Viscosity solutions for second order IPDEs
without monotonicity condition: a new
result
M-A Morlais (j.w.w. S. Hamadène)
LMM, University of Le Mans, France.
Workshop on Stochastic Analysis & Related Topics
University of Jyväskylä, Finlande 21 & 22/12/2015.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
0. Outlines
1
Standard IPDEs & link with BSDEs with jumps ;
2
Some applications in control problem and/or nance
3
Another framework for IPDEs: The IPDE without
monotonicity condition ;
4
The main result: Existence and uniqueness of the solution ;
5
Extensions and perspectives ;
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
1. Standard IPDEs
Let us consider the following IPDE:

2

−∂t u(t, x) − b(t, x)> Dx u(t, x) − 21 Tr σσ > (t, x)Dxx
u(t, x) − Ku(t, x)





−h(t, x, u(t, x), (σ > Dx u)(t, x), Bu(t, x)) = 0, (t, x) ∈ [0, T ] × Rk ;





 u(T , x) = g (x)
(1)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
where: K (u) and B(u) are non local operators such that:
(i) The operators B and K are given by
Bu(t,
x) :=
R
E γ(t, x, e) (u(t, x + β(t, x, e)) − u(t, x)) λ(de) ;
Ku(t,
x) :=
R
u(t,
x + β(t, x, e)) − u(t, x) − β(t, x, e)> Dx u(t, x) λ(de).
E
(2)
(ii) λ is a Lévy measure on E = R` − {0} ;
(iii) β(t, x, e) , γ(t, x, e) given functions ;
(iv) b(t, x), σ(t, x), h(t, x, y , z, ζ) and g (x) are also given
functions ;
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Remarks:
(i) This IPDE is of non local type due to the operators Bu and Ku ;
(ii) Bu and Ku are dened only if u is regular enough.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Remarks:
(i) This IPDE is of non local type due to the operators Bu and Ku ;
(ii) Bu and Ku are dened only if u is regular enough.
In general,"regularity" in (ii) means
(a) u continuous in (t, x ),
(b) u loc. Lipschitz w.r.t. x (uniformly in t ) and/or + with (at
most) polynomial growth w.r.t x ;
Requirements Add (growth + integrab.) conditions on β , γ
especially if λ(E ) = +∞ (λ: Lévy measure).
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Barles et al.'s denition of a viscosity solution of
(1)
Denition
A continuous function u(t, x) is a viscosity sub-solution (resp.
super-solution) of the IPDE (1) if:
(i) ∀x ∈ Rk , u(T , x) ≤ g (x) (resp. u(T , x) ≥ g (x)) ;
(ii) For any (t, x) ∈ (0, T ) × Rk and any function φ of class
C , ([0, T ] × Rk ) such that (t, x) is a global maximum (resp.
minimum) point of u − φ and (u − φ)(t, x) = 0, one has
1 2
−∂t φ(t, x) − LX φ(t, x)
−h(t, x, u(t, x), σ > (t, x)Dx φ(t, x), Bφ(t, x)) ≤ 0 ;
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Denition (continued)
(resp.
−∂t φ(t, x) − LX φ(t, x)
−h(t, x, u(t, x), σ > (t, x)Dx φ(t, x), Bφ(t, x)) ≥ 0 ; )
(iii) The function u(t, x) is a viscosity solution of (1) if it is both a
viscosity sub-solution and viscosity super-solution.
where
LX φ(t, x) :=
1
2
Tr[(σσ > )(t, x)Dxx φ(t, x)] + b(t, x)> Dx φ(t, x) + K φ(t, x).
2
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
2. Connection with BSDEs with jumps
Let (t, x) ∈ [0, T ] × Rk and (Xst,x )s≤T be the solution of the
following standard SDE of diusion-jump type:
Xst,x = x +
+
Rs
t
b(r , Xrt,x )dr +
Rs
t
σ(r , Xrt,x )dBr
t,x
t,x
= x if s ≤ t
E β(r , Xr − , e)µ̃(dr , de), s ∈ [t, T ]; Xs
Rs R
t
(3)
where:
(i) B := (Bt )t≥ a d -dimensional Brownian motion ;
0
(ii) µ is an independant Poisson random measure on R+ × E with
compensator ν(dt, de) = dtλ(de) ; λ is σ -nite measure on (E , E)
and integrating the function (1 ∧ |e| )e∈E .
2
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
(iii) β : (t, x, e) ∈ [0, T ] × Rk × E → β(t, x, e) ∈ Rk a function
such that,
(a) |β(t, x, e)| ≤ C (1 ∧ |e|);
(b) |β(t, x, e) − β(t, x 0 , e)| ≤ C |x − x 0 |(1 ∧ |e|);
(c) the mapping (t, x) ∈ [0, T ] × Rk → β(t, x, e) ∈ Rk
is continuous uniformly wrt e.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
(4)
Next let us consider the following BSDE with jumps:

RT
t,x
t,x
t,x
t,x
t,x
t,x

 Ys = g (XT ) + s f (r , Xr , Yr , Zr , Ur )dr


−
RT
s
Zrt,x dBr
−
RT R
t,x
E Ur (e)µ̃(dr , de),
s
(5)
s ≤ T.
Theorem (Tang-Li '94, Barles et al. '97)
If :
(i) f (t, x, y , z, u) is measurable and Lipschitz in
(y , z, u) ∈ R1+d × L2 (E , λ) uniformly wrt to (t, x) ;
(ii) g (XTt,x ) is square integrable.
Then the solution (Y t,x , Z t,x , U t,x ) of (5) exists and is unique such
that
E [ sup
s≤T
|Yst,x |2
Z
+
T
|Zst,x |2 ds
Z
+
0
M-A Morlais (j.w.w. S. Hamadène)
T
Z
ds
0
|U t,x (s, e)|2 λ(de)] < ∞.
E
Viscosity solutions for second order IPDEs
Assumption [A1]
(i) (a) the function h is continuous and
(y , z, v ) ∈ R1+d × L2 (E , λ) 7→ h(t, x, y , z,
R
E
γ(t, x, e)v (e)λ(de))
is Lipschitz ;
(b) the mapping ζ 7→ h(t, x, y , z, ζ) is non-decreasing ;
(c) the mapping (t, x) 7→ h(t, x, 0, 0, 0) is of polynomial
growth ;
(ii) g is continuous and of polynomial growth ;
(iii) the function γ(t, x, e) veries:
(a) |γ(t, x, e)| ≤ C (1 ∧ |e|);
(b) |γ(t, x, e) − γ(t, x 0 , e)| ≤ C |x − x 0 |(1 ∧ |e|);
(c) the mapping (t, x) ∈ [0, T ] × Rk → γ(t, x, e) ∈ Rk
is continuous uniformly wrt e;
(d) γ(t, x, e) ≥ 0.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
(6)
The link between IPDE (1) and BSDEs with jumps of type (5) is
given by:
Theorem (BBP '97)
Assume [A1] and let (Y t,x , Z t,x , U t,x ) be the solution of
 t,x
Ys = g (XTt,x )+





 R
R
T
h(r , Xrt,x , Yrt,x , Zrt,x , E γ(r , Xrt,x , e)Urt,x (e)λ(de))dr
s





RT R
RT

− s Zrt,x dBr − s E Urt,x (e)µ̃(dr , de), s ≤ T .
(7)
Then there exists a deterministic continuous function with
polynomial growth u(t, x) such that
∀s ∈ [t, T ], Yst,x = u(s, Xst,x ).
(8)
Moreover u(t, x) is the unique viscosity solution of the IPDE (5) in
the class of continuous functions with polynomial growth.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
3. BSDE (with jumps)/ connections with
stochastic control problem
1
2
"Classical" stochastic control problem
in a Brownian ltration: (Ω, F, P, Ft , B)):
Let X u controlled SDE s.t.
Xtu
Z
=x+
t
b(s, Xsu )ds
0
Z
+
t
σ(s, Xsu )dBsu
0
with B u new Brownian
Motion under probab. measure Pu with
Rt
u
density Zt = Et ( us dBs ) Let
0
V := sup J(u) = sup EP
u∈A
u
u
Z
g (XTu ) +
T
f (s, Xsu )ds .
0
u 7→ J(u): objective functional; u := (u(s, ω)) the control, A
admissible set of controls.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
3. BSDE (with jumps)/ connections with
stochastic control problem
1
Let Ytu solving the BSDE
dYsu = −f (s, Xsu )ds + Zsu dBsu , 0 ≤ s ≤ T , YTu = g (XTu ).
2
Y0 = ess supu Y0u = V = sup J(u).
u
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
3. BSDE (with jumps)/ connections with
stochastic control problem
1
Let Ytu solving the BSDE
dYsu = −f (s, Xsu )ds + Zsu dBsu , 0 ≤ s ≤ T , YTu = g (XTu ).
2
Y0 = ess supu Y0u = V = sup J(u).
u
More generally: let
and terminal conditions
(f α )
and
(ξ α )
family of controlled drivers
Yt (sup f α , sup ξ α ) := ess supYt (f α , ξ α )
α
3
Extensions to the case of BSDE with jumps:
Comparison theorem and applications: Becherer ('06) Royer
('06) Sulem-Oksendal ('09) Sulem-Quenez (INRIA Report '13)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
3. BSDE (with jumps)/ connections with nance
Dual representation of convex dynamic risk measure
(Sulem-Quenez '13):
Set-up: Wiener Poisson ltration
f a given "driver" F (ω, t, · · · ) its polar function
F (ω, t, α1 , α2 ) :=
sup
(π,l)∈R2 ×L2
[f (ω, t, π, l) − α1 π − hα2 , liν ]
ν
For any α := (α , α ), (α "admissible " pair of proc.) let Z α be the
density dened by
1
Ztα
2
Z
:= E· (
1
αs dWs +
[0,t]
Z
[0,t]×R∗
αs2 (e)Ñ(ds, de))
and (Qα )α family of prob. measures.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
2bis. BSDE (with jumps)/ connections with
nance
Theorem
Assume L2ν separable. f Lipschitz driver independent of x
continuous and concave w.r.t (π, l).
Let ρ(., T ) be the convex risk measure
Z
ρ(ξ, T ) := sup EQα −ξ −
α
T
F (s, αs , αs )ds ,
1
2
0
then for each ξ in L2 (FT ) there exists some ᾱ admissible
F (ω, t, α¯1 , α¯2 ) := f (t, π, l) − α¯1 π − hα¯2 , liν
with the triple (Y , π, l ) solving BSDE with concave driver f and
terminal condition ξ .
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
3. BSDE (with jumps)/ connections with
stochastic control or nance
1
Main tool: the comparison theorem for BSDE (with or without
jumps)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
3. BSDE (with jumps)/ connections with
stochastic control or nance
1
2
Main tool: the comparison theorem for BSDE (with or without
jumps)
Advantages of using BSDEs:
(i) Obtention of the so-called DPP (Dynamic Programming
Principle) in stoch. control problems;
(ii) Very general results (possibly in non Markovian setting);
(iii) Extensions to reected BSDE and BSDE with contraints
either on Z or on U itself, eventually second order BSDEs with
jumps (in non dominated models),...
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
3. The IPDE (1) without monotonicity condition
Objective: Weaken the monotonicity condition of h wrt to ζ and
γ ≥ 0.
3.1 First case
Assumption [A2]:
(i) The Lévy measure λ is nite i.e. λ(E ) < ∞ (as a rst step).
(ii) the same assumptions as in [A1] but without monotonicity of h
wrt ζ neither γ ≥ 0.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Proposition
Assume [A2]. Then:
(i) There exists (Y t,x , Z t,x , U t,x ) solution of the BSDEJ:
 t,x
Ys = g (XTt,x )+





 R
R
T
h(r , Xrt,x , Yrt,x , Zrt,x , E γ(r , Xrt,x , e)Urt,x (e)λ(de))dr
s





RT
RT R

− s Zrt,x dBr − s E Urt,x (e)µ̃(dr , de), s ≤ T .
(9)
(ii) There exists a deterministic continuous function with
polynomial growth u(t, x) such that
∀s ∈ [t, T ], Yst,x = u(s, Xst,x ).
(10)
(iii) ds ⊗ dP ⊗ dλ on [t, T ] × Ω × E ,
t,x
t,x
t,x
Ust,x (e) = u(s, Xs−
+ β(s, Xs−
, e)) − u(s, Xs−
).
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
(11)
Sketch of the proof
(i)-(ii) Existence is obtained through the result by Tang-Li or Barles
et al.. Polynomial growth and continuity are obtained as in Barles
et al..
(iii)
(a) Since u is polynomial growth and β bounded then
Z TZ
E[
0
t,x
t,x
t,x 2
{|Ust,x (e)|2 +|u(s, Xs−
+β(s, Xs−
, e)−u(s, Xs−
)| }λ(de)ds] < ∞
E
since X t,x has uniform moments of any order.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
As λ(E ) is nite then
Z TZ
t,x
t,x
t,x
E[
{|Ust,x (e)|+|u(s, Xs−
+β(s, Xs−
, e)−u(s, Xs−
)|}λ(de)ds] < ∞.
0
E
Therefore ∀s ∈ [t, T ],
Z sZ
t
Urt,x (e)µ̃(dr , de)
Z sZ
Z sZ
t,x
=
Ur (e)µ(dr , de)−
Urt,x (e)λ(de)dr .
E
t
E
t
E
But Y t,x is solution of the BSDEJ (9) then for any s ∈ [t, T ],
X
{Yrt,x
−
Yrt,x
−}
Z
s
Z
=
t<r ≤s
M-A Morlais (j.w.w. S. Hamadène)
t
Urt,x (e)µ(dr , de).
E
Viscosity solutions for second order IPDEs
On the other hand Yst,x = u(s, Xst,x ) and u is continuous then
Rs R
t,x
E (u(r , Xr −
t
t,x
+ β(s, Xs−
, e)) − u(r , Xrt,x
− ))µ(dr , de)
=
t,x
t<r ≤s {Yr
P
− Yrt,x
− }, s ∈ [t, T ].
It follows that ∀s ∈ [t, T ],
Z
s
Z
t
E
t,x
t,x
t,x
(u(r , Xrt,x
− +β(r , Xr − , e))−u(r , Xr − )−Ur (e))µ(dr , de) = 0.
Taking quadratic variation of this process + expectation,
Z
E[
t
T
Z
dr
E
t,x
t,x
t,x
2
|u(r , Xrt,x
− +β(r , Xr − , e))−u(r , Xr − )−Ur (e)| λ(de)] = 0
gives the desired equality.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Second denition of a viscosity solution
Denition
A deterministic continuous function u(t, x) is a viscosity
sub-solution (resp. super-solution) of the IPDE (1) if:
(i) ∀x ∈ Rk , u(T , x) ≤ gi (x) (resp. u(T , x) ≥ gi (x)) ;
(ii) For any (t, x) ∈ (0, T ) × Rk and any function φ of class
C , ([0, T ] × Rk ) such that (t, x) is a global maximum (resp.
minimum) point of u − φ and (u − φ)(t, x) = 0, one has
1 2
−∂t φ(t, x) − LX φ(t, x)
−h(t, x, u(t, x), σ > (t, x)Dx φ(t, x), Bu(t, x)) ≤ 0 (resp. ≥ 0).
(iii) The function u(t, x) is a viscosity solution of (1) if it is both a
viscosity sub-solution and viscosity super-solution.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Remark: When h does not depend on Bu , the two denitions
coincide.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
The rst result
Theorem
Assume [A2]. Then the function u(t, x) of (10) is a viscosity
solution of (1). Moreover it is unique in the class of continuous
functions with polynomial growth.
Sketch of the proof: Existence Let us consider the following
BSDEJ:
 t,x
YT = g (XTt,x ) and ∀s ≤ T ,






R t,x


t,x
t,x

 d Ys = Zs dBs + E Us (e)µ̃(ds, de)

t,x


−h(s, Xst,x , Yt,x

s , Zs ,





 R
t,x
t,x
t,x
t,x
E γ(s, Xs , e){u(s, Xs− + β(s, Xs− , e)) − u(s, Xs− )}λ(de))ds.
(12)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
The solution of this equation exists and is unique.
Besides the generator of this BSDEJ does not depend on the jump
part and since
Z
(t, x, y , z) 7−→ h(t, x, y , z,
γ(t, x, e){u(t, x+β(t, x, e))−u(t, x)}λ(de))
E
verify [A1] then there exists a deterministic continuous function of
polynomial growth u such that for any (t, x) ∈ [0, T ] × Rk ,
t,x
∀s ∈ [t, T ], Yt,x
s = u(s, Xs ).
Then u is the unique viscosity solution of

2

(t, x)
−∂t u(t, x) − b(t, x)> Dx u(t, x) − 12 Tr σσ > (t, x)Dxx
u





K u(t, x) − h(t, x, u(t, x), (σ > Dx u)(t, x), Bu(t, x)) = 0, (t, x) ∈ [0, T ]





 u(T , x) = g (x)
(13)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Next (Y t,x , Z t,x , U t,x ) verify:
 t,x
Ys = g (XTt,x )+





 R
R
T
h(r , Xrt,x , Yrt,x , Zrt,x , E γ(r , Xrt,x , e)Urt,x (e)λ(de))dr
s





RT
RT R

− s Zrt,x dBr − s E Urt,x (e)µ̃(dr , de), s ≤ T .
and
(14)
t,x
t,x
t,x
Ust,x (e) = u(s, Xs−
+ β(s, Xs−
, e)) − u(s, Xs−
)
Plug this in the second term of the previous BSDEJ + use
uniqueness of the sol. for such BSDEJ to get
t,x
∀s ∈ [t, T ], Yt,x
s = Ys .
and then
u=u
which implies that u is a viscosity solution of (1) in the sense of the
second denition, i.e.,
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Uniqueness
Aim: Show uniqueness inside the class of continuous functions with
polynomial growth. Let ū be another sol. of the IPDE (1) in the
sense of the second denition.

2

−∂t ū(t, x) − b(t, x)> Dx ū(t, x) − 21 Tr σσ > (t, x)Dxx
ū(t, x)





−K ū(t, x) − h(t, x, ū(t, x), (σ > Dx ū)(t, x), B ū(t, x)) = 0;





 ū(T , x) = g (x)
(15)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Let us consider the following BSDEJ:
 t,x
ȲT = g (XTt,x ) and ∀s ≤ T ,







t,x
t,x
t,x
t,x


 d Ȳs = −h(s, Xs , Ȳs , Z̄s ,
R

t,x
t,x
t,x
t,x



E γ(s, Xs , e){ū(s, Xs− + β(s, Xs− , e)) − ū(s, Xs− )}λ(de))ds





R

+Z̄st,x dBs + E Ūst,x (e)µ̃(ds, de).
(16)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Therefore, there exists a deterministic continuous function v (t, x)
of PG such that
∀s ∈ [t, T ], Ȳst,x = v (s, Xst,x ).
and
t,x
t,x
t,x
Ūst,x (e) = v (s, Xs−
+ β(s, Xs−
, e)) − v (s, Xs−
)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Additionally by Barles et al.'s result v (t, x) is the unique solution in
the class of continuous functions of PG of:

2

−∂t v (t, x) − b(t, x)> Dx v (t, x) − 12 Tr σσ > (t, x)Dxx
v (t, x)





−Kv (t, x) − h(t, x, v (t, x), (σ > Dx v )(t, x), B ū(t, x)) = 0;





 v (T , x) = g (x)
(17)
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Finally by the uniqueness of the solution of this latter we have
v = ū and then
t,x
t,x
t,x
Ūst,x (e) = v (s, Xs−
+ β(s, Xs−
, e)) − v (s, Xs−
)
t,x
t,x
t,x
= ū(s, Xs−
+ β(s, Xs−
, e)) − ū(s, Xs−
).
Plug now this equality in the BSDEJ (16) and using
To deduce by uniqueness of the solution of the BSDEJ that
Ȳ t,x = Y t,x
and then u = ū .
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
The second result
3.2 Second case (Hamadène - August 2015)
Assumption [A3]:
(i) The Lévy measure λ is not necessarily nite ;
(ii) the same assumptions as in [A1] but without monotonicity of h
wrt ζ neither γ ≥ 0.
(iii) The functions g and x ∈ Rk 7→ h(t, x, y , z, ζ) verify:
|Φ(x) − Φ(x 0 )| ≤ C (1 + |x|p + |x 0 |p )|x − x 0 |.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
The second result: main estimate
Proposition
Let u be the deterministic function dened in (7), then
|u(t, x) − u(t, x 0 )| ≤ C (1 + |x|p + |x 0 |p )|x − x 0 |, ∀t, x, x 0 .
For later use let dene U
U := {u : (t, x) 7→ u(t, x) continuous and verifying (18)}.
Proof: Just apply Itô's formula with
0
(Yst,x − Yst,x )2 , s ≤ T .
and use the properties of [A3] and u(t, x) = Ytt,x .
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
(18)
The second result (cont')
Denition (ii)
A deterministic continuous function u(t, x) of U is a viscosity
sub-solution (resp. super-solution) of the IPDE (1) if:
(i) ∀x ∈ Rk , u(T , x) ≤ gi (x) (resp. u(T , x) ≥ gi (x)) ;
(ii) For any (t, x) ∈ (0, T ) × Rk and any function φ of class
C , ([0, T ] × Rk ) such that (t, x) is a global maximum (resp.
minimum) point of u − φ and (u − φ)(t, x) = 0, one has
1 2
−∂t φ(t, x) − LX φ(t, x)
−h(t, x, u(t, x), σ > (t, x)Dx φ(t, x), Bu(t, x)) ≤ 0 (resp. ≥ 0).
(iii) The function u(t, x) is a viscosity solution of (1) if it is both a
viscosity sub-solution and viscosity super-solution.
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
The second result (cont')
Theorem
Assume [A3]. Then the function u(t, x) of (10) is a viscosity
solution of (1) (in the sense of Denition (ii)). Moreover it is
unique in the class U .
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Some extensions and perspectives
1
(Theoretical) extensions:
(i) IPDEs with one (already done) or two (to be developed)
obstacles;
(ii) Some multidimensional cases ;
(Application: connection with optimal switching problems..).
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs
Some extensions and perspectives
1
(Theoretical) extensions:
(i) IPDEs with one (already done) or two (to be developed)
obstacles;
(ii) Some multidimensional cases ;
(Application: connection with optimal switching problems..).
2
Perspectives to develop:
(i) Applications both in economics and nance: (ii) Numerics
for IPDEs (using the FK representation).
M-A Morlais (j.w.w. S. Hamadène)
Viscosity solutions for second order IPDEs