Large deviations for multivalued backward stochastic
differential equations
Ibrahim DAKAOU
Université de Maradi NIGER
ICPAM Research School
Analysis and Probability
Abidjan March 17-28, 2014
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
1 / 47
Outline
1
Introduction
2
Multivalued BSDEs
3
Large deviations
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
2 / 47
Outline
1
Introduction
2
Multivalued BSDEs
3
Large deviations
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
2 / 47
Outline
1
Introduction
2
Multivalued BSDEs
3
Large deviations
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
2 / 47
Introduction
Outline
1
Introduction
2
Multivalued BSDEs
3
Large deviations
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
3 / 47
Introduction
Preliminaries
Let X1 , X2 , . . . , Xn be i.i.d. random variables.
Law of Large Numbers
E(X1 ) = µ ∈ R, Var (X1 ) = σ 2 ∈]0, +∞[.
Xn =
n
1X
Xi ; X n
−→ µ
n
n i=1
µ = 0, Γ = {x : |x| ≥ α}; P(X n ∈ Γ) ≤
Var (X n )
α2
Large Deviation Principle (LDP)
Exponentially
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
4 / 47
Introduction
Preliminaries
Let X1 , X2 , . . . , Xn be i.i.d. random variables.
Law of Large Numbers
E(X1 ) = µ ∈ R, Var (X1 ) = σ 2 ∈]0, +∞[.
Xn =
n
1X
Xi ; X n
−→ µ
n
n i=1
µ = 0, Γ = {x : |x| ≥ α}; P(X n ∈ Γ) ≤
Var (X n )
α2
Large Deviation Principle (LDP)
Exponentially
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
4 / 47
Introduction
Preliminaries
Let X1 , X2 , . . . , Xn be i.i.d. random variables.
Law of Large Numbers
E(X1 ) = µ ∈ R, Var (X1 ) = σ 2 ∈]0, +∞[.
Xn =
n
1X
Xi ; X n
−→ µ
n
n i=1
µ = 0, Γ = {x : |x| ≥ α}; P(X n ∈ Γ) ≤
Var (X n )
α2
Large Deviation Principle (LDP)
Exponentially
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
4 / 47
Introduction
Preliminaries
Let X1 , X2 , . . . , Xn be i.i.d. random variables.
Law of Large Numbers
E(X1 ) = µ ∈ R, Var (X1 ) = σ 2 ∈]0, +∞[.
Xn =
n
1X
Xi ; X n
−→ µ
n
n i=1
µ = 0, Γ = {x : |x| ≥ α}; P(X n ∈ Γ) ≤
Var (X n )
α2
Large Deviation Principle (LDP)
Exponentially
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
4 / 47
Introduction
Preliminaries
Let X1 , X2 , . . . , Xn be i.i.d. random variables.
Law of Large Numbers
E(X1 ) = µ ∈ R, Var (X1 ) = σ 2 ∈]0, +∞[.
Xn =
n
1X
Xi ; X n
−→ µ
n
n i=1
µ = 0, Γ = {x : |x| ≥ α}; P(X n ∈ Γ) ≤
Var (X n )
α2
Large Deviation Principle (LDP)
Exponentially
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
4 / 47
Introduction
Example : LDP for i.i.d. sequences
Let (Xn )n be i.i.d. random variables with P(X1 = 0) = P(X1 = 1) = 12 .
∀a >
∆
(
Λ(x) =
1
1
, lim log P Sn ≥ an = −Λ(a)
n→∞
2
n
log 2 + x log x + (1 − x) log(1 − x), x ∈ [0, 1]
∞
Cramer (1938)
Let (Xn )n be i.i.d. R-valued random variables satisfying E(etX1 ) < ∞,
∀t ∈ R. Then, for all a > E(X1 ),
lim
n→∞
1
∆
log P Sn ≥ an = −Λ(a); Λ(x) = sup xt − log E(etX1 ) .
n
t∈R
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
5 / 47
Introduction
Example : LDP for i.i.d. sequences
Let (Xn )n be i.i.d. random variables with P(X1 = 0) = P(X1 = 1) = 12 .
∀a >
∆
(
Λ(x) =
1
1
, lim log P Sn ≥ an = −Λ(a)
n→∞
2
n
log 2 + x log x + (1 − x) log(1 − x), x ∈ [0, 1]
∞
Cramer (1938)
Let (Xn )n be i.i.d. R-valued random variables satisfying E(etX1 ) < ∞,
∀t ∈ R. Then, for all a > E(X1 ),
lim
n→∞
1
∆
log P Sn ≥ an = −Λ(a); Λ(x) = sup xt − log E(etX1 ) .
n
t∈R
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
5 / 47
Introduction
Example : LDP for i.i.d. sequences
Let (Xn )n be i.i.d. random variables with P(X1 = 0) = P(X1 = 1) = 12 .
∀a >
∆
(
Λ(x) =
1
1
, lim log P Sn ≥ an = −Λ(a)
n→∞
2
n
log 2 + x log x + (1 − x) log(1 − x), x ∈ [0, 1]
∞
Cramer (1938)
Let (Xn )n be i.i.d. R-valued random variables satisfying E(etX1 ) < ∞,
∀t ∈ R. Then, for all a > E(X1 ),
lim
n→∞
1
∆
log P Sn ≥ an = −Λ(a); Λ(x) = sup xt − log E(etX1 ) .
n
t∈R
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
5 / 47
Introduction
Example : LDP for i.i.d. sequences
Let (Xn )n be i.i.d. random variables with P(X1 = 0) = P(X1 = 1) = 12 .
∀a >
∆
(
Λ(x) =
1
1
, lim log P Sn ≥ an = −Λ(a)
n→∞
2
n
log 2 + x log x + (1 − x) log(1 − x), x ∈ [0, 1]
∞
Cramer (1938)
Let (Xn )n be i.i.d. R-valued random variables satisfying E(etX1 ) < ∞,
∀t ∈ R. Then, for all a > E(X1 ),
lim
n→∞
1
∆
log P Sn ≥ an = −Λ(a); Λ(x) = sup xt − log E(etX1 ) .
n
t∈R
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
5 / 47
Introduction
SDEs and BSDEs
Xts,x,ε = x +
Z t
s
β(Xrs,x,ε )dr +
ϕts,x
Yts,x,ε = g(XTs,x,ε ) +
Z t
=x+
s
Z T
t
s
β(ϕs,x
r )dr , s ≤ t ≤ T
f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
ψts,x = g(ϕTs,x ) +
I. DAKAOU (UM)
√ Z t
ε
σ(Xrs,x,ε )dWr , s ≤ t ≤ T
Z T
t
Z T
t
Zrs,x,ε dWr
s,x
f (r , ϕs,x
r , ψr , 0)dr , s ≤ t ≤ T
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
6 / 47
Introduction
SDEs and BSDEs
Xts,x,ε = x +
Z t
s
β(Xrs,x,ε )dr +
ϕts,x
Yts,x,ε = g(XTs,x,ε ) +
Z t
=x+
s
Z T
t
s
β(ϕs,x
r )dr , s ≤ t ≤ T
f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
ψts,x = g(ϕTs,x ) +
I. DAKAOU (UM)
√ Z t
ε
σ(Xrs,x,ε )dWr , s ≤ t ≤ T
Z T
t
Z T
t
Zrs,x,ε dWr
s,x
f (r , ϕs,x
r , ψr , 0)dr , s ≤ t ≤ T
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
6 / 47
Introduction
SDEs and BSDEs
Xts,x,ε = x +
Z t
s
β(Xrs,x,ε )dr +
ϕts,x
Yts,x,ε = g(XTs,x,ε ) +
Z t
=x+
s
Z T
t
s
β(ϕs,x
r )dr , s ≤ t ≤ T
f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
ψts,x = g(ϕTs,x ) +
I. DAKAOU (UM)
√ Z t
ε
σ(Xrs,x,ε )dWr , s ≤ t ≤ T
Z T
t
Z T
t
Zrs,x,ε dWr
s,x
f (r , ϕs,x
r , ψr , 0)dr , s ≤ t ≤ T
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
6 / 47
Introduction
SDEs and BSDEs
Xts,x,ε = x +
Z t
s
β(Xrs,x,ε )dr +
ϕts,x
Yts,x,ε = g(XTs,x,ε ) +
Z t
=x+
s
Z T
t
s
β(ϕs,x
r )dr , s ≤ t ≤ T
f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
ψts,x = g(ϕTs,x ) +
I. DAKAOU (UM)
√ Z t
ε
σ(Xrs,x,ε )dWr , s ≤ t ≤ T
Z T
t
Z T
t
Zrs,x,ε dWr
s,x
f (r , ϕs,x
r , ψr , 0)dr , s ≤ t ≤ T
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
6 / 47
Introduction
SDEs and BSDEs
Xts,x,ε = x +
Z t
s
β(Xrs,x,ε )dr +
ϕts,x
Yts,x,ε = g(XTs,x,ε ) +
Z t
=x+
s
Z T
t
s
β(ϕs,x
r )dr , s ≤ t ≤ T
f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
ψts,x = g(ϕTs,x ) +
I. DAKAOU (UM)
√ Z t
ε
σ(Xrs,x,ε )dWr , s ≤ t ≤ T
Z T
t
Z T
t
Zrs,x,ε dWr
s,x
f (r , ϕs,x
r , ψr , 0)dr , s ≤ t ≤ T
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
6 / 47
Introduction
LDP for SDEs and BSDEs
Freidlin and Wentzell (1984)
(X s,x,ε )ε>0 converges in probability, as ε goes to 0, to (ϕs,x
t )s≤t≤T and
satisfies a LDP.
Rainero (2006)
ε ∈]0, 1], (s, x) ∈ [0, T ] × K, where K is a compact subset of Rd .
Then, there exists C > 0 independent of ε, s and x such that
E
sup
s≤t≤T
|Yts,x,ε
−
ψts,x |2
Z T
+
s
!
kZrs,x,ε k2 dr
≤ Cε,
(Y s,x,ε )ε>0 satisfies in C([0, T ], Rd ) a LDP.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
7 / 47
Introduction
LDP for SDEs and BSDEs
Freidlin and Wentzell (1984)
(X s,x,ε )ε>0 converges in probability, as ε goes to 0, to (ϕs,x
t )s≤t≤T and
satisfies a LDP.
Rainero (2006)
ε ∈]0, 1], (s, x) ∈ [0, T ] × K, where K is a compact subset of Rd .
Then, there exists C > 0 independent of ε, s and x such that
E
sup
s≤t≤T
|Yts,x,ε
−
ψts,x |2
Z T
+
s
!
kZrs,x,ε k2 dr
≤ Cε,
(Y s,x,ε )ε>0 satisfies in C([0, T ], Rd ) a LDP.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
7 / 47
Introduction
LDP for SDEs and BSDEs
Freidlin and Wentzell (1984)
(X s,x,ε )ε>0 converges in probability, as ε goes to 0, to (ϕs,x
t )s≤t≤T and
satisfies a LDP.
Rainero (2006)
ε ∈]0, 1], (s, x) ∈ [0, T ] × K, where K is a compact subset of Rd .
Then, there exists C > 0 independent of ε, s and x such that
E
sup
s≤t≤T
|Yts,x,ε
−
ψts,x |2
Z T
+
s
!
kZrs,x,ε k2 dr
≤ Cε,
(Y s,x,ε )ε>0 satisfies in C([0, T ], Rd ) a LDP.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
7 / 47
Introduction
LDP for SDEs and BSDEs
Freidlin and Wentzell (1984)
(X s,x,ε )ε>0 converges in probability, as ε goes to 0, to (ϕs,x
t )s≤t≤T and
satisfies a LDP.
Rainero (2006)
ε ∈]0, 1], (s, x) ∈ [0, T ] × K, where K is a compact subset of Rd .
Then, there exists C > 0 independent of ε, s and x such that
E
sup
s≤t≤T
|Yts,x,ε
−
ψts,x |2
Z T
+
s
!
kZrs,x,ε k2 dr
≤ Cε,
(Y s,x,ε )ε>0 satisfies in C([0, T ], Rd ) a LDP.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
7 / 47
Introduction
LDP for Reflected BSDEs
For every s ≤ t ≤ T ,

















R
√ R
− ρs,x,ε
Xts,x,ε = x + st β(Xrs,x,ε )dr + ε st σ(Xrs,x,ε )dWr + ρs,x,ε
s
t
R
s,x,ε
s,x,ε
t
s,x,ε
ρt
= 0 ∇φ(Xr
)d | ρ
|r ,
Rt
s,x,ε
s,x,ε
|ρ
|t = 0 1{Xrs,x,ε ∈∂Θ} d | ρ
|r
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr − tT Zrs,x,ε dWr
R
R
− tT Urs,x,ε dr , (Yts,x,ε , Uts,x,ε ) ∈ ∂h, E( 0T h(Yrs,x,ε )dr ) < ∞
(1.1)
R
R
where φ is a function of class C 2 with bounded partial derivatives up to
2, Θ = {x : φ(x) > 0}, ∂Θ = {x : φ(x) = 0}, h : Rd −→ (−∞, +∞] is a
proper lower semicontinuous convex function and ∂h is the
subdifferential operator.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
8 / 47
Introduction
LDP for Reflected BSDEs
For every s ≤ t ≤ T ,

















R
√ R
− ρs,x,ε
Xts,x,ε = x + st β(Xrs,x,ε )dr + ε st σ(Xrs,x,ε )dWr + ρs,x,ε
s
t
R
s,x,ε
s,x,ε
t
s,x,ε
ρt
= 0 ∇φ(Xr
)d | ρ
|r ,
Rt
s,x,ε
s,x,ε
|ρ
|t = 0 1{Xrs,x,ε ∈∂Θ} d | ρ
|r
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr − tT Zrs,x,ε dWr
R
R
− tT Urs,x,ε dr , (Yts,x,ε , Uts,x,ε ) ∈ ∂h, E( 0T h(Yrs,x,ε )dr ) < ∞
(1.1)
R
R
where φ is a function of class C 2 with bounded partial derivatives up to
2, Θ = {x : φ(x) > 0}, ∂Θ = {x : φ(x) = 0}, h : Rd −→ (−∞, +∞] is a
proper lower semicontinuous convex function and ∂h is the
subdifferential operator.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
8 / 47
Introduction
LDP for Reflected BSDEs
For every s ≤ t ≤ T ,

















R
√ R
− ρs,x,ε
Xts,x,ε = x + st β(Xrs,x,ε )dr + ε st σ(Xrs,x,ε )dWr + ρs,x,ε
s
t
R
s,x,ε
s,x,ε
t
s,x,ε
ρt
= 0 ∇φ(Xr
)d | ρ
|r ,
Rt
s,x,ε
s,x,ε
|ρ
|t = 0 1{Xrs,x,ε ∈∂Θ} d | ρ
|r
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr − tT Zrs,x,ε dWr
R
R
− tT Urs,x,ε dr , (Yts,x,ε , Uts,x,ε ) ∈ ∂h, E( 0T h(Yrs,x,ε )dr ) < ∞
(1.1)
R
R
where φ is a function of class C 2 with bounded partial derivatives up to
2, Θ = {x : φ(x) > 0}, ∂Θ = {x : φ(x) = 0}, h : Rd −→ (−∞, +∞] is a
proper lower semicontinuous convex function and ∂h is the
subdifferential operator.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
8 / 47
Introduction
LDP for Reflected BSDEs
For every s ≤ t ≤ T ,













ϕs,x
= x + st β(ϕrs,x )dr + ρs,x
− ρs,x
s
t
t
R
R
s,x
t
s,x | , | ρs,x | = t 1 s,x
s,x |
ρs,x
=
∇φ(ϕ
)d
|
ρ
r
r
t
r
t
0
0 {ϕr ∈∂Θ} d | ρ
R
s,x
ψts,x = g(ϕTs,x ) + tT f (r , ϕs,x
r , ψr , 0)dr −
R
(ψts,x , Uts,x ) ∈ ∂h, E( 0T h(ψrs,x )dr ) < ∞
R
RT
t
Urs,x dr
(1.2)
Essaky (2008)
The author proved, as ε goes to 0, the convergence of
(X s,x,ε , ρs,x,ε , Y s,x,ε , Z s,x,ε , U s,x,ε ) solution of system (1.1) to
(ϕs,x , ρs,x , ψ s,x , 0, U s,x ) solution of system (1.2).
He also established a LDP for the law of (Y s,x,ε )ε>0 .
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
9 / 47
Introduction
LDP for Reflected BSDEs
For every s ≤ t ≤ T ,













ϕs,x
= x + st β(ϕrs,x )dr + ρs,x
− ρs,x
s
t
t
R
R
s,x
t
s,x | , | ρs,x | = t 1 s,x
s,x |
ρs,x
=
∇φ(ϕ
)d
|
ρ
r
r
t
r
t
0
0 {ϕr ∈∂Θ} d | ρ
R
s,x
ψts,x = g(ϕTs,x ) + tT f (r , ϕs,x
r , ψr , 0)dr −
R
(ψts,x , Uts,x ) ∈ ∂h, E( 0T h(ψrs,x )dr ) < ∞
R
RT
t
Urs,x dr
(1.2)
Essaky (2008)
The author proved, as ε goes to 0, the convergence of
(X s,x,ε , ρs,x,ε , Y s,x,ε , Z s,x,ε , U s,x,ε ) solution of system (1.1) to
(ϕs,x , ρs,x , ψ s,x , 0, U s,x ) solution of system (1.2).
He also established a LDP for the law of (Y s,x,ε )ε>0 .
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
9 / 47
Introduction
LDP for Reflected BSDEs
For every s ≤ t ≤ T ,













ϕs,x
= x + st β(ϕrs,x )dr + ρs,x
− ρs,x
s
t
t
R
R
s,x
t
s,x | , | ρs,x | = t 1 s,x
s,x |
ρs,x
=
∇φ(ϕ
)d
|
ρ
r
r
t
r
t
0
0 {ϕr ∈∂Θ} d | ρ
R
s,x
ψts,x = g(ϕTs,x ) + tT f (r , ϕs,x
r , ψr , 0)dr −
R
(ψts,x , Uts,x ) ∈ ∂h, E( 0T h(ψrs,x )dr ) < ∞
R
RT
t
Urs,x dr
(1.2)
Essaky (2008)
The author proved, as ε goes to 0, the convergence of
(X s,x,ε , ρs,x,ε , Y s,x,ε , Z s,x,ε , U s,x,ε ) solution of system (1.1) to
(ϕs,x , ρs,x , ψ s,x , 0, U s,x ) solution of system (1.2).
He also established a LDP for the law of (Y s,x,ε )ε>0 .
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
9 / 47
Introduction
LDP for Reflected BSDEs
For every s ≤ t ≤ T ,













ϕs,x
= x + st β(ϕrs,x )dr + ρs,x
− ρs,x
s
t
t
R
R
s,x
t
s,x | , | ρs,x | = t 1 s,x
s,x |
ρs,x
=
∇φ(ϕ
)d
|
ρ
r
r
t
r
t
0
0 {ϕr ∈∂Θ} d | ρ
R
s,x
ψts,x = g(ϕTs,x ) + tT f (r , ϕs,x
r , ψr , 0)dr −
R
(ψts,x , Uts,x ) ∈ ∂h, E( 0T h(ψrs,x )dr ) < ∞
R
RT
t
Urs,x dr
(1.2)
Essaky (2008)
The author proved, as ε goes to 0, the convergence of
(X s,x,ε , ρs,x,ε , Y s,x,ε , Z s,x,ε , U s,x,ε ) solution of system (1.1) to
(ϕs,x , ρs,x , ψ s,x , 0, U s,x ) solution of system (1.2).
He also established a LDP for the law of (Y s,x,ε )ε>0 .
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
9 / 47
Introduction
LDP
Freidlin and Wentzell (1984)
Schilder (1966)
Brownian motion
Rainero (2006)
BSDEs
SDEs
Essaky (2008)
RBSDEs (Sub-differential operator)
Multivalued BSDEs
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
10 / 47
Introduction
LDP
Freidlin and Wentzell (1984)
Schilder (1966)
Brownian motion
Rainero (2006)
BSDEs
SDEs
Essaky (2008)
RBSDEs (Sub-differential operator)
Multivalued BSDEs
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
10 / 47
Introduction
LDP
Freidlin and Wentzell (1984)
Schilder (1966)
Brownian motion
Rainero (2006)
BSDEs
SDEs
Essaky (2008)
RBSDEs (Sub-differential operator)
Multivalued BSDEs
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
10 / 47
Introduction
LDP
Freidlin and Wentzell (1984)
Schilder (1966)
Brownian motion
Rainero (2006)
BSDEs
SDEs
Essaky (2008)
RBSDEs (Sub-differential operator)
Multivalued BSDEs
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
10 / 47
Introduction
LDP
Freidlin and Wentzell (1984)
Schilder (1966)
Brownian motion
Rainero (2006)
BSDEs
SDEs
Essaky (2008)
RBSDEs (Sub-differential operator)
Multivalued BSDEs
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
10 / 47
Introduction
LDP
Freidlin and Wentzell (1984)
Schilder (1966)
Brownian motion
Rainero (2006)
BSDEs
SDEs
Essaky (2008)
RBSDEs (Sub-differential operator)
Multivalued BSDEs
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
10 / 47
Multivalued BSDEs
Outline
1
Introduction
2
Multivalued BSDEs
3
Large deviations
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
11 / 47
Multivalued BSDEs
Notations
Inner product and Euclidean norm
h., .i
|.|
kzk2 = tr (zz ∗ )
Let A be a multivalued operator on Rd
D(A) = {x ∈ Rd : A(x) 6= ∅}
Gr (A) = {(x, y ) ∈ R2d : x ∈ Rd , y ∈ A(x)}
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
12 / 47
Multivalued BSDEs
Notations
Inner product and Euclidean norm
h., .i
|.|
kzk2 = tr (zz ∗ )
Let A be a multivalued operator on Rd
D(A) = {x ∈ Rd : A(x) 6= ∅}
Gr (A) = {(x, y ) ∈ R2d : x ∈ Rd , y ∈ A(x)}
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
12 / 47
Multivalued BSDEs
Notations
Inner product and Euclidean norm
h., .i
|.|
kzk2 = tr (zz ∗ )
Let A be a multivalued operator on Rd
D(A) = {x ∈ Rd : A(x) 6= ∅}
Gr (A) = {(x, y ) ∈ R2d : x ∈ Rd , y ∈ A(x)}
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
12 / 47
Multivalued BSDEs
Notations
Inner product and Euclidean norm
h., .i
|.|
kzk2 = tr (zz ∗ )
Let A be a multivalued operator on Rd
D(A) = {x ∈ Rd : A(x) 6= ∅}
Gr (A) = {(x, y ) ∈ R2d : x ∈ Rd , y ∈ A(x)}
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
12 / 47
Multivalued BSDEs
Notations
Inner product and Euclidean norm
h., .i
|.|
kzk2 = tr (zz ∗ )
Let A be a multivalued operator on Rd
D(A) = {x ∈ Rd : A(x) 6= ∅}
Gr (A) = {(x, y ) ∈ R2d : x ∈ Rd , y ∈ A(x)}
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
12 / 47
Multivalued BSDEs
Multivalued operator
Monotone
hy1 − y2 , x1 − x2 i ≥ 0, ∀(x1 , y1 ), (x2 , y2 ) ∈ Gr (A)
At most one solution
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
Maximal monotone
(x, y ) ∈ Gr (A) ⇔ {hy − v , x − ui ≥ 0, ∀(u, v ) ∈ Gr (A)}
Uniqueness
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
13 / 47
Multivalued BSDEs
Multivalued operator
Monotone
hy1 − y2 , x1 − x2 i ≥ 0, ∀(x1 , y1 ), (x2 , y2 ) ∈ Gr (A)
At most one solution
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
Maximal monotone
(x, y ) ∈ Gr (A) ⇔ {hy − v , x − ui ≥ 0, ∀(u, v ) ∈ Gr (A)}
Uniqueness
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
13 / 47
Multivalued BSDEs
Multivalued operator
Monotone
hy1 − y2 , x1 − x2 i ≥ 0, ∀(x1 , y1 ), (x2 , y2 ) ∈ Gr (A)
At most one solution
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
Maximal monotone
(x, y ) ∈ Gr (A) ⇔ {hy − v , x − ui ≥ 0, ∀(u, v ) ∈ Gr (A)}
Uniqueness
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
13 / 47
Multivalued BSDEs
Multivalued operator
Monotone
hy1 − y2 , x1 − x2 i ≥ 0, ∀(x1 , y1 ), (x2 , y2 ) ∈ Gr (A)
At most one solution
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
Maximal monotone
(x, y ) ∈ Gr (A) ⇔ {hy − v , x − ui ≥ 0, ∀(u, v ) ∈ Gr (A)}
Uniqueness
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
13 / 47
Multivalued BSDEs
Multivalued operator
Monotone
hy1 − y2 , x1 − x2 i ≥ 0, ∀(x1 , y1 ), (x2 , y2 ) ∈ Gr (A)
At most one solution
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
Maximal monotone
(x, y ) ∈ Gr (A) ⇔ {hy − v , x − ui ≥ 0, ∀(u, v ) ∈ Gr (A)}
Uniqueness
∀ λ > 0, z ∈ Rd , z ∈ x + λA(x)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
13 / 47
Multivalued BSDEs
Well-known results about maximal monotone operator
Let λ > 0.
The single-valued map Jλ = (I + λA)−1 from Rd to Rd is a contraction.
Yosida’s approximation
Aλ = λ1 (I − Jλ ) is a single-valued map from Rd to Rd which is maximal,
monotone and Lipschitz continuous with Lipschitz constant λ1 .
For all x ∈ D(A), A(x) is a closed convex subset of Rd .
A◦ (x) = ProjA(x) (0).
For all x ∈ D(A),
|Aλ (x)| ≤ |A◦ (x)|, lim Aλ (x) = A◦ (x).
λ→0
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
14 / 47
Multivalued BSDEs
Well-known results about maximal monotone operator
Let λ > 0.
The single-valued map Jλ = (I + λA)−1 from Rd to Rd is a contraction.
Yosida’s approximation
Aλ = λ1 (I − Jλ ) is a single-valued map from Rd to Rd which is maximal,
monotone and Lipschitz continuous with Lipschitz constant λ1 .
For all x ∈ D(A), A(x) is a closed convex subset of Rd .
A◦ (x) = ProjA(x) (0).
For all x ∈ D(A),
|Aλ (x)| ≤ |A◦ (x)|, lim Aλ (x) = A◦ (x).
λ→0
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
14 / 47
Multivalued BSDEs
Well-known results about maximal monotone operator
Let λ > 0.
The single-valued map Jλ = (I + λA)−1 from Rd to Rd is a contraction.
Yosida’s approximation
Aλ = λ1 (I − Jλ ) is a single-valued map from Rd to Rd which is maximal,
monotone and Lipschitz continuous with Lipschitz constant λ1 .
For all x ∈ D(A), A(x) is a closed convex subset of Rd .
A◦ (x) = ProjA(x) (0).
For all x ∈ D(A),
|Aλ (x)| ≤ |A◦ (x)|, lim Aλ (x) = A◦ (x).
λ→0
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
14 / 47
Multivalued BSDEs
Well-known results about maximal monotone operator
Let λ > 0.
The single-valued map Jλ = (I + λA)−1 from Rd to Rd is a contraction.
Yosida’s approximation
Aλ = λ1 (I − Jλ ) is a single-valued map from Rd to Rd which is maximal,
monotone and Lipschitz continuous with Lipschitz constant λ1 .
For all x ∈ D(A), A(x) is a closed convex subset of Rd .
A◦ (x) = ProjA(x) (0).
For all x ∈ D(A),
|Aλ (x)| ≤ |A◦ (x)|, lim Aλ (x) = A◦ (x).
λ→0
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
14 / 47
Multivalued BSDEs
Well-known results about maximal monotone operator
Let λ > 0.
The single-valued map Jλ = (I + λA)−1 from Rd to Rd is a contraction.
Yosida’s approximation
Aλ = λ1 (I − Jλ ) is a single-valued map from Rd to Rd which is maximal,
monotone and Lipschitz continuous with Lipschitz constant λ1 .
For all x ∈ D(A), A(x) is a closed convex subset of Rd .
A◦ (x) = ProjA(x) (0).
For all x ∈ D(A),
|Aλ (x)| ≤ |A◦ (x)|, lim Aλ (x) = A◦ (x).
λ→0
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
14 / 47
Multivalued BSDEs
Well-known results about maximal monotone operator
Let λ > 0.
The single-valued map Jλ = (I + λA)−1 from Rd to Rd is a contraction.
Yosida’s approximation
Aλ = λ1 (I − Jλ ) is a single-valued map from Rd to Rd which is maximal,
monotone and Lipschitz continuous with Lipschitz constant λ1 .
For all x ∈ D(A), A(x) is a closed convex subset of Rd .
A◦ (x) = ProjA(x) (0).
For all x ∈ D(A),
|Aλ (x)| ≤ |A◦ (x)|, lim Aλ (x) = A◦ (x).
λ→0
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
14 / 47
Multivalued BSDEs
Well-known results about maximal monotone operator
Let λ > 0.
The single-valued map Jλ = (I + λA)−1 from Rd to Rd is a contraction.
Yosida’s approximation
Aλ = λ1 (I − Jλ ) is a single-valued map from Rd to Rd which is maximal,
monotone and Lipschitz continuous with Lipschitz constant λ1 .
For all x ∈ D(A), A(x) is a closed convex subset of Rd .
A◦ (x) = ProjA(x) (0).
For all x ∈ D(A),
|Aλ (x)| ≤ |A◦ (x)|, lim Aλ (x) = A◦ (x).
λ→0
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
14 / 47
Multivalued BSDEs
Multivalued forward-backward SDE
Let us fix s ≥ 0 and x ∈ Rd .
For every s ≤ t ≤ T ,







R
√ R
Xts,x,ε = x + st β(Xrs,x,ε )dr + ε st σ(Xrs,x,ε )dWr
R
R
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr − tT Zrs,x,ε dWr
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A)
(2.1)
We interest to the following backward equation
(
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A), s ≤ t ≤ T
R
RT
t
Zrs,x,ε dWr
(2.2)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
15 / 47
Multivalued BSDEs
Multivalued forward-backward SDE
Let us fix s ≥ 0 and x ∈ Rd .
For every s ≤ t ≤ T ,







R
√ R
Xts,x,ε = x + st β(Xrs,x,ε )dr + ε st σ(Xrs,x,ε )dWr
R
R
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr − tT Zrs,x,ε dWr
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A)
(2.1)
We interest to the following backward equation
(
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A), s ≤ t ≤ T
R
RT
t
Zrs,x,ε dWr
(2.2)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
15 / 47
Multivalued BSDEs
Multivalued forward-backward SDE
Let us fix s ≥ 0 and x ∈ Rd .
For every s ≤ t ≤ T ,







R
√ R
Xts,x,ε = x + st β(Xrs,x,ε )dr + ε st σ(Xrs,x,ε )dWr
R
R
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr − tT Zrs,x,ε dWr
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A)
(2.1)
We interest to the following backward equation
(
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A), s ≤ t ≤ T
R
RT
t
Zrs,x,ε dWr
(2.2)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
15 / 47
Multivalued BSDEs
Multivalued forward-backward SDE
Let us fix s ≥ 0 and x ∈ Rd .
For every s ≤ t ≤ T ,







R
√ R
Xts,x,ε = x + st β(Xrs,x,ε )dr + ε st σ(Xrs,x,ε )dWr
R
R
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr − tT Zrs,x,ε dWr
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A)
(2.1)
We interest to the following backward equation
(
Yts,x,ε = g(XTs,x,ε ) + tT f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε )dr −
+KTs,x,ε − Kts,x,ε , Yts,x,ε ∈ D(A), s ≤ t ≤ T
R
RT
t
Zrs,x,ε dWr
(2.2)
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
15 / 47
Multivalued BSDEs
Assumptions
Let f : [0, T ] × Rd × Rd × Rd×k −→ Rd and g : Rd −→ Rd be
continuous functions satisfying the following assumptions :
(A1) g satisfies a Lipschitz condition.
(A2) There exist constants µ ∈ R, K > 0 such that
∀t, ∀(x, x 0 ), ∀y , ∀(z, z 0 ),
|f (t, x, y , z) − f (t, x 0 , y , z 0 )| ≤ K (|x − x 0 | + ||z − z 0 ||)
∀t, ∀x, ∀(y , y 0 ), ∀z,
hy − y 0 , f (t, x, y , z) − f (t, x, y 0 , z)i ≤ µ|y − y 0 |2
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
16 / 47
Multivalued BSDEs
Assumptions
Let f : [0, T ] × Rd × Rd × Rd×k −→ Rd and g : Rd −→ Rd be
continuous functions satisfying the following assumptions :
(A1) g satisfies a Lipschitz condition.
(A2) There exist constants µ ∈ R, K > 0 such that
∀t, ∀(x, x 0 ), ∀y , ∀(z, z 0 ),
|f (t, x, y , z) − f (t, x 0 , y , z 0 )| ≤ K (|x − x 0 | + ||z − z 0 ||)
∀t, ∀x, ∀(y , y 0 ), ∀z,
hy − y 0 , f (t, x, y , z) − f (t, x, y 0 , z)i ≤ µ|y − y 0 |2
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
16 / 47
Multivalued BSDEs
Assumptions
Let f : [0, T ] × Rd × Rd × Rd×k −→ Rd and g : Rd −→ Rd be
continuous functions satisfying the following assumptions :
(A1) g satisfies a Lipschitz condition.
(A2) There exist constants µ ∈ R, K > 0 such that
∀t, ∀(x, x 0 ), ∀y , ∀(z, z 0 ),
|f (t, x, y , z) − f (t, x 0 , y , z 0 )| ≤ K (|x − x 0 | + ||z − z 0 ||)
∀t, ∀x, ∀(y , y 0 ), ∀z,
hy − y 0 , f (t, x, y , z) − f (t, x, y 0 , z)i ≤ µ|y − y 0 |2
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
16 / 47
Multivalued BSDEs
Assumptions
(A3) There exists constant c > 0 such that ∀t, ∀x, ∀y , ∀z,
|g(x)| + |f (t, x, y , z)| ≤ c(1 + |x| + |y | + ||z||)
(A4) A the multivalued operator satisfies
Int(D(A)) 6= ∅, g(x) ∈ D(A),
∀x ∈ D(A), |A◦ (x)| ≤ δ(1 + |x|), δ > 0.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
17 / 47
Multivalued BSDEs
Solution
Definition
A solution of (2.2) is a triple of progressively measurable processes
{(Yts,x,ε , Zts,x,ε , Kts,x,ε ) : s ≤ t ≤ T } with values in Rd × Rd×k × Rd ,
such that
1
E
sup
s≤t≤T
|Yts,x,ε |2
Z T
+
s
!
||Zts,x,ε ||2 dt
< +∞,
2
K s,x,ε is continuous and has bounded variation with Kss,x,ε = 0 a.s.,
3
Y s,x,ε is continuous and takes values in D(A),
4
For any optional process (ν, υ) with values in Gr (A), the measure
hYrs,x,ε − νr , dKrs,x,ε + υr dr i is almost surely negative on [s, T ].
N’zi and Ouknine (1997) ⇒ Existence and Uniqueness
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
18 / 47
Multivalued BSDEs
Solution
Definition
A solution of (2.2) is a triple of progressively measurable processes
{(Yts,x,ε , Zts,x,ε , Kts,x,ε ) : s ≤ t ≤ T } with values in Rd × Rd×k × Rd ,
such that
1
E
sup
s≤t≤T
|Yts,x,ε |2
Z T
+
s
!
||Zts,x,ε ||2 dt
< +∞,
2
K s,x,ε is continuous and has bounded variation with Kss,x,ε = 0 a.s.,
3
Y s,x,ε is continuous and takes values in D(A),
4
For any optional process (ν, υ) with values in Gr (A), the measure
hYrs,x,ε − νr , dKrs,x,ε + υr dr i is almost surely negative on [s, T ].
N’zi and Ouknine (1997) ⇒ Existence and Uniqueness
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
18 / 47
Multivalued BSDEs
Solution
Definition
A solution of (2.2) is a triple of progressively measurable processes
{(Yts,x,ε , Zts,x,ε , Kts,x,ε ) : s ≤ t ≤ T } with values in Rd × Rd×k × Rd ,
such that
1
E
sup
s≤t≤T
|Yts,x,ε |2
Z T
+
s
!
||Zts,x,ε ||2 dt
< +∞,
2
K s,x,ε is continuous and has bounded variation with Kss,x,ε = 0 a.s.,
3
Y s,x,ε is continuous and takes values in D(A),
4
For any optional process (ν, υ) with values in Gr (A), the measure
hYrs,x,ε − νr , dKrs,x,ε + υr dr i is almost surely negative on [s, T ].
N’zi and Ouknine (1997) ⇒ Existence and Uniqueness
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
18 / 47
Multivalued BSDEs
Solution
Definition
A solution of (2.2) is a triple of progressively measurable processes
{(Yts,x,ε , Zts,x,ε , Kts,x,ε ) : s ≤ t ≤ T } with values in Rd × Rd×k × Rd ,
such that
1
E
sup
s≤t≤T
|Yts,x,ε |2
Z T
+
s
!
||Zts,x,ε ||2 dt
< +∞,
2
K s,x,ε is continuous and has bounded variation with Kss,x,ε = 0 a.s.,
3
Y s,x,ε is continuous and takes values in D(A),
4
For any optional process (ν, υ) with values in Gr (A), the measure
hYrs,x,ε − νr , dKrs,x,ε + υr dr i is almost surely negative on [s, T ].
N’zi and Ouknine (1997) ⇒ Existence and Uniqueness
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
18 / 47
Multivalued BSDEs
Solution
Definition
A solution of (2.2) is a triple of progressively measurable processes
{(Yts,x,ε , Zts,x,ε , Kts,x,ε ) : s ≤ t ≤ T } with values in Rd × Rd×k × Rd ,
such that
1
E
sup
s≤t≤T
|Yts,x,ε |2
Z T
+
s
!
||Zts,x,ε ||2 dt
< +∞,
2
K s,x,ε is continuous and has bounded variation with Kss,x,ε = 0 a.s.,
3
Y s,x,ε is continuous and takes values in D(A),
4
For any optional process (ν, υ) with values in Gr (A), the measure
hYrs,x,ε − νr , dKrs,x,ε + υr dr i is almost surely negative on [s, T ].
N’zi and Ouknine (1997) ⇒ Existence and Uniqueness
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
18 / 47
Multivalued BSDEs
Problem
For every s ≤ t ≤ T ,







ϕts,x = x + st β(ϕs,x
r )dr
RT
s,x
s,x
s,x
s,x
s,x
ψt = g(ϕT ) + t f (r , ϕs,x
r , ψr , 0)dr + KT − Kt
ψts,x ∈ D(A)
R
(2.3)
Our aim
Convergence
LDP
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
19 / 47
Multivalued BSDEs
Problem
For every s ≤ t ≤ T ,







ϕts,x = x + st β(ϕs,x
r )dr
RT
s,x
s,x
s,x
s,x
s,x
ψt = g(ϕT ) + t f (r , ϕs,x
r , ψr , 0)dr + KT − Kt
ψts,x ∈ D(A)
R
(2.3)
Our aim
Convergence
LDP
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
19 / 47
Multivalued BSDEs
Problem
For every s ≤ t ≤ T ,







ϕts,x = x + st β(ϕs,x
r )dr
RT
s,x
s,x
s,x
s,x
s,x
ψt = g(ϕT ) + t f (r , ϕs,x
r , ψr , 0)dr + KT − Kt
ψts,x ∈ D(A)
R
(2.3)
Our aim
Convergence
LDP
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
19 / 47
Multivalued BSDEs
Problem
For every s ≤ t ≤ T ,







ϕts,x = x + st β(ϕs,x
r )dr
RT
s,x
s,x
s,x
s,x
s,x
ψt = g(ϕT ) + t f (r , ϕs,x
r , ψr , 0)dr + KT − Kt
ψts,x ∈ D(A)
R
(2.3)
Our aim
Convergence
LDP
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
19 / 47
Large deviations
Outline
1
Introduction
2
Multivalued BSDEs
3
Large deviations
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
20 / 47
Large deviations
Large deviations
Definition
Let χ be a topological space. Let Y ε = (Ytε , s ≤ t ≤ T ) be a family of
processes which depends on a parameter ε.
1
2
A rate function is a lower semicontinuous function Λ defined on χ
with values in [0, +∞]. (i.e for all α ≥ 0, {x ∈ χ : Λ(x) ≤ α} is a
closed subset of χ). A good rate function.
Y ε = (Ytε , s ≤ t ≤ T ) is said to satisfy a LDP with a good rate
function Λ if the following condition hold for every Borel set
A ⊆ C([s, T ], Rd )
lim inf ε ln(P(Y ε ∈ A)) ≥ − inf Λ(Ψ)
ε→0
Ψ∈Å
lim sup ε ln(P(Y ε ∈ A)) ≤ − inf Λ(Ψ)
ε→0
I. DAKAOU (UM)
Ψ∈Ā
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
21 / 47
Large deviations
Large deviations
Definition
Let χ be a topological space. Let Y ε = (Ytε , s ≤ t ≤ T ) be a family of
processes which depends on a parameter ε.
1
2
A rate function is a lower semicontinuous function Λ defined on χ
with values in [0, +∞]. (i.e for all α ≥ 0, {x ∈ χ : Λ(x) ≤ α} is a
closed subset of χ). A good rate function.
Y ε = (Ytε , s ≤ t ≤ T ) is said to satisfy a LDP with a good rate
function Λ if the following condition hold for every Borel set
A ⊆ C([s, T ], Rd )
lim inf ε ln(P(Y ε ∈ A)) ≥ − inf Λ(Ψ)
ε→0
Ψ∈Å
lim sup ε ln(P(Y ε ∈ A)) ≤ − inf Λ(Ψ)
ε→0
I. DAKAOU (UM)
Ψ∈Ā
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
21 / 47
Large deviations
Large deviations
Definition
Let χ be a topological space. Let Y ε = (Ytε , s ≤ t ≤ T ) be a family of
processes which depends on a parameter ε.
1
2
A rate function is a lower semicontinuous function Λ defined on χ
with values in [0, +∞]. (i.e for all α ≥ 0, {x ∈ χ : Λ(x) ≤ α} is a
closed subset of χ). A good rate function.
Y ε = (Ytε , s ≤ t ≤ T ) is said to satisfy a LDP with a good rate
function Λ if the following condition hold for every Borel set
A ⊆ C([s, T ], Rd )
lim inf ε ln(P(Y ε ∈ A)) ≥ − inf Λ(Ψ)
ε→0
Ψ∈Å
lim sup ε ln(P(Y ε ∈ A)) ≤ − inf Λ(Ψ)
ε→0
I. DAKAOU (UM)
Ψ∈Ā
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
21 / 47
Large deviations
Large deviations
Definition
Let χ be a topological space. Let Y ε = (Ytε , s ≤ t ≤ T ) be a family of
processes which depends on a parameter ε.
1
2
A rate function is a lower semicontinuous function Λ defined on χ
with values in [0, +∞]. (i.e for all α ≥ 0, {x ∈ χ : Λ(x) ≤ α} is a
closed subset of χ). A good rate function.
Y ε = (Ytε , s ≤ t ≤ T ) is said to satisfy a LDP with a good rate
function Λ if the following condition hold for every Borel set
A ⊆ C([s, T ], Rd )
lim inf ε ln(P(Y ε ∈ A)) ≥ − inf Λ(Ψ)
ε→0
Ψ∈Å
lim sup ε ln(P(Y ε ∈ A)) ≤ − inf Λ(Ψ)
ε→0
I. DAKAOU (UM)
Ψ∈Ā
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
21 / 47
Large deviations
Large deviations
Definition
Let χ be a topological space. Let Y ε = (Ytε , s ≤ t ≤ T ) be a family of
processes which depends on a parameter ε.
1
2
A rate function is a lower semicontinuous function Λ defined on χ
with values in [0, +∞]. (i.e for all α ≥ 0, {x ∈ χ : Λ(x) ≤ α} is a
closed subset of χ). A good rate function.
Y ε = (Ytε , s ≤ t ≤ T ) is said to satisfy a LDP with a good rate
function Λ if the following condition hold for every Borel set
A ⊆ C([s, T ], Rd )
lim inf ε ln(P(Y ε ∈ A)) ≥ − inf Λ(Ψ)
ε→0
Ψ∈Å
lim sup ε ln(P(Y ε ∈ A)) ≤ − inf Λ(Ψ)
ε→0
I. DAKAOU (UM)
Ψ∈Ā
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
21 / 47
Large deviations
Large deviations
Definition
Let χ be a topological space. Let Y ε = (Ytε , s ≤ t ≤ T ) be a family of
processes which depends on a parameter ε.
1
2
A rate function is a lower semicontinuous function Λ defined on χ
with values in [0, +∞]. (i.e for all α ≥ 0, {x ∈ χ : Λ(x) ≤ α} is a
closed subset of χ). A good rate function.
Y ε = (Ytε , s ≤ t ≤ T ) is said to satisfy a LDP with a good rate
function Λ if the following condition hold for every Borel set
A ⊆ C([s, T ], Rd )
lim inf ε ln(P(Y ε ∈ A)) ≥ − inf Λ(Ψ)
ε→0
Ψ∈Å
lim sup ε ln(P(Y ε ∈ A)) ≤ − inf Λ(Ψ)
ε→0
I. DAKAOU (UM)
Ψ∈Ā
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
21 / 47
Large deviations
Large deviations
Definition
Let χ be a topological space. Let Y ε = (Ytε , s ≤ t ≤ T ) be a family of
processes which depends on a parameter ε.
1
2
A rate function is a lower semicontinuous function Λ defined on χ
with values in [0, +∞]. (i.e for all α ≥ 0, {x ∈ χ : Λ(x) ≤ α} is a
closed subset of χ). A good rate function.
Y ε = (Ytε , s ≤ t ≤ T ) is said to satisfy a LDP with a good rate
function Λ if the following condition hold for every Borel set
A ⊆ C([s, T ], Rd )
lim inf ε ln(P(Y ε ∈ A)) ≥ − inf Λ(Ψ)
ε→0
Ψ∈Å
lim sup ε ln(P(Y ε ∈ A)) ≤ − inf Λ(Ψ)
ε→0
I. DAKAOU (UM)
Ψ∈Ā
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
21 / 47
Large deviations
Contraction principle
Varadhan (1984)
Let χ and Υ be Polish spaces and let F ε : χ → Υ, ∀ε > 0, be
continuous maps. Assume F ε converges, as ε goes to 0, to a
continuous map F 0 , uniformly on each compact subset of χ.
Let Λ : χ −→ [0, +∞] be a good rate function.
For all y ∈ Υ, set Π(y ) = inf{Λ(x) : x ∈ χ, y = F 0 (x)}, with the usual
convention inf ∅ = +∞.
Then, Π is a good rate function on Υ.
Moreover, if Λ is associated to a LDP for a family of measures (µε )ε>0
on χ when ε tends to 0, then (µε (F ε )−1 )ε>0 satisfies a LDP on Υ with
rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
22 / 47
Large deviations
Contraction principle
Varadhan (1984)
Let χ and Υ be Polish spaces and let F ε : χ → Υ, ∀ε > 0, be
continuous maps. Assume F ε converges, as ε goes to 0, to a
continuous map F 0 , uniformly on each compact subset of χ.
Let Λ : χ −→ [0, +∞] be a good rate function.
For all y ∈ Υ, set Π(y ) = inf{Λ(x) : x ∈ χ, y = F 0 (x)}, with the usual
convention inf ∅ = +∞.
Then, Π is a good rate function on Υ.
Moreover, if Λ is associated to a LDP for a family of measures (µε )ε>0
on χ when ε tends to 0, then (µε (F ε )−1 )ε>0 satisfies a LDP on Υ with
rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
22 / 47
Large deviations
Contraction principle
Varadhan (1984)
Let χ and Υ be Polish spaces and let F ε : χ → Υ, ∀ε > 0, be
continuous maps. Assume F ε converges, as ε goes to 0, to a
continuous map F 0 , uniformly on each compact subset of χ.
Let Λ : χ −→ [0, +∞] be a good rate function.
For all y ∈ Υ, set Π(y ) = inf{Λ(x) : x ∈ χ, y = F 0 (x)}, with the usual
convention inf ∅ = +∞.
Then, Π is a good rate function on Υ.
Moreover, if Λ is associated to a LDP for a family of measures (µε )ε>0
on χ when ε tends to 0, then (µε (F ε )−1 )ε>0 satisfies a LDP on Υ with
rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
22 / 47
Large deviations
Contraction principle
Varadhan (1984)
Let χ and Υ be Polish spaces and let F ε : χ → Υ, ∀ε > 0, be
continuous maps. Assume F ε converges, as ε goes to 0, to a
continuous map F 0 , uniformly on each compact subset of χ.
Let Λ : χ −→ [0, +∞] be a good rate function.
For all y ∈ Υ, set Π(y ) = inf{Λ(x) : x ∈ χ, y = F 0 (x)}, with the usual
convention inf ∅ = +∞.
Then, Π is a good rate function on Υ.
Moreover, if Λ is associated to a LDP for a family of measures (µε )ε>0
on χ when ε tends to 0, then (µε (F ε )−1 )ε>0 satisfies a LDP on Υ with
rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
22 / 47
Large deviations
Contraction principle
Varadhan (1984)
Let χ and Υ be Polish spaces and let F ε : χ → Υ, ∀ε > 0, be
continuous maps. Assume F ε converges, as ε goes to 0, to a
continuous map F 0 , uniformly on each compact subset of χ.
Let Λ : χ −→ [0, +∞] be a good rate function.
For all y ∈ Υ, set Π(y ) = inf{Λ(x) : x ∈ χ, y = F 0 (x)}, with the usual
convention inf ∅ = +∞.
Then, Π is a good rate function on Υ.
Moreover, if Λ is associated to a LDP for a family of measures (µε )ε>0
on χ when ε tends to 0, then (µε (F ε )−1 )ε>0 satisfies a LDP on Υ with
rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
22 / 47
Large deviations
Contraction principle
Varadhan (1984)
Let χ and Υ be Polish spaces and let F ε : χ → Υ, ∀ε > 0, be
continuous maps. Assume F ε converges, as ε goes to 0, to a
continuous map F 0 , uniformly on each compact subset of χ.
Let Λ : χ −→ [0, +∞] be a good rate function.
For all y ∈ Υ, set Π(y ) = inf{Λ(x) : x ∈ χ, y = F 0 (x)}, with the usual
convention inf ∅ = +∞.
Then, Π is a good rate function on Υ.
Moreover, if Λ is associated to a LDP for a family of measures (µε )ε>0
on χ when ε tends to 0, then (µε (F ε )−1 )ε>0 satisfies a LDP on Υ with
rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
22 / 47
Large deviations
Convergence for backward equation
Theorem
For any ε ∈]0, 1] and all x ∈ Rd , there exists a constant C > 0,
independent of (s, x, ε), such that
sup
E
s≤t≤T
≤ CE
|Yts,x,ε
|XTs,x,ε
−
−
ψts,x |2
2
ϕs,x
T |
1
+
2
Z T
+
s
Z T
s
!
kZrs,x,ε k2 dr
!
|Xrs,x,ε
−
2
ϕs,x
r | dr
Corollary
ε ∈]0, 1], (s, x) ∈ [0, T ] × K, where K is a compact subset of Rd .
!
E
sup
s≤t≤T
I. DAKAOU (UM)
|Yts,x,ε
−
ψts,x |2
≤ Cε
Large deviations for multivalued BSDEs
(3.1)
Abidjan March 27, 2014
23 / 47
Large deviations
Convergence for backward equation
Theorem
For any ε ∈]0, 1] and all x ∈ Rd , there exists a constant C > 0,
independent of (s, x, ε), such that
sup
E
s≤t≤T
≤ CE
|Yts,x,ε
|XTs,x,ε
−
−
ψts,x |2
2
ϕs,x
T |
1
+
2
Z T
+
s
Z T
s
!
kZrs,x,ε k2 dr
!
|Xrs,x,ε
−
2
ϕs,x
r | dr
Corollary
ε ∈]0, 1], (s, x) ∈ [0, T ] × K, where K is a compact subset of Rd .
!
E
sup
s≤t≤T
I. DAKAOU (UM)
|Yts,x,ε
−
ψts,x |2
≤ Cε
Large deviations for multivalued BSDEs
(3.1)
Abidjan March 27, 2014
23 / 47
Large deviations
Convergence for backward equation
Theorem
For any ε ∈]0, 1] and all x ∈ Rd , there exists a constant C > 0,
independent of (s, x, ε), such that
sup
E
s≤t≤T
≤ CE
|Yts,x,ε
|XTs,x,ε
−
−
ψts,x |2
2
ϕs,x
T |
1
+
2
Z T
+
s
Z T
s
!
kZrs,x,ε k2 dr
!
|Xrs,x,ε
−
2
ϕs,x
r | dr
Corollary
ε ∈]0, 1], (s, x) ∈ [0, T ] × K, where K is a compact subset of Rd .
!
E
sup
s≤t≤T
I. DAKAOU (UM)
|Yts,x,ε
−
ψts,x |2
≤ Cε
Large deviations for multivalued BSDEs
(3.1)
Abidjan March 27, 2014
23 / 47
Large deviations
Cépa (1994)
Lemma of monotony
Skip details
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
24 / 47
Large deviations
Cépa (1994)
Lemma of monotony
Cépa (1994)
Let A be a multivalued maximal monotone operator on Rd . Let (Y, K)
and (Y’, K’) be continuous functions on R+ with values in Rd , such that
1
K, K’ are bounded variation,
2
Y, Y’ are with values in D(A),
3
the measures
hYr − νr , dKr + υr dr i and hYr0 − νr , dKr0 + υr dr i
are almost surely negative on R+ for each pair of continuous
functions (ν, υ) satisfying
(νr , υr ) ∈ Gr (A), ∀r ∈ R+ .
Then hYr − Yr0 , dKr − dKr0 i is negative on R+ .
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
24 / 47
Large deviations
Proof of Theorem
Applying Itô’s formula to |Yts,x,ε − ψts,x |2
More details
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
25 / 47
Large deviations
Proof of Theorem
Using Lemma of monotony, assumptions (A1)-(A3), Young’s inequality,
2
2uv ≤ λu 2 + vλ for λ > 0
More details
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
26 / 47
Large deviations
Proof of Theorem
Taking λ ≥ 2 such that 2µ + (1 + λ)K 2 ≥ 0, by Gronwall’s lemma
More details
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
27 / 47
Large deviations
Proof of Theorem
Therefore, it follows from the Burkhölder-Davis-Gundy inequality that
E
sup
s≤t≤T
≤ CE
|Yts,x,ε
|XTs,x,ε
−
−
ψts,x |2
2
ϕs,x
T |
1
+
2
Z T
+
s
Z T
s
!
kZrs,x,ε k2 dr
!
|Xrs,x,ε
−
ϕrs,x |2 dr
.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
28 / 47
Large deviations
Large deviations for backward equation
We want to prove that the process Y s,x,ε satisfies, as ε goes to 0, a
LDP.
Definition
Let (Yts,x,ε , s ≤ t ≤ T ) and (ψts,x , s ≤ t ≤ T ) be the solutions of the
backward equations of systems (2.1) and (2.3) respectively. Let ε > 0,
we define the continuous functions u ε , u 0 with values in D(A) by
∆
u ε (t, x) = Ytt,x,ε , t ∈ [0, T ], x ∈ Rd
∆
u 0 (t, x) = ψtt,x , t ∈ [0, T ], x ∈ Rd
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
29 / 47
Large deviations
Large deviations for backward equation
We want to prove that the process Y s,x,ε satisfies, as ε goes to 0, a
LDP.
Definition
Let (Yts,x,ε , s ≤ t ≤ T ) and (ψts,x , s ≤ t ≤ T ) be the solutions of the
backward equations of systems (2.1) and (2.3) respectively. Let ε > 0,
we define the continuous functions u ε , u 0 with values in D(A) by
∆
u ε (t, x) = Ytt,x,ε , t ∈ [0, T ], x ∈ Rd
∆
u 0 (t, x) = ψtt,x , t ∈ [0, T ], x ∈ Rd
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
29 / 47
Large deviations
Large deviations for backward equation
We want to prove that the process Y s,x,ε satisfies, as ε goes to 0, a
LDP.
Definition
Let (Yts,x,ε , s ≤ t ≤ T ) and (ψts,x , s ≤ t ≤ T ) be the solutions of the
backward equations of systems (2.1) and (2.3) respectively. Let ε > 0,
we define the continuous functions u ε , u 0 with values in D(A) by
∆
u ε (t, x) = Ytt,x,ε , t ∈ [0, T ], x ∈ Rd
∆
u 0 (t, x) = ψtt,x , t ∈ [0, T ], x ∈ Rd
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
29 / 47
Large deviations
Large deviations for backward equation
We want to prove that the process Y s,x,ε satisfies, as ε goes to 0, a
LDP.
Definition
Let (Yts,x,ε , s ≤ t ≤ T ) and (ψts,x , s ≤ t ≤ T ) be the solutions of the
backward equations of systems (2.1) and (2.3) respectively. Let ε > 0,
we define the continuous functions u ε , u 0 with values in D(A) by
∆
u ε (t, x) = Ytt,x,ε , t ∈ [0, T ], x ∈ Rd
∆
u 0 (t, x) = ψtt,x , t ∈ [0, T ], x ∈ Rd
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
29 / 47
Large deviations
Large deviations for backward equation
We want to prove that the process Y s,x,ε satisfies, as ε goes to 0, a
LDP.
Definition
Let (Yts,x,ε , s ≤ t ≤ T ) and (ψts,x , s ≤ t ≤ T ) be the solutions of the
backward equations of systems (2.1) and (2.3) respectively. Let ε > 0,
we define the continuous functions u ε , u 0 with values in D(A) by
∆
u ε (t, x) = Ytt,x,ε , t ∈ [0, T ], x ∈ Rd
∆
u 0 (t, x) = ψtt,x , t ∈ [0, T ], x ∈ Rd
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
29 / 47
Large deviations
Large deviations for backward equation
Definition
Let s ∈ [0, T ] and ε ≥ 0, we define the following applications :
∆
F ε (φ) = [t 7−→ u ε (t, φt )], t ∈ [s, T ], φ ∈ C([s, T ], Rd )
Proposition
Y s,x,ε = F ε (X s,x,ε ), ψ s,x = F 0 (ϕs,x )
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
30 / 47
Large deviations
Large deviations for backward equation
Definition
Let s ∈ [0, T ] and ε ≥ 0, we define the following applications :
∆
F ε (φ) = [t 7−→ u ε (t, φt )], t ∈ [s, T ], φ ∈ C([s, T ], Rd )
Proposition
Y s,x,ε = F ε (X s,x,ε ), ψ s,x = F 0 (ϕs,x )
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
30 / 47
Large deviations
Large deviations for backward equation
Definition
Let s ∈ [0, T ] and ε ≥ 0, we define the following applications :
∆
F ε (φ) = [t 7−→ u ε (t, φt )], t ∈ [s, T ], φ ∈ C([s, T ], Rd )
Proposition
Y s,x,ε = F ε (X s,x,ε ), ψ s,x = F 0 (ϕs,x )
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
30 / 47
Large deviations
Large deviations for backward equation
Definition
Let s ∈ [0, T ] and ε ≥ 0, we define the following applications :
∆
F ε (φ) = [t 7−→ u ε (t, φt )], t ∈ [s, T ], φ ∈ C([s, T ], Rd )
Proposition
Y s,x,ε = F ε (X s,x,ε ), ψ s,x = F 0 (ϕs,x )
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
30 / 47
Large deviations
Proof of Proposition
We use a property of the SDE and the uniqueness of the solution of
the Multivalued BSDE.
More details
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
31 / 47
Large deviations
LDP for Multivalued BSDEs
Definition
Let Λ be the rate function associated to a large deviation principle for a
family of processes (X s,x,ε )ε>0 . For any Φ ∈ C([s, T ], D(A)), we set
∆
n
o
Π(Φ) = inf Λ(Ψ)| Φt = F 0 (Ψ)(t) = u 0 (t, Ψt ), ∀t ∈ [s, T ]
with the usual convention inf ∅ = +∞.
Theorem
Y s,x,ε satisfies, as ε goes to 0, a LDP with good rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
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Large deviations
LDP for Multivalued BSDEs
Definition
Let Λ be the rate function associated to a large deviation principle for a
family of processes (X s,x,ε )ε>0 . For any Φ ∈ C([s, T ], D(A)), we set
∆
n
o
Π(Φ) = inf Λ(Ψ)| Φt = F 0 (Ψ)(t) = u 0 (t, Ψt ), ∀t ∈ [s, T ]
with the usual convention inf ∅ = +∞.
Theorem
Y s,x,ε satisfies, as ε goes to 0, a LDP with good rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
32 / 47
Large deviations
LDP for Multivalued BSDEs
Definition
Let Λ be the rate function associated to a large deviation principle for a
family of processes (X s,x,ε )ε>0 . For any Φ ∈ C([s, T ], D(A)), we set
∆
n
o
Π(Φ) = inf Λ(Ψ)| Φt = F 0 (Ψ)(t) = u 0 (t, Ψt ), ∀t ∈ [s, T ]
with the usual convention inf ∅ = +∞.
Theorem
Y s,x,ε satisfies, as ε goes to 0, a LDP with good rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
32 / 47
Large deviations
LDP for Multivalued BSDEs
Definition
Let Λ be the rate function associated to a large deviation principle for a
family of processes (X s,x,ε )ε>0 . For any Φ ∈ C([s, T ], D(A)), we set
∆
n
o
Π(Φ) = inf Λ(Ψ)| Φt = F 0 (Ψ)(t) = u 0 (t, Ψt ), ∀t ∈ [s, T ]
with the usual convention inf ∅ = +∞.
Theorem
Y s,x,ε satisfies, as ε goes to 0, a LDP with good rate function Π.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
32 / 47
Large deviations
Proof of Theorem
To apply the contraction principle, we need to prove the following
Lemmas
Lemma 1
For any ε > 0, F ε is continuous.
Lemma 2
F ε converges uniformly to F 0 on every compact subset of C([s, T ], Rd ).
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
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Large deviations
Proof of Theorem
To apply the contraction principle, we need to prove the following
Lemmas
Lemma 1
For any ε > 0, F ε is continuous.
Lemma 2
F ε converges uniformly to F 0 on every compact subset of C([s, T ], Rd ).
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
33 / 47
Large deviations
Proof of Theorem
To apply the contraction principle, we need to prove the following
Lemmas
Lemma 1
For any ε > 0, F ε is continuous.
Lemma 2
F ε converges uniformly to F 0 on every compact subset of C([s, T ], Rd ).
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
33 / 47
Large deviations
Proof of Theorem
To apply the contraction principle, we need to prove the following
Lemmas
Lemma 1
For any ε > 0, F ε is continuous.
Lemma 2
F ε converges uniformly to F 0 on every compact subset of C([s, T ], Rd ).
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
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Large deviations
Proof of Lemma 1
Let (φn )n be a sequence in C([s, T ], Rd ), φ its limit for the uniform
norm, and ζ > 0.
More details
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
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Large deviations
Proof of Lemma 1
Since u ε is continuous, it follows that u ε is uniformly continuous on any
compact subset of [s, T ] × B(0, M).
More details
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
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Large deviations
Proof of Lemma 2
Let K be a compact subset of C([s, T ], Rd ), and φ ∈ K.
More details
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Large deviations for multivalued BSDEs
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Large deviations
Proof of Lemma 2
Since φ is continuous, it follows that L = {φr : φ ∈ K, r ∈ [s, T ]} is a
compact subset of Rd .
More details
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Large deviations
Proof of Theorem
In view of Lemmas 1 and 2, the proof of theorem is completed by
virtue of the contraction principle.
I. DAKAOU (UM)
Large deviations for multivalued BSDEs
Abidjan March 27, 2014
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Large deviations
Proof of Theorem
In view of Lemmas 1 and 2, the proof of theorem is completed by
virtue of the contraction principle.
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Large deviations for multivalued BSDEs
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END
Thanks
Thank you very much ! ! !
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Proof of Theorem
Applying Itô’s formula to |Yts,x,ε − ψts,x |2
|Yts,x,ε − ψts,x |2 +
Z T
t
kZrs,x,ε k2 dr
2
= |g(XTs,x,ε ) − g(ϕs,x
T )|
Z T
+2
t
−2
Z T
t
Z T
+2
t
s,x
hYrs,x,ε − ψrs,x , f (r , Xrs,x,ε , Yrs,x,ε , Zrs,x,ε ) − f (r , ϕs,x
r , ψr , 0)idr
hYrs,x,ε − ψrs,x , Zrs,x,ε dWr i
hYrs,x,ε − ψrs,x , dKrs,x,ε − dKrs,x i.
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Proof of Theorem
Using Lemma of monotony, assumptions (A1)-(A3), Young’s inequality,
2
2uv ≤ λu 2 + vλ for λ > 0
|Yts,x,ε
E
≤ E
−
|g(XTs,x,ε )
+E
ψts,x |2
−
Z T
+
t
2
g(ϕs,x
T )|
Z T
(2µ + (1 + λ)K
t
+E
!
kZrs,x,ε k2 dr
Z T
1
t
λ
2
)|Yrs,x,ε
−
ψrs,x |2
+
|Xrs,x,ε
−
2
ϕs,x
r |
!
dr
!
kZrs,x,ε k2 dr
.
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Proof of Theorem
Taking λ ≥ 2 such that 2µ + (1 + λ)K 2 ≥ 0, by Gronwall’s lemma
sup E
s≤t≤T
≤ CE
|Yts,x,ε
|XTs,x,ε
Z T
+CE
s
−
−
ψts,x |2
2
ϕs,x
T |
1
+ E
2
Z T
s
!
kZrs,x,ε k2 dr
!
|Xrs,x,ε
−
2
ϕs,x
r | dr
.
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Proof of Proposition
We use a property of the SDE and the uniqueness of the solution of
the Multivalued BSDE.
t,Xts,x,ε ,ε
Yrs,x,ε = Yr
, s≤t ≤r ≤T
Taking r = t, we deduce that
Yts,x,ε = u ε (t, Xts,x,ε )
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Proof of Lemma 1
Let (φn )n be a sequence in C([s, T ], Rd ), φ its limit for the uniform
norm, and ζ > 0.
Then, there exists M > 0 such that for all n ∈ N,
kφn k∞ ≤ M, kφk∞ ≤ M
where
∆
kθk∞ = sup |θr |, ∀θ ∈ C([s, T ], Rd ).
r ∈[s,T ]
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Proof of Lemma 1
Since u ε is continuous, it follows that u ε is uniformly continuous on any
compact subset of [s, T ] × B(0, M).
There exists η > 0 such that |r − r 0 | < η and |z − z 0 | < η,
z, z 0 ∈ B(0, M) imply
|u ε (r , z) − u ε (r 0 , z 0 )| ≤ ζ
There exists n0 such that ∀ n ≥ n0 , kφn − φk∞ ≤ η. Therefore,
|u ε (r , φnr ) − u ε (r , φr )| ≤ ζ
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Proof of Lemma 2
Let K be a compact subset of C([s, T ], Rd ), and φ ∈ K.
kF ε (φ) − F 0 (φ)k∞ =
sup |F ε (φ)(r ) − F 0 (φ)(r )|
r ∈[s,T ]
=
sup |u ε (r , φr ) − u 0 (r , φr )|
r ∈[s,T ]
=
sup |Yrr ,φr ,ε − ψrr ,φr |.
r ∈[s,T ]
So,
kF ε (φ) − F 0 (φ)k2∞ = sup |Yrr ,φr ,ε − ψrr ,φr |2 .
r ∈[s,T ]
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Proof of Lemma 2
Since φ is continuous, it follows that L = {φr : φ ∈ K, r ∈ [s, T ]} is a
compact subset of Rd .
Thanks to Corollary, there exists a constant C depends only of T and
L, such that for every r ∈ [s, T ], all x ∈ L, all ε > 0, we have
|Yrr ,x,ε − ψrr ,x |2 = E |Yrr ,x,ε − ψrr ,x |2
!
≤ E
sup
t∈[r ,T ]
|Ytr ,x,ε
−
ψtr ,x |2
≤ Cε
Consequently, sup kF ε (φ) − F 0 (φ)k2∞ ≤ Cε.
φ∈K
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