1/37
Some results on multidimensional BSDE with local
conditions on the coefficient
Khaled Bahlali
Khaled Bahlali
IMATH, Université de Toulon
Et CNRS, I2M, Aix-Marseille Université
CIMPA, Tlemcen, 12-24 Avril 2014
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
1 / 37
12-2
2/37
1. BSDEs–Introduction
• The goal of this first slide consists to present the backward stochastic
differential equations with some of their relations with mathematical
finance and partial differential equations (PDEs). I will also present the
few known results on BSDEs with local conditions on the generator, with
application to PDEs.
• The second slide is devoted to BSDEs with logarithmic growth.
• In the third slide, we show that the classical Quadratic BSDE with
exponential integrability of the terminal datum can be derived from BSDE
with logarithmic nonlinearity YLnY .
• The fourth and final slide is devoted to a class of BSDEs with
L2 -integrable terminal data. A Krylov’s type estimate as well as an
Itô-Krylov’s change of variable formula are also established for BSDEs with
measurable generator.
• Une bonne partie des travaux exposés a été réalisée en collaboration
avec : M. Eddahbi, E. Essaky, M. Hassani, O. Kebiri, N. Khelfallah, Y.
Ouknine et E. Pardoux.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
2 / 37
12-2
3/37
1. BSDEs–Introduction
Our Approach 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 in the
following theorem.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
3 / 37
12-2
4/37
1. BSDEs–Introduction
(Ω, F, (Ft )t≤1 , P) be a complete probability space
Ft = σ(Bs , 0 ≤ s ≤ t) ∨ N be a filtration
Consider the following ODE
Khaled Bahlali
(
dYt = 0, t ∈ [0, T ],
YT = ξ ∈ IR.
(1)
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
4 / 37
12-2
4/37
1. BSDEs–Introduction
(Ω, F, (Ft )t≤1 , P) be a complete probability space
Ft = σ(Bs , 0 ≤ s ≤ t) ∨ N be a filtration
Consider the following terminal value problem
Khaled Bahlali
(
dYt = 0, t ∈ [0, T ],
YT = ξ ∈ L2 (Ω, FT ; IR).
(1)
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
4 / 37
12-2
4/37
1. BSDEs–Introduction
(Ω, F, (Ft )t≤1 , P) be a complete probability space
Ft = σ(Bs , 0 ≤ s ≤ t) ∨ N be a filtration
Consider the following terminal value problem
(
dYt = 0, t ∈ [0, T ],
YT = ξ ∈ L2 (Ω, FT ; IR).
(1)
We want to FIND Ft -ADAPTED solution Y for equation (1).
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
4 / 37
12-2
4/37
1. BSDEs–Introduction
(Ω, F, (Ft )t≤1 , P) be a complete probability space
Ft = σ(Bs , 0 ≤ s ≤ t) ∨ N be a filtration
Consider the following
(
terminal value problem
dYt = 0, t ∈ [0, T ],
YT = ξ ∈ L2 (Ω, FT ; IR).
(1)
We want to FIND Ft -ADAPTED solution Y for equation (1).
This is IMPOSSIBLE, since the only solution is
Yt = ξ, for all t ∈ [0, T ],
(2)
which is not Ft −adapted.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
4 / 37
12-2
4/37
1. BSDEs–Introduction
(Ω, F, (Ft )t≤1 , P) be a complete probability space
Ft = σ(Bs , 0 ≤ s ≤ t) ∨ N be a filtration
Consider the following terminal value problem
(
dYt = 0, t ∈ [0, T ],
YT = ξ ∈ L2 (Ω, FT ; IR).
(1)
We want to FINDFt -ADAPTED solution Y for equation (1).
This is IMPOSSIBLE, since the only solution is
Yt = ξ, for all t ∈ [0, T ],
(2)
which is not Ft −adapted.
A natural way of making (2) Ft −adapted is to redefine Y. as follows
Yt = IE(ξ|Ft ), t ∈ [0, T ].
(3)
Then Y. is Ft −adapted and satisfies YT = ξ, but not equation (1).
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
4 / 37
12-2
5/37
1. BSDEs–Introduction
MRT =⇒ there exists an Ft −adapted process Z square integrable s.t
t
Z
Zs dBs .
Yt = Y0 +
(4)
0
It follows that
Z
YT = ξ = Y0 +
T
Zs dBs .
(5)
0
Combining (4) and (5), one has
Z
Yt = ξ −
T
Zs dBs ,
(6)
t
whose differential form is
n
dYt = Zt dBt , t ∈ [0, T ],
YT = ξ.
(7)
Comparing (1) and (7), the term ”Zt dBt ” has been added.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
5 / 37
12-2
6/37
1. BSDEs–Introduction
BSDE is an equation of the following type:
Khaled Bahlali
Z
T
Z
f (s, Ys , Zs )ds −
Yt = ξ +
t
T
Zs dBs ,
0 ≤ t ≤ T.
(8)
t
T : TERMINAL TIME
f : Ω × [0, T ] × IRd × IRd×n → IRd : GENERATOR or COEFFICIENT
ξ : TERMINAL CONDITION FT −adapted process with value in IRd .
UNKNOWNS ARE : Y ∈ IRd and Z ∈ IRd×n .
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
6 / 37
12-2
7/37
1. BSDEs–Introduction
Denote by L the set of IRd × IRd×n –valued processes (Y , Z ) defined on IR+ × Ω which are
Ft –adapted and such that:
k(Y , Z )k2 = IE
sup |Yt |2 +
0≤t≤T
Z
T
|Zs |2 ds
< +∞.
0
The couple (L, k.k) is then a Banach space.
Definition
A solution of equation (8) is a pair of processes (Y , Z ) which belongs to the space (L, k.k) and
satisfies equation (8).
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
7 / 37
12-2
8/37
2. BSDEs with Lipshitz coefficient
Consider the following assumptions:
For all (y, z) ∈ IRd × IRd×n : (ω, t) −→ f (ω, t, y, z) is Ft − progressively measurable
f (., 0, 0) ∈ L2 ([0, T ] × Ω, IRd )
f is Lipschitz : ∃K > 0 and ∀y, y 0 ∈ IRd , z, z 0 ∈ IRd×n and (ω, t) ∈ Ω × [0, T ] s.t
| f (ω, t, y, z) − f (ω, t, y 0 , z 0 ) |≤ K | y − y 0 | + | z − z 0 | .
ξ ∈ L2 (Ω, FT ; IRd )
Theorem [5]
Suppose that the above assumptions hold true. Then, there exists a unique
solution for BSDE (14).
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
8 / 37
12-2
8/37
2. BSDEs with Lipshitz coefficient
Consider the following assumptions:
For all (y, z) ∈ IRd × IRd×n : (ω, t) −→ f (ω, t, y, z) is Ft − progressively measurable
f (., 0, 0) ∈ L2 ([0, T ] × Ω, IRd )
f is Lipschitz : ∃K > 0 and ∀y, y 0 ∈ IRd , z, z 0 ∈ IRd×n and (ω, t) ∈ Ω × [0, T ] s.t
| f (ω, t, y, z) − f (ω, t, y 0 , z 0 ) |≤ K | y − y 0 | + | z − z 0 | .
ξ ∈ L2 (Ω, FT ; IRd )
Theorem [5]
Suppose that the above assumptions hold true. Then, there exists a unique
solution for BSDE (14).
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
8 / 37
12-2
9/37
3. APPLICATIONS OF BSDE : FINANCE & PDE
Consider a market where only two basic assets are traded.
BOND :
STOCK :
Consider a European call option whose payoff is
(XT − K )+ .
The option pricing problem is : fair price of this option at time t = 0?
Suppose that this option has a price y at time t = 0. Then the fair price for the option at time
t = 0 should be such a y that the corresponding optimal investment would result in a wealth
process Yt satisfying
YT = (XT − K )+ .
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
9 / 37
12-2
9/37
3. APPLICATIONS OF BSDE : FINANCE & PDE
Consider a market where only two basic assets are traded.
BOND : dXt0 = rXt0 dt
STOCK :
Consider a European call option whose payoff is
(XT − K )+ .
The option pricing problem is : fair price of this option at time t = 0?
Suppose that this option has a price y at time t = 0. Then the fair price for the option at time
t = 0 should be such a y that the corresponding optimal investment would result in a wealth
process Yt satisfying
YT = (XT − K )+ .
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
9 / 37
12-2
9/37
3. APPLICATIONS OF BSDE : FINANCE & PDE
Consider a market where only two basic assets are traded.
BOND : dXt0 = rXt0 dt
STOCK : dXt = bXt dt + σXt dBt
Consider a European call option whose payoff is
(XT − K )+ .
The option pricing problem is : fair price of this option at time t = 0?
Suppose that this option has a price y at time t = 0. Then the fair price for the option at time
t = 0 should be such a y that the corresponding optimal investment would result in a wealth
process Yt satisfying
YT = (XT − K )+ .
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
9 / 37
12-2
10/37
3. APPLICATIONS OF BSDE : FINANCE & PDE
Denote by
Rt : the amount that the writer invests in the stock
Yt − Rt : the remaining amount which is invested in the bond
Rt determines a strategy of the investment which is called a portfolio.
By setting Zt = σRt , we obtain the following BSDE

dXt = bXt dt + σXt dBt


b−r



 dYt = (rYt + σ Zt ) dt + Zt dBt , t ∈ [0, T ],
|
{z
}
f (t,Yt ,Zt )

 X = x, Y = (X − K )+ .

0
T


| T {z }

(9)
ξ
Pardoux & Peng result =⇒ there exits a unique solution (Yt , Zt ). The option price at time
t = 0 is given by Y0 , and the portfolio is given by Rt =
Khaled Bahlali
Zt
σ
.
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
10 / 37
12-2
11/37
3. APPLICATIONS OF BSDE : PDE
Let u be the solution of the following system of semi-linear parabolic PDE’s:
∂u
(t, x) + 12 Tr(σσ ∗ ∆u)(t, x) + b∇u(t, x) + f (t, x, u(t, x), ∇uσ(t, x)) = 0
∂t
u(T , x) = g(x).
Introducing {(Y s,x , Z s,x ) ; s ≤ t ≤ T } the adapted solution of the backward stochastic
differential equation
−dYt = f (t, Xts,x , Yt , Zt )ds − Zt∗ dBt
t,x
YT = g(XT
),
(10)
(11)
where (X s,x ) denotes the solution of the following stochastic differential equation
Khaled Bahlali
n
dXt = b(t, Xt )dt + σ(t, Xt )dBt
Xs = x.
(12)
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
11 / 37
12-2
12/37
3. APPLICATIONS OF BSDE : FINANCE & PDE
Then we have :
u is a classical solution of PDE (10) =⇒
(Yts,x = u(t, Xts,x ), Zts,x = ∇u(t, Xts,x )σ(s, Xts,x ))
is a solution the BSDE (11).
There exists a solution to the BSDE (11)=⇒u(t, x) = Ytt,x , is a viscosity solution of PDE
(10). This formula is a generalization of Feynman-Kac formula.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
12 / 37
12-2
13/37
4. BSDEs with locally Lipschitz coefficient
Consider the following assumptions:
(A1) f is continuous in (y, z) for almost all (t, ω).
(A2) There exist K > 0 and 0 ≤ α ≤ 1 such that
| f (t, ω, y, z) |≤ K (1+ | y |α + | z |α ).
(A3) For each N > 0, there exists LN such that:
| f (t, y, z) − f (t, y 0 , z 0 ) | ≤ LN (| y − y 0 | + | z − z 0 |)
| y |, | y 0 |, | z |, | z 0 |≤ N .
(A4) ξ ∈ L2 (Ω, FT ; IRd )
Theorem [1, 2]
Assume moreover
that there exists a positive constant L such that
√
LN = L + log N then there exists a unique solution for the BSDE (1).
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
13 / 37
12-2
13/37
4. BSDEs with locally Lipschitz coefficient
Consider the following assumptions:
(A1) f is continuous in (y, z) for almost all (t, ω).
(A2) There exist K > 0 and 0 ≤ α ≤ 1 such that
| f (t, ω, y, z) |≤ K (1+ | y |α + | z |α ).
(A3) For each N > 0, there exists LN such that:
| f (t, y, z) − f (t, y 0 , z 0 ) | ≤ LN (| y − y 0 | + | z − z 0 |)
| y |, | y 0 |, | z |, | z 0 |≤ N .
(A4) ξ ∈ L2 (Ω, FT ; IRd )
Theorem [1, 2]
Assume moreover
that there exists a positive constant L such that
√
LN = L + log N then there exists a unique solution for the BSDE (1).
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
13 / 37
12-2
14/37
5. BSDEs with locally monotone coefficient–Question
Question
Let f (y) := −y log | y |. Suppose that ξ ∈ L2 (FT ) or ξ ∈ Lp (FT ), p > 1 and consider the
following BSDE with logarithmic nonlinearity
Z
Yt = ξ −
T
Z
Ys log | Ys |ds −
t
T
Zs dWs .
t
Does this equation has a unique solution?
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
14 / 37
12-2
15/37
5. BSDEs with locally monotone coefficient-Assumptions
Consider the following assumptions:
(H1) f is continuous in (y, z) for almost all (t, ω),
(H2) There exist M > 0, γ < 12 and η ∈ L1 ([0, T ] × Ω) such that,
hy, f (t, ω, y, z)i ≤ η + M |y|2 + γ|z|2
(H3)
P − a.s., a.e. t ∈ [0, T ].
0
”Almost” quadratic growth : ∃M1 > 0, 0 ≤ α < 2, α0 > 1 and η ∈ Lα ([0, T ] × Ω) s.t :
Khaled Bahlali
| f (t, ω, y, z) |≤ η + M1 (| y |α + | z |α ).
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
15 / 37
12-2
16/37
5. BSDEs with locally monotone coefficient
(H4) There exists a real valued sequence (AN )N >1 and constants M > 1, r > 1 such that:
i) ∀N > 1, 1 < AN ≤ N r .
ii) lim AN = ∞.
N →∞
iii) Locally monotone condition : For every
N ∈ IN , ∀y, y 0 , z, z 0 such that | y |, | y 0 |, | z |, | z 0 |≤ N , we have
hy − y 0 , f (t, y, z) − f (t, y 0 , z 0 )i
≤ M | y − y 0 |2 logAN + M | y − y 0 || z − z 0 |
p
log AN + MA−1
.
N
Theorem [3]
Let ξ be a square integrable random variable. Assume that (H1)–(H4) are
satisfied. Then the BSDE has a unique solution.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
16 / 37
12-2
17/37
1. Lp -solutions to BSDEs with super-Motivation
Let’s mention some considerations which have motivated the present work.
The growth conditions on the nonlinearity constitute a critical case. Indeed, it is
R tknown
that for any ε > 0, the solutions of the ordinary differential equation Xt = x +
Xs1+ε ds
0
explode at a finite time.
The logarithmic nonlinearities appear in some PDEs arising in physics.
In terms of continuous-state branching processes, the logarithmic nonlinearity u log u
corresponds to the Neveu branching mechanism. This process was introduced by Neveu.
For instance, the super-process with Neveu’s branching mechanism is related to the
Cauchy problem,
( ∂u
− ∆u + u log u = 0 on (0, ∞) × IRd
∂t
(13)
u(0+ ) = ϕ > 0
Hence, our result can be seen as an alternative approach to the PDEs.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
17 / 37
12-2
18/37
5. BSDEs with locally monotone coefficient–Example
Example
Let f (y) := −y log | y | then for all ξ ∈ L2 (FT ) the following BSDE has a unique solution
Z
T
Z
Ys log | Ys |ds −
Yt = ξ −
t
T
Zs dWs .
t
Indeed, f satisfies (H.1)-(H.3) since hy, f (y)i ≤ 1 and | f (y) |≤ 1 +
(H.4) is satisfied for every N > e and AN = N .
Khaled Bahlali
1
| y |1+ε for all ε > 0.
ε
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
18 / 37
12-2
19/37
5. BSDEs with locally monotone coefficient–Idea of the proof
We define a family of semi-norms ρn (f )
Z
ρn (f ) = IE
n∈IN
by,
T
sup
0
|f (s, y, z)|ds.
|y|,|z|≤n
We Approximate f by a sequence (fn )n>1 of Lipschitz functions :
Lemma
Let f be a process which satisfies (H.1)–(H.3). T hen there exists a sequence of
processes (fn ) such that,
(a) For each n, fn is bounded and globally Lipschitz in (y, z) a.e. t and
P-a.s.ω.
There exists M 0 > 0, such that:
(b) supn |fn (t, ω, y, z)| ≤ η + M 0 + M1 (| y |α + | z |α ). P-a.s., a.e.
t ∈ [0, T ].
(c)
sup < y, fn (t, ω, y, z) >≤ η + M 0 + M |y|2 + γ|z|2
n
(d) For every N , ρN (fn − f ) −→ 0 as n −→ ∞.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
19 / 37
12-2
20/37
5. BSDEs with locally monotone coefficient–Key steps of the proof
We consider the following BSDE
Ytn = ξ +
Z
T
fn (s, Ysn , Zsn )ds −
t
Z
T
Zsn dWs ,
0 ≤ t ≤ T.
t
Lemma
There exits a universal constant ` such that
T
Z
e 2Ms |Zsfn |2 ds ≤
a) IE
0
1
1 − 2γ
e 2MT IE | ξ |2 +2IE
Z
T
e 2Ms (η + M 0 )ds = K1
0
b) IE sup (e 2Mt | Ytfn |2 ) ≤ `K1 = K2
0≤t≤T
T
2Ms
Z
c) IE
e
0
Z
|fn (s, Ysfn , Zsfn )|α ds ≤
T
4α−1 IE
e 2Ms ((η + M 0 )α + 4)ds + M1α K1 + TM1α K2
= K3
0
Z
d) IE
T
2
).
α
(IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université
e 2Ms |f (s, Ysfn , Zsfn )|α ds ≤ K3 , where α = min(α0 ,
Khaled Bahlali0 UTLN
CIMPA, Tlemcen,
20 / 37
12-2
21/37
**************************************************
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
21 / 37
12-2
21/37
5. BSDEs with locally monotone coefficient–Key steps of the proof
Hence the following convergences hold true
Y n * Y , weakly star in L2 (Ω, L∞ [0, T ])
Z n * Z , weakly in L2 (Ω × [0, T ])
fn (., Y n , Z n ) * Γ. weakly in Lα (Ω × [0, T ]),
Moreover
Z
T
T
Z
Γs ds −
Yt = ξ +
Zs dWs , ∀t ∈ [0, T ].
t
t
We Apply Itô’s formula to (|Y n − Y m |2 + ε)p for some 0 < p < 1, instead of
|Y n − Y m |2 :
lim
n→+∞
IE sup
|Ytn
Z
β
− Yt | + IE
0≤t≤T
|Zsn
− Zs |ds
= 0, 1 < β < 2.
0
Z
We Identify Γs by proving that : lim E
Khaled Bahlali
T
n
T
|fn (s, Ysn , Zsn ) − f (s, Ys , Zs )|ds = 0.
0
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
21 / 37
12-2
22/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Definition
Let p > 1 is an arbitrary fixed real number and all the considered processes are (Ft )-predictable.
Definition
A solution of equation (8 is an (Ft )-adapted and IRd+dr -valued process (Y , Z ) such that
Z
IE sup |Yt |p + IE
|Zs |2 ds
t≤T
Z
Z
T
0
Z
f (s, Ys , Zs )ds −
Yt = ξ +
t
T
|f (s, Ys , Zs )|ds < +∞
+ IE
0
and satisfies
Khaled Bahlali
2p
T
T
Zs dBs ,
0 ≤ t ≤ T.
(14)
t
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
22 / 37
12-2
23/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Assumptions
We consider the following assumptions on (ξ, f ):
(H.0)

1 ∧ (p − 1)
0
1
0
2

[

 There are M ∈ L (Ω; L Z([0, T ]; IR+ )), K ∈ L (Ω; L ([0, T ]; IR+ )) and γ ∈]0,
2
T
p
λs ds

2

K2
 such that: IE | ξ |p e 0
< ∞, where λs := 2Ms + s
2γ
(H.1) f is continuous in (y, z) for almost all (t, ω).
(H.2)

Z s
 2p


Z

T
λr dr






e 0
ηs ds < ∞
There are η and f 0 ∈ L0 (Ω × [0, T ]; IR+ ) satisfying IE 



0



Z s


p

Z T 1
λr dr
2



0
and IE 
e
fs0 ds < ∞, where λ is defined in assumption (H.0),



0






such that:




0
2
hy, f (t, y, z)i ≤ ηt + ft |y| + Mt |y| + Kt |y||z|.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
23 / 37
12-2
24/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Assumptions
(
(H.3)
There are η ∈ Lq (Ω × [0, T ]; IR+ )) (for some q > 1) and α ∈]1, p[, α0 ∈]1, p ∧ 2[
such that:
0
| f (t, ω, y, z) | ≤ η t + | y |α + | z |α .

0
There are v ∈ Lq (Ω × [0, T ]; IR+ )) (for some q 0 > 0) and K 0 ∈ IR+ such that



for
every
N
∈
I
N
and every y, y 0 z, z 0 satisfying | y |, | y 0 |, | z |, | z 0 |≤ N





1vt (ω)≤N hy − y 0 , f (t, ω, y, z) − f (t, ω, y 0 , z 0 )i
(H.4)
p
log AN


≤ K 0 log AN | y − y 0 |2 + K 0 log AN | y − y 0 || z − z 0 | +K 0


AN



 where AN is µa increasing sequence and satisfies AN > 1, limN →∞ AN = ∞
Khaled Bahlali
and AN ≤ N
for some µ > 0.
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
24 / 37
12-2
25/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Existence and Uniqueness
Theorem [4]
If (H.0)-(H.4) hold then (8) has a unique solution (Y , Z ). Moreover we have
p
IE sup | Yt | e
p
2
Rt
0
λs ds
Rs
e
+ IE
t
0
λr dr
2p
2
| Zs | ds
0
(
≤C
T
Z
p
IE | ξ | e
p
2
RT
0
λs ds
Z
+ IE
T
Rs
e
0
λr dr
0
2p
ηs ds
Z
+ IE
T
e
1
2
Rs
λr dr 0
0
fs ds
p )
.
0
for some constant C depending only on p and γ.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
25 / 37
12-2
26/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Examples
Example 1
Let f (y) := −y log | y | then for all ξ ∈ Lp (FT ) the following BSDE has a unique solution
Z
T
Z
Ys log | Ys |ds −
Yt = ξ −
t
T
Zs dWs .
t
Indeed, f satisfies (H.1)-(H.3) since hy, f (y)i ≤ 1 and | f (y) |≤ 1 +
(H.4) is satisfied for every N > e with vs = 0 and AN = N .
Khaled Bahlali
1
| y |1+ε for all ε > 0.
ε
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
26 / 37
12-2
27/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Examples
Example 2
Let g(y) := y log
T
|y|
and h ∈ C(IRdr ; IR+ ) C 1 (IRdr − {0}; IR+ ) be such that :
1+ | y |
h(z) =
|z|
|z|
p
p− log |z|
log |z|
if |z| < 1 − ε0
if |z| > 1 + ε0 ,
where ε0 ∈]0, 1[. Finally, we put f (y, z) := g(y)h(z). Then for every ξ ∈ Lp (FT ) the following
BSDE has a unique solution
Z
t
(H.4) is satisfied for every N >
Khaled Bahlali
√
T
Z
f (Ys , Zs )ds −
Yt = ξ +
T
Zs dWs .
t
e with vs = 0 and AN = N
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
27 / 37
12-2
28/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Examples
Example 3
Let (Xt )t≤T be an (Ft )−adapted and IRk −valued process satisfying :
Z
t
Xt = X0 +
Z
0
t
σ(s, Xs )dWs ,
b(s, Xs )ds +
0
where X0 ∈ IRk and σ, b : [0, T ] × IRk → IRkr × IRk are measurable functions such that
kσ(s, x)k ≤ c and |b(s, x)| ≤ c(1 + |x|), for some constant c.
Z
Consider the following BSDE Yt = g(XT ) +
T
(| Xs |q Ys − Ys log | Ys |)ds −
t
0
Z
T
Zs dWs .
t
where q ∈]0, 2[ and g is a measurable function satisfying | g(x) |≤ c exp c | x |q , for some
constants c > 0, q 0 ∈ [0, 2[.
The previous BSDE has a unique solution (Y , Z ) such that
Z
IE sup | Yt |p +IE
t
2p
T
| Zs |2 ds
(H.4) is satisfied with vs = exp | Xs |q and AN = N .
Khaled Bahlali
≤ C exp (C | X0 |2 ).
0
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
28 / 37
12-2
29/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Examples
Example 4
Let (ξ, f ) satisfying (H.0)-(H.3) and
(H’.4)

Z


There are a positive process C satisfying IE


T
eq
0
Cs
0
ds < ∞ and K 0 ∈ IR+ such that:
hy − y 0 , f (t, ω, y, z) − f (t, ω, y 0 , z 0 )i ≤ K 0 | y − y 0 |2 Ct (ω)+ | log(| y − y 0 |) |



p

0
0
0
+K | y − y || z − z |
Then the following BSDE has a unique solution
Z
T
Z
f (s, Ys , Zs )ds −
Yt = ξ +
t
Ct (ω)+ | log | z − z 0 | |.
T
Zs dWs .
t
(H.4) is satisfied with vs = exp (Cs ) and AN = N .
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
29 / 37
12-2
30/37
6. Lp -solutions to BSDEs with locally monotone coefficient-Idea of the proof
We Approximate f by a sequence (fn )n>1 of Lipschitz functions :
fn (t, y, z) =
(c1 e)2 1{Λ
Zt
≤ n}
ψ(n −2 |y|2 )ψ(n −2 |z|2 )×
Z
f (t, y − u, z − v)Πdi=1 ψ(mui )Πdi=1 Πrj=1 ψ(mvij )dudv,
m (d+dr)
IRd
IRdr
n 2p
and Λt := ηt + η t + ft0 + Mt + Kt +
ht
such that 0 < ht ≤ 1.
We consider the following BSDE
with m :=
Ytn
Z
= ξ1{|ξ|≤n} +
T
fn (s, Ysn , Zsn )ds
Z
−
t
where ht is a predictable process
T
Zsn dWs ,
0 ≤ t ≤ T.
t
We Apply Itô’s formula to ({|Y n − Y m |2 +
|Y n
1
ht
Y m |2
1
AN
β
}) 2 for some 1 < β < p ∧ 2, instead of
−
we have:
For every p0 < p,
0
0
(Y n , Z n ) → (Y , Z ) strongly in Lp (Ω; C([0, T ]; IRd )) × Lp (Ω; L2 ([0, T ]; IRdr )).
2
p
p
For every β̂ < 0 ∧ ∧ 0 ∧ q
α
α α
Khaled Bahlali
Z
T
|fn (s, Ysn , Zsn ) − f (s, Ys , Zs )|β̂ ds = 0.
n→∞
0
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
30 / 37
12-2
lim IE
31/37
1. Lp -solutions to BSDEs with super-linear growth coefficient-Application to PDEs
Consider
the following system of semilinear PDE (P (g,F) )
(
∂u(t, x)
+ Lu(t, x) + F(t, x, u(t, x), σ ∗ ∇u(t, x)) = 0 t ∈]0, T [, x ∈ IRk
∂t
u(T , x) = g(x) x ∈ IRk
where L :=
1
2
X
X
i,j
i
(σσ ∗ )ij ∂ij2 +
bi ∂i ,
σ ∈ Cb3 (IRk , IRkr ),
b ∈ Cb2 (IRk , IRk ).
Let
H1+ :=
[
v ∈ C([0, T ]; Lβ (IRk , e −δ|x| dx; IRd )) :
Z
0
δ≥0,β>1
T
Z
|σ ∗ ∇v(s, x)|β e −δ|x| dxds < ∞
IRk
Definition
A (weak) solution of (P (g,F) ) is a function u ∈ H1+ such that for every ϕ ∈ Cc1
RT
t
< u(s),
∂ϕ(s)
∂s
=< g, ϕ(T ) > +
> ds+ < u(t), ϕ(t) >
RT
RT
< F(s, ., u(s), σ ∗ ∇u(s)), ϕ(s) > ds +
< Lu(s), ϕ(s) > ds,
tR
t
where < f (s), h(s) >=
Khaled Bahlali
IRk
f (s, x)h(s, x)dx.
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
31 / 37
12-2
32/37
1. Lp -solutions to BSDEs with super-linear growth coefficient-Application to PDE
(A.0) g(x) ∈ Lp (IRk , e −δ|x| dx; IRd ).
(A.1) F(t, x, ., .) is continuous
(A.2)
f


(A.3)
a.e. (t, x)

p
are η 0 ∈ L 2 ∨1 ([0, T ] × IRk , e −δ|x| dtdx; IR+ )),

 There
0
p
k −δ|x|
0
0
∈ L ([0, T ] × IR , e
dtdx; IR+ )), and M , M ∈ IR+ such that
0
hy, F(t, x, y, z)i ≤ η 0 (t, x) + f 0 (t, x)|y| + (M + M 0 |x|)|y|2 +
p
M + M 0 |x||y||z|.

0
q
k −δ|x| dtdx; IR )) (for some q > 1), α ∈]1, p[

+
 There 0are η ∈ L ([0, T ] × IR , e
and α ∈]1, p ∧ 2[ such that


0
|F(t, x, y, z)| ≤ η 0 (t, x) + |y|α + |z|α .
(A.4)

∈ IN and every e r|x| , | y |, | y 0 |, | z |, | z 0 |≤ N ,
 There 0are K , r ∈ IR+ such that0 ∀N
0 )i
hy − y ; F(t,
x,
y,
z)
−
F(t,
x,
y
,
z
 ≤ K log N 1 + |y − y 0 |2 + √K log N |y − y 0 ||z − z 0 |.
N
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
32 / 37
12-2
33/37
1. Existence and uniqueness of solutions to PDE
Consider the diffusion process with infinitesimal operator L
Xst,x = x +
Z
s
b(Xrt,x )dr +
Z
t
s
σ(Xrt,x )dWr ,
t≤s≤T
t
Theorem [4]
Under assumption (A.0)-(A.4) we have
1) The PDE (P (g,F) ) has a unique solution u on [0, T ]
2) Z
For all t ∈ [0, T ] there exists Dt ⊂ IRk such that
i)
1 dx = 0
Dtc
t,x
t,x
ii) For all t ∈ [0, T ] and all x ∈ Dt (E (ξ ,f ) ) has a unique solution (Y t,x , Z t,x ) on [t, T ]
t,x
where ξ t,x := g(XT
) and f t,x (s, y, z) := 1{s>t} F(s, Xst,x , y, z)
3) For all t ∈ [0, T ]
u(s, Xst,x ), σ ∗ ∇u(s, Xst,x ) = Yst,x , Zst,x
Khaled Bahlali
a.e.(s, x, ω)
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
33 / 37
12-2
34/37
1. Existence and uniqueness of solutions to PDE-Idea of the proof
We Approximate F by a sequence (Fn )n>1 of Lipschitz functions :
Fn (t, x, y, z) = (n 2p e |x| )(d+dr) (c1 e)2 1{η0 (t,x)+η0 (t,x)+f 00 (t,x)+|x|≤n} ψ(n −2 |y|2 )ψ(n −2 |
R
R
IRd
IRdr
F(t, x, y − u, z − v)Πdi=1 ψ(n 2p e |x| ui )Πdi=1 Πrj=1 ψ(n 2p e |x| vij )dudv,
t,x
We consider (Y t,x,n , Z t,x,n ) be the unique solution of BSDE (1) avec ξnt,x := gn (XT
)
t,x
t,x
and fn (s, y, z) := 1{s>t} Fn (s, Xs , y, z), with gn (x) := g(x)1{|g(x)|≤n} .
There exists a unique solution u n of PDE (P (gn ,Fn ) )
(
∂u n (t, x)
+ Lu n (t, x) + Fn (t, x, u n (t, x), σ ∗ ∇u n (t, x)) = 0t ∈]0, T [, x ∈ IRk
∂t
n
u (T , x) = gn (x) x ∈ IRk
such that for all t
u n (s, Xst,x ) = Yst,x,n
and
σ ∗ ∇u n (s, Xst,x ) = Zst,x,n
a.e (s, ω, x).
We have the following convergence :
Khaled Bahlali
Z
0
| u n (t, x) − u m (t, x) |p e −δ
lim sup
n,m 0≤t≤T
T
Z
Z
lim
n,m
0
0
|x|
dx = 0
IRk
| σ ∗ ∇u n (t, x) − σ ∗ ∇u m (t, x) |p
0
∧2
e −δ
0
|x|
dtdx = 0.
IRk
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
34 / 37
12-2
35/37
1. Existence and uniqueness of solutions to PDE-Idea of the proof
For uniqueness we prove that the system of semilinear PDEs
(
∂u(t, x)
+ Lu(t, x) + f (t, x, u(t, x), ∇u(t, x)) = 0,
t ∈]0, T [, x ∈ IRk
∂t
k
u(T , x) = g(x),
x ∈ IR
has a unique solution if and only if 0 is the unique solution of the linear system
Khaled Bahlali
(
∂u(t, x)
+ Lu(t, x) = 0,
∂t
u(T , x) = 0,
x ∈ IRk
t ∈]0, T [, x ∈ IRk
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
35 / 37
12-2
36/37
K. Bahlali, Backward stochastic differential equations with locally Lipschitz
coefficient. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 5, 481-486.
K. Bahlali Existence and uniqueness of solutions for BSDEs with locally
Lipschitz coefficient. Electron. Comm. Probab., 7, 169–179, 2002.
K. Bahlali, E.H. Essaky, M. Hassani, E. Pardoux
Existence, uniqueness and stability of backward stochastic differential
equations with locally monotone coefficient. C. R. Acad. Sci., Paris 335,
no. 9, 757–762, 2002.
K. Bahlali, E.H. Essaky, M. Hassani, E. Pardoux
p-integrable solutions to multidimensional BSDEs
and degenerate systems of PDEs with logarithmic nonlinearities. Preprint.
Pardoux, É.; Peng, S. G. Adapted solution of a backward stochastic
differential equation. Systems Control Lett. 14 (1990), no. 1, 55-61.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
36 / 37
12-2
37/37
Pardoux, E. Backward stochastic differential equations and viscosity
solutions of systems of semilinear parabolic and elliptic PDEs of second
order, in Stochastic analysis and related topics VI, Geilo 1996, Progr.
Probab., vol. 42, Birkhäuser, Boston, MA, pp. 79–127, 1998
Pardoux, E. BSDEs, weak convergence and homogenization of semilinear
PDEs in F. H Clarke and R. J. Stern (eds.), Nonlinear Analysis, Differential
Equations and Control, 503-549K luwer Academic Publishers., 1999.
Khaled Bahlali
UTLN (IMATH, Université
Some
de results
Toulon
on multidimensional
Et CNRS, I2M,BSDE
Aix-Marseille
with local conditions
Université CIMPA, Tlemcen,
37 / 37
12-2