QBSDEs with unbounded terminal data
K HALED B AHLALI
IMATH, Université de Toulon
& CNRS, I2M, Aix-marseille Université
CIMPA, Tlemcen, 12-24 avril 2014
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
1/
19
1
Some previous results in Quadratic BSDEs
2
The summarized way
Essaky-Hassani result on reflected BSDEs
3
4
5
A simple QBSDE
Proof
Usual QBSDE
References
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
Previous results in BSDEs and Quadratic BSDEs
The BSDE under consideration is
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds −
Z T
t
Zs dWs
(1)
Usually, the BSDE eq (ξ, H ) is called quadratic if there exist α, β, γ > 0
such that for every (t, y , z ),
|H (t, y , z )| ≤ α + β|y | + γ|z |2 .
(2)
Kobylanski, 1997, and 2000,
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
Previous results in BSDEs and Quadratic BSDEs
The BSDE under consideration is
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds −
Z T
t
Zs dWs
(1)
Usually, the BSDE eq (ξ, H ) is called quadratic if there exist α, β, γ > 0
such that for every (t, y , z ),
|H (t, y , z )| ≤ α + β|y | + γ|z |2 .
(2)
Kobylanski, 1997, and 2000,
Dermoune-Hamadene-Ouknine, 1997,
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
Previous results in BSDEs and Quadratic BSDEs
The BSDE under consideration is
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds −
Z T
t
Zs dWs
(1)
Usually, the BSDE eq (ξ, H ) is called quadratic if there exist α, β, γ > 0
such that for every (t, y , z ),
|H (t, y , z )| ≤ α + β|y | + γ|z |2 .
(2)
Kobylanski, 1997, and 2000,
Dermoune-Hamadene-Ouknine, 1997,
Lepeltier–San Martin 1998,
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
Previous results in BSDEs and Quadratic BSDEs
The BSDE under consideration is
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds −
Z T
t
Zs dWs
(1)
Usually, the BSDE eq (ξ, H ) is called quadratic if there exist α, β, γ > 0
such that for every (t, y , z ),
|H (t, y , z )| ≤ α + β|y | + γ|z |2 .
(2)
Kobylanski, 1997, and 2000,
Dermoune-Hamadene-Ouknine, 1997,
Lepeltier–San Martin 1998,
Briand-Hu, 2006 and 2008,
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
Previous results in BSDEs and Quadratic BSDEs
The BSDE under consideration is
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds −
Z T
t
Zs dWs
(1)
Usually, the BSDE eq (ξ, H ) is called quadratic if there exist α, β, γ > 0
such that for every (t, y , z ),
|H (t, y , z )| ≤ α + β|y | + γ|z |2 .
(2)
Kobylanski, 1997, and 2000,
Dermoune-Hamadene-Ouknine, 1997,
Lepeltier–San Martin 1998,
Briand-Hu, 2006 and 2008,
Tevzadze, 2008 existence and uniqueness for quadratic-Lipshitz
and bounded terminal condition.
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
Previous results in BSDEs and Quadratic BSDEs
The BSDE under consideration is
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds −
Z T
t
Zs dWs
(1)
Usually, the BSDE eq (ξ, H ) is called quadratic if there exist α, β, γ > 0
such that for every (t, y , z ),
|H (t, y , z )| ≤ α + β|y | + γ|z |2 .
(2)
Kobylanski, 1997, and 2000,
Dermoune-Hamadene-Ouknine, 1997,
Lepeltier–San Martin 1998,
Briand-Hu, 2006 and 2008,
Tevzadze, 2008 existence and uniqueness for quadratic-Lipshitz
and bounded terminal condition.
Barrieu-El Karoui, 2012-2013, with more or less similar conditions,
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
Previous results in BSDEs and Quadratic BSDEs
The BSDE under consideration is
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds −
Z T
t
Zs dWs
(1)
Usually, the BSDE eq (ξ, H ) is called quadratic if there exist α, β, γ > 0
such that for every (t, y , z ),
|H (t, y , z )| ≤ α + β|y | + γ|z |2 .
(2)
Kobylanski, 1997, and 2000,
Dermoune-Hamadene-Ouknine, 1997,
Lepeltier–San Martin 1998,
Briand-Hu, 2006 and 2008,
Tevzadze, 2008 existence and uniqueness for quadratic-Lipshitz
and bounded terminal condition.
Barrieu-El Karoui, 2012-2013, with more or less similar conditions,
Essaky-Hassani, 2011 and 2013, for stochastic quadratic BSDE
and others conditions.
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
2/
19
The summarized way
Our Aproach consists to derive the existence of solutions for the BSDE
without reflection from solutions of a suitable 2-barriers Reflected
BSDE.
To this end, we use the result of Essaky & Hassani which establishes
the existence of solutions for reflected QBSDEs without assuming any
integrability condition on the terminal datum.
For the self-contained, we state the Essaky & Hassani result of in the
following theorem.
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
3/
19
Theorem
(Essaky–Hassani (JDE 2013)). Let L and U be continuous processes
and ξ be a FT measurable random variable. Assume that
1)
LT ≤ ξ ≤ UT .
2)
there exists a semimartingale which passes between the barriers
L and U.
3) H is continuous in (y , z ) and satisfies for every (s, ω ), every
y ∈ [Ls (ω ), Us (ω )] and every z ∈ Rd .
|f (s, ω, y , z )| ≤ η s (ω ) + Cs2(ω ) |z |2
where η ∈ L1 ([0, T ] × Ω) and C is a continuous process.
Then, the following RBSDE has a minimal and a maximal solution.

RT
RT
RT
RT
+
−

H
(
s,
Y
,
Z
)
ds
+
dK
−
dK
−
Zs dBs
(
i
)
Y
=
ξ
+

s
s
t
s
s
t
t
t
t



 (ii ) ∀t ≤ T , Lt ≤ Yt ≤ Ut ,
RT
RT
(iii ) t (Yt − Lt )dKt+ = t (Ut − Yt )dKt− = 0, a.s.,



(iv ) K0+ = K0− = 0, K + , K − , are continuous nondecreasing.



(v ) dK + ⊥dK −
(3)
CIMPA, Tlemcen, 12-24 avril 2014
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
4/
19
The equation eq (ξ, a + b |y | + 12 z 2 )
Theorem
Let a and b be positive real numbers. Assume that ξ satifies
(HExp) E exp[(e2bT + 1)|ξ |] < ∞
Then, the BSDE
Yt = ξ +
Z T
t
1
(a + b |Ys | + |z |2 )ds −
2
Z T
t
Zs dWs , 0 ≤ t ≤ T
(4)
has a unique solution (Y , Z ) such that
E
sup exp([e2bs + 1]|Ys |) < ∞ and
0≤s ≤T
K HALED B AHLALI (UTLN)
E
QBSDEs with unbounded terminal data
Z T
0
|Zs |2 ds < ∞.
(5)
CIMPA, Tlemcen, 12-24 avril 2014
5/
19
Proof simple QBSDE
Proof. We put g (y , z ) := a + b |y | + 12 |z |2 . Let (Y , Z ) be a solution to
BSDE (4). By Itô’s formula we have,
e Yt = e YT +
ξ
=e +
Z T
t
T
Z
e
t
=eξ+
Z T
t
1
2
Z T
1
1
(a + b |Ys | + |Zs |2 ) − |Zs |2 ds −
2
2
Z T
e Ys g (s, Ys , Zs )ds −
Ys
e Ys (a + b |Ys |)ds −
Z T
t
Z T
t
e Ys Zs dWs −
t
t
e Ys |Zs |2 ds
e Ys Zs dWs
e Ys Zs dWs
We set, Yt = e Yt , Zt = e Yt Zt and ξ = e ξ .
It is not difficult to see that Y > 0 and (Y , Z ) satisfies the BSDE
Yt
= ξ+
Z T
t
(aYs + bYs | ln Y s |)ds −
Z T
t
Zs dWs
(6)
Hence,
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QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
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Proof simple BSDE, continued 1
according to Theorem on BSDEs with logarithmic nonlinearity (Slide
2), the previous equation has a unique solution which satisfy the
inequalities
E
sup |Yt |
e2c0 t +1
<∞
and
E
Z T
t ∈[0,T ]
0
|Zs |2 ds < ∞
(7)
Therefore, the BSDE (4) has a unique solution too. Indeed,
Let (Y 1 , Z 1 ) and (Y 2 , Z 2 ) be two solutions to BSDE (4).
i
i
Then, for i = 1, 2, Ȳ i > 0 and Y¯ i = e Yt and Z¯ i = e Yt Z i are solutions
t
t
t
to the logarithmic BSDE (6) which satisfy inequalities (7).
Hence, according to Theorem on Logarithmic BSDEs ,
(Y¯1 , Z¯1 ) = (Y¯2 , Z¯2 ) from which we deduce that (Y 1 , Z 1 ) = (Y 2 , Z 2 ).
It remains to prove inequalities (5).
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
7/
19
Proof, simple BSDE, continued 2
It is not difficult to show that E sup0≤s ≤T exp([e2bs + 1]|Ys |) < ∞.
To prove that Z belongs to M2 , we need the following simple Lemma.
Lemma
The function u (x ) := ex − x − 1 satisfies the differential equation,
u 00 (x ) − u 0 (x ) = 1 and has the following properties:
(i ) u belongs to C 2 (R),
(ii ) for every x ≥ 0, u (x ) ≥ 0 and u 0 (x ) ≥ 0.
(iii ) The map x 7−→ v (x ) := u (|x |) belongs to C 2 (R), v 0 (0) = 0 and
u 0 (|x |) ≤ e |x | .
We now prove that Z belongs to M2 .
Rt
For N > 0, let τ N := inf{t > 0 : |Yt | + 0 |u 0 (|Ys |)|2 |Zs |2 ds ≥ N } ∧ T .
Set sgn(x ) = 1 if x ≥ 0 and sgn(x ) = −1 if x < 0.
Itô’s formula gives,
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
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19
Proof, simple BSDE, continued 3
Z t ∧τ N
u (|Yt ∧τN |) = u (|Y0 |) +
sgn(Ys )u 0 (|Ys |)Zs dWs
0
Z t ∧τ N 1 00
+
u (|Ys |)|Zs |2 − sgn(Ys )u 0 (|Ys |)g (s, Ys , Zs ) ds
2
0
Passing to expectation and using the previous simple Lemma, we get
for any N > 0
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
CIMPA, Tlemcen, 12-24 avril 2014
9/
19
Proof, simple BSDE, continued 4
Eu (|Y0 |) = Eu (|Yt ∧τN |) + E
1
− E
2
Z t ∧τ N
0
sgn(Ys )u 0 (|Ys |)g (s, Ys , Zs )ds
Z t ∧τ N
u 00 (|Ys |)|Zs |2 ds
Z t ∧τ N ≤ Eu (|Yt ∧τN |) + E
u 0 (|Ys |)(a + b |Ys |) −
0
Z t ∧τ N ≤ Eu (|Yt ∧τN |) + E
u 0 (|Ys |)(a + b |Ys |) −
0
Z t ∧τ N ≤ Eu (|Yt ∧τN |) + E
u 0 (|Ys |)(a + b |Ys |) −
0
0
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
1
2
|Zs | ds
2
1
2
|Zs | ds
2
1
2
|Zs | ds
2
CIMPA, Tlemcen, 12-24 avril 2014
10 /
19
Proof, simple BSDE, continued 5
We successively use the previous simple Lemma and the previous
exponential integrability for Y to show that for every N
1
E
2
Z t ∧τ N
0
Z T
(a + b |Ys |)u 0 (|Ys |) ds
0
Z Th
i
≤ Ee|Y0 | + E
(ae|Ys | + b |Ys |e|Ys | ds
|Zs |2 ds ≤ u (|Y0 |) + E
0
≤K
0
where K 0 is a constant not depending on N.
RT
Using Fatou’s lemma, we deduce that E 0 |Zs |2 ds < ∞. The proof is
complete.
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QBSDEs with unbounded terminal data
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Theorem
Theorem
Assume that there exist positive constants a and b such that ξ satisfies
(HExp). Let the generator H (t, ω, y , z ) be continuous in (y , z ) for a.e
(t, ω ) and satisfies
1
H (t, y , z ) ≤ a + b |y | + |z |2
2
(8)
Then, the BSDE (ξ, H ) has at least one solution (Y , Z ) which satisfies
(5).
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
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19
Proof, H dominé
Proof. We put g (t, y , z ) := a + b |y | + 12 |z |2 .
Let ξ + := max (ξ, 0) and ξ − := min(ξ, 0).
According to the previous Theorem, the BSDE with the parameters
(ξ + , g ) as well as the BSDE with the parameters (−ξ − , −g ) have
unique solutions satisfying inequalities (5).
We denote by (Y g , Z g ) [resp. (Y −g , Z −g )] the unique solution of
eq (ξ + , g ) [resp. eq (−ξ − , −g )].
Using then the Essaky-Hassani result with
−g
L = Y −g , U = Y g , η t = a + b (|Yt
| + |Ytg |) + 21 c 2 , and Ct = 1 ,
we deduce the existence of solution (Y , Z , K + , K − ) to the following
Reflected BSDE, such that (Y , Z ) belongs to C × L2 .
K HALED B AHLALI (UTLN)
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Proof, H dominé, continued 1



(i )






(ii )



(iii )




(iv )







(v )
Yt = ξ +
Z T
t
H (s, Ys , Zs )ds +
∀ t ≤ T , Yt−g ≤ Yt ≤ Ytg ,
Z T
0
K0+
(Yt − Yt−g )dKt+ =
= K0− = 0,
Z T
0
K +, K −
Z T
t
dKs+ −
Z T
t
dKs− −
Z T
t
Zs dWs ,
(Ytg − Yt )dKt− = 0, a.s.,
are continuous nondecreasing.
dK + ⊥dK −
(9)
Now, if we show that dK + = dK − = 0, then the proof is finished. We
shall prove this property.
g
Since Yt is a solution to BSDE eq (ξ, g ), then Tanaka’s formula
g
applied to (Yt − Yt )+ shows that,
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Proof, H dominé, continued 2
(Ytg − Yt )+ = (Y0g − Y0 )+ +
+
Z t
0
Z t
0
g
g
1{Ysg >Ys } [f (s, Ys , Zs ) − g (s, Ys , Zs )]ds
1{Ysg >Ys } (dKs+
− dKs− ) +
Z t
0
g
1{Ysg >Ys } (Zs − Zs )dWs
+ L0t (Y g − Y )
where L0t (Y g − Y ) denotes the local time at time t and level 0 of the
semimartingale (Y g − Y ).
g
g
Since Y g ≥ Y , then (Yt − Yt )+ = (Yt − Yt ).
g
g
Therefore, identifying the terms of (Yt − Yt )+ with those of (Yt − Yt )
and using the fact that 1 − 1{Ysg >Ys } = 1{Ysg ≤Ys } = 1{Ysg =Ys } , we show
that (Z − Z g )1{Ysg =Ys } a.e.
Using the previous equalities, we deduce that,
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Proof, H dominé, continued 3
Z t
0
1{Ysg =Ys } (dKs+ − dKs− ) =
Z t
0
g
g
1{Ysg =Ys } [g (s, Ys , Zs ) − f (s, Ys , Zs )]ds
+ L0t (Y g − Y )
Since
0≤
Rt
0
1{Ysg =Ys } dKs+ = 0, it holds that
Z t
0
=−
g
g
1{Ysg =Ys } [g (s, Ys , Zs ) − f (s, Ys , Zs )]ds + L0t (Y g − Y )
Z t
0
1{Ysg =Ys } dKs− ≤ 0
This shows that ,
Rt
0
≥0
1{Ysg =Ys } dKs− = 0, and hence dK − = 0.
Arguing symmetrically, one can show that dK + = 0.
Therefore, (Y , Z ) satisfies the initial non reflected BSDE.
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Proof, H dominé, continued 4
Since both Y g and Y −g satisfies inequality (5), so it is for Y .
Arguing as in the previous Theorem, one can show that
RT
E 0 |Zs |2 ds < ∞. This complete the proof.
K HALED B AHLALI (UTLN)
QBSDEs with unbounded terminal data
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Proof, simple BSDE, continued
K HALED B AHLALI (UTLN)
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N.V. Krylov 1969 (Theorya), 1987 (livre, Springer), 1987 (Math.
Sbornik)
E.Pardoux, S. Peng, 1990(SCL)
S. Peng, 1992 (Stochastics)
K. Bahlali,1999 (Stochastics)
K. Bahlali, B. Mezerdi, 1996 (ROSE)
K. Bahlali, A. Elouaflin, E. Pardoux, 2009 (EJP)
M. Kobylanski, 1997(CRAS) and 2000(AOP).
Dermoune, Hamadene, Ouknine, 1997 (Stochastics)
J-P. Lepeltier, J. San Martin, 1998(Stochastics).
E. Pardoux BSDEs, 1999 (Nonlinear Analysis, Differential Equations
and Control).
N. El Karoui et al. 2008(CRAS), 2012-13(AOP).
P. Briand, Y. Hu 2006(PTRF) and 08(PTRF).
Tevzadze, 2008 (SPA).
E. Essaky, M. Hassani 2011 (BULSM), 2013(JDE).
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