
B, an integer-valued random measure χ and an (Ω, F, P)Markov càdlàg process F = (X, N ) on [0, T ] with initial condition (x, i) at time 0, such that:
• dening ν as the integer-valued random measure on I which counts the transitions νt (j) of N to
state j between time 0 and time t, the P-compensatrix ν
e of ν is given by
So under suitable conditions (see [10, 11]), there exists a stochastic basis
d-dimensional
with a
Brownian motion
de
νt (j) = dνt (j) − 1{Nt 6=j} λ(t, Ft , j) dt
(with
(30)
λ(s, Ft , j) = λNt ,j (s, Xt )), whence the following canonical special semimartingale
N:
X
X
dNt =
λ(t, Ft , j)(j − Nt ) dt +
(j − Nt− ) de
νt (j) , t ∈ [0, T ] ;
representa-
tion for
j∈I
• the P-compensatrix χ
e
of
(31)
j∈I
χ
is given by
χ
e(dt, dy) = χ(dt, dy) − f (t, Ft , y)m(dy)dt,
and the
Rd -valued
process
X
satises, for
t ∈ [0, T ] :
Z
dXt = b(t, Ft ) dt + σ(t, Ft ) dBt +
Rd
δ(t, Ft− , y) χ
e(dy, dt) .
(32)
p ∈ [2, +∞):
≤ Cp 1 + |x|p .
Besides the following estimates are available, for any
kXkpS p
d
(33)
We then have the following variant of the Itô formula (see, e.g., Jacod [24, Theorem 3.89 page 109]),
where
∂u
denotes the row-gradient of
u = ui (t, x)
with respect to
x:
du(t, Ft ) = Gu(t, Ft )dt + ∂u(t, Ft )σ(t, Ft )dBt
Z
+
u(t, Xt− + δ(t, Ft− , y), Nt− ) − u(t, Ft− ) χ
e(dy, dt)
d
R
X
+
(u(t, Xt− , j) − u(t, Ft− ))de
νt (j) , t ≥ 0
(34)
j∈I
for any system
u = (ui )i∈I
of functions
ui = ui (t, x)
of class
C 1,2
on
[0, T ] × Rd .
In particular
(Ω, F, P, F ) is a solution to the time-dependent local martingale problem with generator G
condition (t, x, i) (see Ethier and Kurtz [19, sections 7.A and 7.B]).
Finally, still under suitable conditions (see [11]), every
and initial
(Ω, F, P)-square integrable martingale M
in
this model admits a representation
Z
Mt = M0 +
t
Z tZ
Zs dBs +
0
for some
Z ∈ Hd2 , Ve ∈ Hχ2
Ves (y)e
χ(dy, ds) +
0
and
Rd
XZ
j∈I
t
fs (j)de
W
νs (j) , t ∈ [0, T ]
(35)
0
f ∈ H2 .
W
ν
6.2.2 Mapping with the Abstract Set-Up
Let
0d stand for the null in Rd . The model F = (X, N ) is thus a rather generic Markovian specication
of our abstract set-up, with (cf. section 2):
• E,
the subset
Rd × {0} ∪ {0d } × I
of
Rd+1 ;
15
S. Crépey and A. Matoussi
• BE ,
the sigma eld generated by
the Borel sigma eld on
• ρ(de)
R
d
B(Rd ) × {0}
{0d } × I on E, where B(Rd )
all parts of I, respectively;
and
and the sigma eld of
and
I
stand for
e = (y, j) ∈ E :
m(dy) if
j=0
f (t, Ft , y)
if
j=0
ρ(de) =
, ζt (e) =
1
if
y = 0d
1{Nt 6=j} λ(t, Ft , j) if y = 0d
• µ, the integer-valued random measure on [0, T ] × E, B([0, T ]) ⊗ BE counting the jumps
d
size y ∈ A and the jumps of N to state j between 0 and t, for any t ≥ 0, A ∈ B(R ), j ∈ I.
and
ζt (e)
respectively given by, for any
of X of
We denote for short:
(E, BE , ρ) = (Rd ⊕ I, B(Rd ) ⊕ I, m(dy) ⊕ 1) .
So, in the present context:
Mρ ≡ M(Rd , B(Rd ), m(dy); R) × Rk
and the compensator of
µ
is given by, for any
(36)
t ≥ 0, A ∈ B(Rd ), j ∈ I,
with
A ⊕ {j} := A × {0} ∪
{0d } × {j} :
Z tZ
Z tZ
ζs (e)ρ(de)ds =
0
A⊕{j}
Z
f (s, Fs , y)m(dy)ds +
0
A
t
1{Ns 6=j} λ(s, Fs , j) ds .
0
Note nally that (35) is a martingale representation of the form (24), with for
(
Vs (de) =
Hence the model
6.3
F
e = (y, j):
Ves (y) if j = 0
fs (j) if y = 0d .
W
has the martingale representation property (H).
Markovian BSDEs
We consider in this model the BSDE naturally connected with the Itô formula (34), namely for
t≥0:
Z
− dYt = g(t, Ft , Yt , Zt , Vt )dt − Zt dBt −
Vet (y)e
χ(dy, dt) −
Rd
with
f ),
V = (Ve , W
X
ft (j)de
W
νt (j)
j∈I
possibly supplemented by suitable barrier and minimality conditions, and for a
suitable driver coecient
g(t, Ft , y, z, v) where v = (e
v , w)
e ∈ M(Rd , B(Rd ), m(dy); R) × Rk
d
i
(cf. (36)).
u on [0, T ] × R × I such that u is Borel-measurable with
x for any i ∈ I. Let us further be given real-valued continuous running cost
functions g
ei (t, x, u, z, r) (where (u, z, r) ∈ Rk × R1⊗d × R), terminal cost functions Ψi (x), and lower
i
i
and upper obstacle functions ` (t, x) and h (t, x), such that:
(M.0) Ψ lies in P ;
(M.1.i) (t, x, i) 7→ gei (t, x, 0, 0, 0) lies in P ;
(M.1.ii) ge is uniformly Λ Lipschitz continuous with respect to (u, z, r), in the sense that Λ is a
d
0 0 0
k
1⊗d
constant such that for every (t, x, i) ∈ [0, T ] × R × I and (u, z, r), (u , z , r ) ∈ R × R
×R:
|e
gi (t, x, u, z, r) − gei (t, x, u0 , z 0 , r0 )| ≤ Λ |u − u0 | + |z − z 0 | + |r − r0 | ;
Let
P
denote the class of functions
polynomial growth in
(M.1.iii) ge is non-decreasing with respect
(M.2.i) ` and h lie in P ;
(M.2.ii) ` ≤ h, `(T, ·) ≤ Ψ ≤ h(T, ·);
to
r;
16
Reflected BSDEs with Jumps
We dene for any
(t, y, z, v) ∈ [t, T ]×R×R1⊗d ×Mρ , with v = (e
v , w)
e ∈ M(Rd , B(Rd ), m(dy); R)×Rk :
g(t, Ft , y, z, v) = ge(t, Ft , u
et , z, ret ) −
X
wj λ(t, Ft , j) ,
(37)
j∈I\{Nt }
where
u
et = u
et (y, w)
e
ret = ret (e
v ) are dened as
Z
y,
j = Nt
(e
ut )j =
, ret =
ve(y)f (t, Ft , y)m(dy) .
y+w
ej , j 6= Nt
Rd
and
(38)
We then consider the data
gt (ω, y, z, v) = g(t, Ft , y, z, v) , ξ = Ψ(FT ) , Lt = `(t, Ft ) , Ut = h(t, Ft ) .
Remark 6.1
(39)
The connection between the Markovian R2BSDEs with data of the form (39) and the
Markovian R2BSDEs which appear in risk-neutral pricing problems in nance (see [7]) is established
in [10] (see also [11]).
Proposition 6.2
The data (39) satisfy assumptions (H.0)(H.1)(H.2)'.
Proof. Given (M.0)(M.1)(M.2) and the estimate (33) on
is straightforward (see [10] for every detail).
Within model
F
X, the verication of (H.0)(H.1)(H.2)'
2
we are able to specify a concrete class of processes
S
which satisfy the conditions
of Proposition 6.1. We thus have the following
Lemma 6.3
[0, T ] × R
d
Let
φ = (φi )i∈I
be a system of real-valued functions
φi = φi (t, x)
of class
C 1,2
on
such that
Z
|φi (t, x + δ i (t, x, y))|m(dy) ∈ P .
φ, Gφ, ∂φσ, (t, x, i) 7→
(40)
Rd
Then the process
S
dened by, for
t ∈ [0, T ] :
St = φ(t, Ft ) ,
is an Itô-Lévy process with square integrable special semimartingale decomposition components, with
related process
Proof.
a
in (25) given as
at = Gφ(t, Ft ),
for
t ∈ [0, T ].
Under our polynomial growth assumptions and given the estimates (33) on
follows by application of the Itô formula (34) to
Example 6.2
X,
the result
2
φ(t, Ft ).
S in Proposition 6.1 is S = X 1 , the rst
d ≥ 1 therein). This corresponds to the case
The standing example we have in mind for
X of
φi (t, x) = x1
F = (X, N )
component of
our model
where
in Lemma 6.3. Note that in this case:
Z
Gφ = b1 , ∂φσ = σ1 ,
(assuming
|φi (t, x + δ i (t, x, y))|m(dy) =
Rd
Z
Rd
|x1 + δ1i (t, x, y)|m(dy) ,
so that (40) reduces to
Z
b1 , σ1 , (t, x, i) 7→
Rd
|δ1i (t, x, y)|m(dy) ∈ P .
(41)
17
S. Crépey and A. Matoussi
Theorem 6.4
Given the data (39) with ` specied as φ ∨ c where φ satises (40) (e.g., φ = x1 ,
assuming (41)) and for some constant c ∈ R ∪ {−∞}, then the related R2BSDE (E ) admits a unique
+
+
solution (Y, Z, V, K). Moreover K
is an Lebesgue-absolutely continuous process with density k
0
satisfying (5). The RBSDE (E ) also admits a unique solution. Finally, given a further stopping
0
time τ ∈ T , the RBSDE with random terminal time (Ē ) (assuming ξ Fτ -measurable, here) and the
τ -R2BSDE (Ē ) also have unique solutions.
F
Proof.
First, our model
6.2.2).
Moreover assumptions (H.0)(H.1)(H.2)' are satised, by Proposition 6.2.
has the martingale representation property (H) (see end of section
a quasimartingale with Rao components in
(see also Example 6.2 in case
Proposition 5.2(i).
Consequently
K+
S 2,
Therefore (E ) admits a unique solution
φ = x1 ).
Finally,
is
(Y, Z, V, K),
by
Moreover all the conditions of Lemma 3.1(ii) are fullled, by Proposition 6.1.
is an Lebesgue-absolutely continuous process with density
k+
satisfying (5). The
2
remaining results follow likewise by application of Proposition 5.2.
A
L
by application of Proposition 6.1 and Lemma 6.3
Proof of Theorem 3.2
In this appendix,
c
denotes a large constant which may change from line to line. We do not track
the dependency of the constants line after line, letting the reader check in the end that the overall
dependency is indeed like stated in Theorem 3.2.
A.1
Proof of the bound estimate
c
We have to show that there exists a constant
t ∈ [0, T ]
and
with the required dependencies such that, for any
n∈N:
T
Z
h
E sup |Ytn |2 +
t∈[0,T ]
|Zsn |2 ds +
0
We omit indices
n
T
Z
Z
0
i
|Vsn (e)|2 ζs (e)ρ(de)ds + (KTn,+ )2 + (KTn,− )2 ≤ c .
E
(42)
in the rest of this section, to alleviate the notation. Standard computations based
on Itô's formula and Gronwall's Lemma yield:
T
hZ
E
Ys2 ds +
0
T
Z
|Zs |2 ds +
0
Z
h
≤ c E ξ2 +
T
Z
Z
0
T
gs2 (0, 0, 0)ds +
|Vs (e)|2 ζs (e)ρ(de)ds
E
T
Z
0
|Ls |dKs+ +
T
g,
Z
gs2 (0, 0, 0)ds +
0
Z
+
0
(43)
we have:
T
T
Z
|Ys |2 ds +
0
T
Z
h
2
≤ E(A−
)
+
cE
ξ2 +
T
i
|Us |dKs− .
0
Besides, using (3) and the Lipschitz continuity property of
Z
T
Z
0
Z
h
h
2
E (KT+ )2 ≤ E (A−
)
+
T
i
|Zs |2 ds
0
i
|Vs (e)|2 ν(ds, de)
E
T
gs2 (0, 0, 0)ds
T
Z
|Ls |dKs+
+
0
T
Z
+
0
i
|Us |dKs− ,
(44)
0
by (43). Moreover, we have likewise by the related R2BSDE:
Z
2
h
+
−
2
E KT − KT
≤ cE ξ +
0
T
gs2 (0, 0, 0)ds
Z
+
0
T
|Ls |dKs+
Z
+
0
T
i
|Us |dKs− .
(45)
18
Reflected BSDEs with Jumps
So, combining (44) and (45):
i
h
E (KT+ )2 + (KT− )2 ≤
T
Z
h
2
cE ξ 2 + (A−
)
+
T
gs2 (0, 0, 0)ds + sup L2s + sup Us2
0≤s≤T
0
i
(46)
0≤s≤T
and nally
T
Z
h
2
E |Yt | +
|Zs |2 ds +
T
Z
0
Z
0
E
T
Z
h
− 2
2
≤ cE ξ + (AT ) +
|Vs (e)|2 ζs (e)ρ(de)ds + (KT+ )2 + (KT− )2
i
gs2 (0, 0, 0)ds + sup L2s + sup Us2 .
0≤s≤T
0
Y2
Applying Itô's formula to
i
(47)
0≤s≤T
again, and taking rst suprema in time, then expectations, we deduce
(42) by the Burkholder inequality.
dAn,− ≤ αtn dt
Moreover, in the case
for some progressively measurable processes
αn
with
kαn kH2
nite, we have by application of Lemma 3.1(ii):
dK n,+ = kt+,n dt
with
kt+,n ≤ 1{Ytn =Lnt } gtn (Ytn , Ztn , Vtn )− + αtn .
kk n,+ kH2 is nite, by the previous results. One may then replace
2
Ls ds in (46) and (47), and then in turn kLn k2S 2 by kLn k2H2 in (12).
0
In particular
sup0≤s≤T L2s
by
RT
A.2
Proof of the error estimate (14)
n
Expliciting indices
and
·
of
g,
p
again, we get by the Itô formula and the Lipschitz continuity property
with ≤ standing for ≤ up to a martingale term :
(Ytn − Ytp )2 +
T
Z
|Zsn − Zsp |2 ds +
Z
t
n
T
Z
t
Z
p 2
|ξ − ξ | + 2
·
|Vsn (e) − Vsp (e)|2 ζs (e)ρ(de)ds ≤
E
T
|gsn (Ysn , Zsn , Vsn )
− gsp (Ysn , Zsn , Vsn )|2 ds
t
T
Z
|Ysn
+c
−
Ysp |2
t
+
T
Z
1
2
Z
t
1
ds +
2
Z
T
|Zsn − Zsp |2 ds
t
|Vsn (e) − Vsp (e)|2 ζs (e)ρ(de)ds + 2
Z
E
T
(Ysn − Ysp )(dKsn − dKsp ) .
t
Now, by the barriers conditions:
T
Z
(Ysn − Ysp )(dKsn − dKsp ) ≤
(48)
t
Z
T
(Lns − Lps )(dKsn,+ − dKsp,+ ) − (Usn − Usp )(dKsn,− − dKsp,− ) .
t
Thus
Z
Z Z
h
i
1 T n
1 T
p 2
n
p 2
E |Yt − Yt | +
|Zs − Zs | ds +
|Vsn (e) − Vsp (e)|2 ζs (e)ρ(de)ds ≤
2 t
2 t E
Z T
h
c E |ξ n − ξ p |2 +
|Ysn − Ysp |2 ds
t
Z
+
T
|gsn (Ysn , Zsn , Vsn ) − gsp (Ysn , Zsn , Vsn )|2 ds
t
i
+ sup |Lns − Lps |(KTn,+ + KTp,+ ) + sup |Usn − Usp |(KTn,− + KTp,− ) .
0≤s≤T
0≤s≤T
(49)
19
S. Crépey and A. Matoussi
kY n,p k2S 2 +
n,p 2
n,p 2
kZ kH2 + kV kH2 by Gronwall's Lemma, estimate (13) and Burkholder inequality. The control
µ
d
n,p 2
over kK
kS 2 follows using the equation for K n,p deduced of the related R2BSDEs.
n,−
Moreover, in the case where dA
≤ αtn dt for some progressively measurable processes αn with
n
kα kH2 nite (see end of section A.1), then the barriers conditions (48) write:
Using arguments already used in the previous section, we get the required control over
T
Z
(Ysn − Ysp )(dKsn − dKsp ) ≤
t
T
Z
(Lns − Lps )(ksn,+ − ksp,+ )ds −
T
Z
t
(Usn − Usp )(dKsn,− − dKsp,− ) .
t
RT
|Lns − Lps |(ksn,+ + ksp,+ )ds instead of sup0≤s≤T |Lns −
0
n,p
therein, which in turn implies (14) with kL
kH2 instead of kLn,p kS 2 therein.
We thus have (49) with
A.3
Lps |(KTn,+ + KTp,+ )
Convergence proof
We now turn to the situation considered in the last part of the Theorem. In this case, we are for
each
n
in the situation of Lemma 3.1(ii), whence
dK n,+ = kt+,n dt
So
kk n,+ kH2
n
n
n
(Y , Z , V )
2
2
in S × Hd ×
with
kt+,n ≤ 1{Ytn =Lnt } gtn (Ytn , Ztn , Vtn )− + αtn .
is bounded, by the results of the previous section (assuming
kαn kH2
bounded).
n
n
n
n
2
2
is bounded in S × Hd × Hµ , by (13). Hence (Y , Z , V , K ) is a Cauchy sequence
2
n
n
n
n
2
2
2
2
Hµ × S , by (14). Therefore (Y , Z , V , K ) S × Hd × Hµ × S 2 -converges to some
2
(Y, Z, V, K).
limiting process
K) solves (E ).
(Y, Z, V,
E (KTn,+ )2 ≤ c, so the K n,+ are bounded in H2 , as are
kk n,+ k2H2 is bounded, as noticed above. Thus by application
Let us show that
By the bound estimate (13), we have that
the
K n,
whence the
K n,− .
Besides,
of the BanachMazur Lemma (see CvitanicKaratzas [12, page 2046] and references therein), there
n ∈ N,
exist, for every
an integer
N (n) ≥ n
N (n)
X
e n,± =
K
wjn ≥ 0
and weights
with
PN (n)
j=n
wjn = 1
such that:
N (n)
e±
wjn K j,± → K
and
X
e
k n,+ =
j=n
wjn k j,+ → e
k+
in
H2
as
n→∞.
j=n
This implies in particular that
e+ =
K
Z
·
e
ku+ du
(cf. CvitanicKaratzas [12, page 2047]). Moreover,
0
since
K n,+ − K n,− = K n
with
K n,± ∈ A2i ,
thus
e+ − K
e− = K
K
e ± = 0),
K
0
with
e± ≥ 0
dK
e ± ∈ A2 , using also the continuity of
K
i
e +, K
e − ), and the process
K. In addition, by passage to the limit, estimate (13) holds for (Y, Z, V, K
e + −K
e − , satises the limiting equation (ii) in (E ). We also have L ≤ Y ≤ U.
(Y, Z, V, K), with K = K
Z T
Finally there comes, using the fact that
(Utn − Ytn )dKtn,− = 0 in the second line:
(and
by passage to the limit in
H2 .
So nally
0
Z
0≤
T
e t− =
(Ut − Yt )dK
0
Z
T
e t− − dKtn,− ) +
(Ut − Yt )(dK
0
Z
=
0
T
(Ut − Yt )dKtn,−
0
T
e t− −
(Ut − Yt )(dK
Z
dKtn,− )
Z
+
0
T
(Ut − Utn + Ytn − Yt )dKtn,− .
20
Reflected BSDEs with Jumps
T
Z
(Ut − Utn + Ytn − Yt )dKtn,−
Now,
converges to 0 in expectation, by
0
(Y, U )
to
K n,− .
and bound estimate (13) on the
measure, of
e− − K
e n,−
K
(S 2 )2 -convergence of (Y n , U n )
H2 ,
Besides, we have convergence in
to 0 (at least, along a suitable subsequence).
hence in
Moreover, by Proposition
1.5(d) in MéminSlominski [29] (see also Prigent [30, Theorem 1.4.2(4) page 102]), the sequence
e − −K
e n,− )n
(K
is predictably uniformly tight (see JacodShiryaev [25, VI.6a page 377]), as converging
in law (to 0) with
e tn,− )
dK
e t− − K
e tn,− )n
(K
2
L
bounded in
for every
t ∈ [0, T ].
e t− −
(Ut − Yt )(dK
Therefore
0
converges in measure (for the Skorokhod topology) to 0 (JacodShiryaev [25, Theorem
Z
T
VI.6.22(c) page 383], see also Prigent [30, Ÿ1.4]), so that nally
Z
T
Z
e t− = 0.
(Ut − Yt )dK
Likewise,
A2i
and such
0
T
e t+ = 0.
(Yt − Lt )dK
0
Since
that
e+ − K
e+
K=K
e ±.
K± ≤ K
with
Z
Thus
T
e ± ∈ A2 ,
K
i
(Ut − Yt )dKt− =
of
K
are also in
T
Z
0
B
K±
so the Jordan components
2
(Yt − Lt )dKt+ = 0.
0
Proof of Proposition 5.2
B.1
Basic Problems
With the exception of Becherer [3], previous works on BSDEs with jumps (see e.g. [33, 2, 21, 18, 22])
deal more specically with the case where the integer-valued random measure
measure.
µ is a Poisson random
Becherer [3] treats the case of a classic BSDE (no barriers) in the present set-up, thus
extending to the case of a random density
ζt (e)
the results of [33, 2].
We leave to the reader the routine task to check that all the results in [21, 18, 22] can be immediately
0
extended to the abstract set-up of the present paper. So our RBSDE (E ) admits a (unique) solution
(see Hamadène and Ouknine [21]). As for (E ), we know by HamadèneHassani [22, Theorem 4.1
and Remark 4.2] that the existence of a solution to (E ) is equivalent to the Mokobodski condition.
In particular, existence holds for (E ) when
Remark B.1
holds when
L
L
or
U
is a quasimartingale with Rao components in
By application of Theorem 3.3(ii) and in view of Remark 3.1(i), existence for (E ) also
(or
U)
is a limit in
S2
of quasimartingales
provided the predictable nite variation components
αn
with
B.2
S 2.
kαn kH2
bounded over
Ln
n,−
A
U n)
(resp.
of
n
L
with Rao components in
(resp.
n,+
A
of
U
n
S 2,
) have densities
n ∈ N.
Extensions with stopping time
Given a further stopping time
τ ∈ T , we now consider the variants of the above problems introduced
in section 2.1.2.
B.2.1 Reected BSDE with random terminal time
By inspection of the arguments of Hamadène and Ouknine [21], it appears that the existence result
0
for (E ) admits an immediate extension to the case of a reected BSDE with random terminal time
τ
(in the sense of Darling and Pardoux [13], but in the rather elementary situation where our stopping
time
τ
is bounded here, cf. Remark 2.4(ii)). So, assuming that
0
solution to the RBSDE (Ē ) also holds true.
ξ
is
Fτ -measurable,
existence of a
21
S. Crépey and A. Matoussi
B.2.2 Upper barrier with delayed activation
We nally consider the
τ -R2BSDE (Ē ).
Note that in applications (see [5, 7, 8]),
as a predictable stopping time. In this case, the upper barrier
Ū
time, and (H.2.i)' (or an immediate adaptation to the case of an
not satised by
Ū .
This is why the
τ -R2BSDE
τ
is typically given
has a jump at a predictable stopping
R ∪ {+∞}-valued
upper barrier) is
deserves a separate treatment.
τ -R2BSDE (Ē ) with data (g, ξ, L, U, τ ) has a solution under the Mokobodski
b
b
b denote the solution to (E ). This solution is indeed known to exist
(Y , Z, Vb , K)
In order to show that the
condition, let then
(and be unique) under the Mokobodski condition, by the results reviewed in section B.1. Let likewise
(Ȳ , Z̄, V̄ , K̄)
denote the solution, known to exist by the result of section B.2.1, to the RBSDE with
random terminal time
K+
τ
and data
(Ybτ , g, L)
on
[0, τ ].
Now, dening
(Y, Z, V, K)
by
Y := Ȳ 1t<τ + Yb 1t≥τ
b− − K
b τ− )1t≥τ
b + + (K̄τ − K
b τ+ )]1t≥τ , K − := (K
:= K̄1t<τ + [K
b t>τ , V := V̄ 1t≤τ + Vb 1t>τ ,
Z := Z̄1t≤τ + Z1
then by construction
(Y, Z, V, K)
Acknowledgements.
is a solution to the
τ -R2BSDE (Ē )
on
[0, T ].
It is our pleasure to thank Monique Jeanblanc for kind advice and useful
discussions throughout the work.
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