stochastic pdeâ s on networks with non local boundary...
TRANSCRIPT
Stochastic PDEs on networks with non–localboundary conditions and application to finance
F. Cordoni, University of Verona - HPA s.r.l.
December 20, 2017,Opening conference VPSMS, Verona
Bibliography
[1] F. C. and L. Di Persio, Gaussian estimates on networks with dynamicstochastic boundary conditions, Infinite Dimensional Analysis, QuantumProbability and Related Topics, 20, (2017): 1750001;[2] F. C. and L. Di Persio, Stochastic reaction–diffusion equations onnetworks with dynamic time–delayed boundary conditions, Journal ofMathematical Analysis and Applications, (2017), 1, 583-603.
Outline
Main motivations
Notation
Non-local Kirchhoff conditionThe perturbed non–linear stochastic problem
Time–Delayed Kirchhoff conditionThe perturbed non–linear stochastic problem
Application to optimal control
Financial applications
Main motivations
Notation
Non-local Kirchhoff conditionThe perturbed non–linear stochastic problem
Time–Delayed Kirchhoff conditionThe perturbed non–linear stochastic problem
Application to optimal control
Financial applications
Main motivations
(i) quantum mechanics;
(ii) electrical circuits;
(iii) traffic flow;
(iv) neurobiology;
(v) smart grid optimization;
(vi) system of interconnected banks.
Main motivations
Notation
Non-local Kirchhoff conditionThe perturbed non–linear stochastic problem
Time–Delayed Kirchhoff conditionThe perturbed non–linear stochastic problem
Application to optimal control
Financial applications
Notation
Let us consider a graph G with:
n ∈ N vertices V = v1, . . . , vn
m ∈ N edges E = e1, . . . , em
Greek letters for vertices vα, vβ , vγ
Latin letters for edges ei , ej , ek
vα
vβ
vγ
ek
ei
ej
Notation
Incidence matrix
I = (ιve) , ιve =
1 · e−−−−→ v ,
−1 ve−−−−→ ·
0 otherwise ;
Adjacency matrix
A = (avw ) , avw =
1 v
e−−−−→ w ,
1 we−−−−→ v ,
0 otherwise ;
vα
vβ
vγ
ek
ei
ej
Notation
Incidence matrix
I = (ιve) , ιve =
1 · e−−−−→ v ,
−1 ve−−−−→ ·
0 otherwise ;
Adjacency matrix
A = (avw ) , avw =
γ(e) v
e−−−−→ w ,
γ(e) we−−−−→ v ,
0 otherwise ;
vα
vβ
vγ
ek
ei
ej
Consider a diffusion equation on thegraph G
uj(t, x) = ∆uj(t, x) ,
uj(t, x) , on the edge ej ,
vα
vβ
vγ
ek
ei
ej
Main aim
Write the diffusion equation as an abstract operatorial problem
Boundary conditions?
Main aim
Write the diffusion equation as an abstract operatorial problem
Boundary conditions?
Continuity in the nodes
uj(t, vα) = ui (t, vα) =: duα(t) , i , j ∈ Γ(vα) ,
Kirchhoff condition∑j∈Γ(vα)
ιαju′j (t, vα) = 0 .
Continuity in the nodes
uj(t, vα) = ui (t, vα) =: duα(t) , i , j ∈ Γ(vα) ,
Kirchhoff condition∑j∈Γ(vα)
ιαju′j (t, vα) = 0 .
Main motivations
Notation
Non-local Kirchhoff conditionThe perturbed non–linear stochastic problem
Time–Delayed Kirchhoff conditionThe perturbed non–linear stochastic problem
Application to optimal control
Financial applications
Reaction–diffusion equation
uj(t, x) =∑m
i=1 (ciju′i )′ (t, x) +
∑mi=1 pijui (t, x) ,
uj(t, vα) = ul(t, vα) =: duα(t) ,∑n
β=1 bαβduβ(t) =
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vα) , α = n0 + 1, . . . , n ,
duα(t) = −
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vβ) +
∑nβ=1 bαβd
uβ(t) , α = 1, . . . , n0 ,
uj(0, x) = u0j (x) ,
dui (0) = d0
i , i = 1, . . . , n0 ,
Continuity condition
uj(t, x) =∑m
i=1 (ciju′i )′ (t, x) +
∑mi=1 pijui (t, x) ,
uj(t, vα) = ul(t, vα) =: duα(t) ,∑n
β=1 bαβduβ(t) =
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vα) , α = n0 + 1, . . . , n ,
duα(t) = −
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vβ) +
∑nβ=1 bαβd
uβ(t) , α = 1, . . . , n0 ,
uj(0, x) = u0j (x) ,
dui (0) = d0
i , i = 1, . . . , n0 ,
Generalized non–local Kirchhoff condition
uj(t, x) =∑m
i=1 (ciju′i )′ (t, x) +
∑mi=1 pijui (t, x) ,
uj(t, vα) = ul(t, vα) =: duα(t) ,∑n
β=1 bαβduβ(t) =
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vα) , α = n0 + 1, . . . , n ,
duα(t) = −
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vβ) +
∑nβ=1 bαβd
uβ(t) , α = 1, . . . , n0 ,
uj(0, x) = u0j (x) ,
duα(0) = d0
α , α = 1, . . . , n0 ,
Dynamic non–local Kirchhoff condition
uj(t, x) =∑m
i=1 (ciju′i )′ (t, x) +
∑mi=1 pijui (t, x) ,
uj(t, vα) = ul(t, vα) =: duα(t) ,∑n
β=1 bαβduβ(t) =
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vα) , α = n0 + 1, . . . , n ,
duα(t) = −
∑mi,j=1
∑nβ=1 δ
αiβj u′j (t, vα) +
∑nβ=1 bαβd
uβ(t) , α = 1, . . . , n0 ,
uj(0, x) = u0j (x) ,
duα(0) = d0
α , α = 1, . . . , n0 ,
The abstract setting
X 2 :=(L2([0, 1])
)m, Rn ,
X 2 := X 2 × Rn ,
⟨(u
du
),
(v
dv
)⟩X 2
:=m∑j=1
∫ 1
0
uj(x)vj(x)dx +n∑
α=1
duαd
vα ,
The differential operator
Au =
(c1,1u′1)′ + p1,1u1 . . . (c1,mu
′1)′ + p1,mum
.... . .
...(cm,1u
′1)′
+ pm,1u1 . . . (cm,mu′m)′
+ pm,mum
,
with domain
D(A) =u ∈
(H2(0, 1)
)m: ∃ du(t) ∈ Rn s.t.
(Φ+)T
du(t) = u(0) ,(Φ−)T
du(t) = u(1) , Φ+δ u′(0)− Φ−δ u
′(1) = B2du(t)
.
Operator matrix
A =
(A 0C B1
),
with
D(A) =
(u
du
)∈ D(A)× Rn : ui (vα) = du
α , ∀ i ∈ Γ(vα), α = 1, . . . , n
.
C : D(C ) := D(A)→ Rn the feedback operator
Cu :=
− m∑i,j=1
n∑β=1
δ1iβju′j (v1), . . . ,−
m∑i,j=1
n∑β=1
δn0iβj u′j (vn0 ), 0, . . . , 0
T
,
Operator matrix
A =
(A 0C B1
),
with
D(A) =
(u
du
)∈ D(A)× Rn : ui (vα) = du
α , ∀ i ∈ Γ(vα), α = 1, . . . , n
.
C : D(C ) := D(A)→ Rn the feedback operator
Cu :=
− m∑i,j=1
n∑β=1
δ1iβju′j (v1), . . . ,−
m∑i,j=1
n∑β=1
δn0iβj u′j (vn0 ), 0, . . . , 0
T
,
The abstract equation
u(t) = Au(t) , t ≥ 0 ,
u(0) = u0 ∈ X 2 .
Does A generate a C0−semigroup?
The abstract equation
u(t) = Au(t) , t ≥ 0 ,
u(0) = u0 ∈ X 2 .
Does A generate a C0−semigroup?
Define the sesquilinear form
a(u, v) := 〈Cu′, v ′〉2 − 〈Pu, v〉2 − 〈B1du, dv 〉n − 〈B2d
u, dv 〉n .
PropositionThe operator associated with the form a is the operator (A,D(A)).Also (A,D(A)) generates an analytic and compact C0−semigroup T (t)on X 2.
Define the sesquilinear form
a(u, v) := 〈Cu′, v ′〉2 − 〈Pu, v〉2 − 〈B1du, dv 〉n − 〈B2d
u, dv 〉n .
PropositionThe operator associated with the form a is the operator (A,D(A)).Also (A,D(A)) generates an analytic and compact C0−semigroup T (t)on X 2.
Gaussian upper bound
TheoremThe semigroup T (t), acting on the space X 2 and associated to a, isultracontractive, namely there exists a constant M > 0 such that
‖T (t)u‖X∞ ≤ Mt−14 ‖u‖X 2 , t ∈ [0,T ], u ∈ X 2 .
TheoremThe semigroup T (t) has an integral kernel Kt
[T (t)g ] (x) =
∫Ω
Kt(x , y)g(y)µ(dy) .
It holds the Gaussian upper bound
0 ≤ Kt(x , y) ≤ cδt− 1
2 e−|x−y|2σt .
Gaussian upper bound
TheoremThe semigroup T (t), acting on the space X 2 and associated to a, isultracontractive, namely there exists a constant M > 0 such that
‖T (t)u‖X∞ ≤ Mt−14 ‖u‖X 2 , t ∈ [0,T ], u ∈ X 2 .
TheoremThe semigroup T (t) has an integral kernel Kt
[T (t)g ] (x) =
∫Ω
Kt(x , y)g(y)µ(dy) .
It holds the Gaussian upper bound
0 ≤ Kt(x , y) ≤ cδt− 1
2 e−|x−y|2σt .
PropositionFor any t ≥ 0, the semigroup T (t) ∈ L2(X 2), moreover there existsM > 0 such that
|T (t)|HS ≤ Mt−14 .
The perturbed non–linear stochastic problem
uj (t, x) =∑m
i=1
(ciju
′i
)′(t, x) +
∑mi=1 pijui (t, x)
+fj (t, x , uj (t, x)) + gj (t, x , uj (t, x))W 1j (t, x) ,
uj (t, vα) = ul (t, vα) =: duα(t) ,∑n
β=1 bαβduβ(t) =
∑mi,j=1
∑nβ=1 δ
αiβj u
′j (t, vα) ,
duα(t) = −
∑mi,j=1
∑nβ=1 δ
αiβj u
′j (t, vβ) +
∑nβ=1 bαβd
uβ(t)+gα(t, du
α(t))W 2α(t, vα) ,
uj (0, x) = u0j (x) ,
duα(0) = d0
α ,
The perturbed non–linear stochastic problem
uj (t, x) =∑m
i=1
(ciju
′i
)′(t, x) +
∑mi=1 pijui (t, x)+
+fj (t, x , uj (t, x))+gj (t, x , uj (t, x))W 1j (t, x) ,
uj (t, vα) = ul (t, vα) =: duα(t) ,∑n
β=1 bαβduβ(t) =
∑mi,j=1
∑nβ=1 δ
αiβj u
′j (t, vα) ,
duα(t) = −
∑mi,j=1
∑nβ=1 δ
αiβj u
′j (t, vβ) +
∑nβ=1 bαβd
uβ(t)+gα(t, du
α(t))W 2α(t, vα) ,
uj (0, x) = u0j (x) ,
duα(0) = d0
α ,
The perturbed non–linear stochastic problem
uj (t, x) =∑m
i=1
(ciju
′i
)′(t, x) +
∑mi=1 pijui (t, x)+
+fj (t, x , uj (t, x)) + gj (t, x , uj (t, x))W 1j (t, x) ,
uj (t, vα) = ul (t, vα) =: duα(t) ,∑n
β=1 bαβduβ(t) =
∑mi,j=1
∑nβ=1 δ
αiβj u
′j (t, vα) ,
duα(t) = −
∑mi,j=1
∑nβ=1 δ
αiβj u
′j (t, vβ) +
∑nβ=1 bαβd
uβ(t) + gα(t, du
α(t))W 2α(t, vα) ,
uj (0, x) = u0j (x) ,
duα(0) = d0
α ,
The abstract equation
du(t) = [Au(t) + F (t,u(t))] dt + G (t,u(t))dW (t) , t ≥ 0 ,
u(0) = u0 ∈ X 2 ,
TheoremThere exists a unique mild solution in the sense that
u(t) = T (t)u0 +
∫ t
0T (t − s)F (s, u(s))ds +
∫ t
0T (t − s)G(s, u(s))dW (s) .
Proof.Main difficulty: treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
Standard assumption G (s,u(s)) ∈ L2(X 2)
TheoremThere exists a unique mild solution in the sense that
u(t) = T (t)u0 +
∫ t
0T (t − s)F (s, u(s))ds +
∫ t
0T (t − s)G(s, u(s))dW (s) .
Proof.Main difficulty: treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
Standard assumption G (s,u(s)) ∈ L2(X 2)
TheoremThere exists a unique mild solution in the sense that
u(t) = T (t)u0 +
∫ t
0T (t − s)F (s, u(s))ds +
∫ t
0T (t − s)G(s, u(s))dW (s) .
Proof.Main difficulty: treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
Standard assumption G (s,u(s)) ∈ L2(X 2)
TheoremThere exists a unique mild solution in the sense that
u(t) = T (t)u0 +
∫ t
0T (t − s)F (s, u(s))ds +
∫ t
0T (t − s)G(s, u(s))dW (s) .
Proof.Main difficulty to treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
We only require G (s,u(s)) ∈ L(X 2)
proof continued...
Recall propositions above:
A generates an analytic C0−semigroup and
|T (t)|HS ≤ Mt−14 .
T (t)G (t,u(t)) ∈ L2(X 2) :
|T (t)G (t,u(t))|L2(X 2) ≤ Ct−14 (1 + |u|) .
proof continued...
Recall propositions above:
A generates an analytic C0−semigroup and
|T (t)|HS ≤ Mt−14 .
T (t)G (t,u(t)) ∈ L2(X 2) :
|T (t)G (t,u(t))|L2(X 2) ≤ Ct−14 (1 + |u|) .
Main motivations
Notation
Non-local Kirchhoff conditionThe perturbed non–linear stochastic problem
Time–Delayed Kirchhoff conditionThe perturbed non–linear stochastic problem
Application to optimal control
Financial applications
Reaction–diffusion equation
uj(t, x) =(cju′j
)′(t, x) ,
uj(t, vα) = ul(t, vα) =: dα(t) ,
dα(t) = −∑m
j=1 φjαu′j (t, vα) + bαd
α(t) +∫ 0
−r dα(t + θ)µ(dθ) ,
uj(0, x) = u0j (x) ,
dα(0) = d0α ,
dα(θ) = η0α(θ) .
Continuity condition
uj(t, x) =(cju′j
)′(t, x) ,
uj(t, vα) = ul(t, vα) =: dα(t) ,
dα(t) = −∑m
j=1 φjαu′j (t, vα) + bαd
α(t) +∫ 0
−r dα(t + θ)µ(dθ) ,
uj(0, x) = u0j (x) ,
dα(0) = d0α ,
dα(θ) = η0α(θ) .
Dynamic time–delayed Kirchhoff condition
uj(t, x) =(cju′j
)′(t, x) ,
uj(t, vα) = ul(t, vα) =: dα(t) ,
dα(t) = −∑m
j=1 φjαu′j (t, vα) + bαd
α(t) +∫ 0
−r dα(t + θ)µ(dθ) ,
uj(0, x) = u0j (x) ,
dα(0) = d0α ,
dα(θ) = η0α(θ) .
Dynamic time–delayed Kirchhoff condition
uj(t, x) =(cju′j
)′(t, x) ,
uj(t, vα) = ul(t, vα) =: dα(t) ,
dα(t) = −∑m
j=1 φjαu′j (t, vα) + bαd
α(t) +∫ 0
−r dα(t + θ)µ(dθ) ,
uj(0, x) = u0j (x) ,
dα(0) = d0α ,
dα(θ) = η0α(θ) .
The abstract setting
X 2 :=(L2([0, 1])
)m, Z 2 := L2([−r , 0];Rn) ,
X 2 := X 2 × Rn , E2 := X 2 × Z 2 ,
Consider the process d : [−r ,T ]→ Rn and define the segment
dt : [−r , 0]→ Rn , [−r , 0] 3 θ 7→ dt(θ) := d(t + θ) ∈ Rn .
The abstract setting
X 2 :=(L2([0, 1])
)m, Z 2 := L2([−r , 0];Rn) ,
X 2 := X 2 × Rn , E2 := X 2 × Z 2 ,
Consider the process d : [−r ,T ]→ Rn and define the segment
dt : [−r , 0]→ Rn , [−r , 0] 3 θ 7→ dt(θ) := d(t + θ) ∈ Rn .
The abstract PDE
u(t) = Amu(t) , t ∈ [0,T ] ,
d(t) = Cu(t) + Φdt + Bd(t) , t ∈ [0,T ] ,
dt = Aθdt , t ∈ [0,T ] ,
Lu(t) = d(t) ,
u(0) = u0 ∈ X 2 , d0 = η ∈ Z 2 , d(0) = d0 ∈ Rn ,
The abstract PDE
Amu(t, x) =
∂∂x
(cj(x) ∂∂x u1(t, x)
)0 0
0. . . 0
0 0 ∂∂x
(cm(x) ∂∂x um(t, x)
) ,
and such that Am : D(Am) ⊂ X 2 → X 2, with domain
D(A) :=u ∈
(H2([0, 1])
)m: ∃d ∈ Rn : Lu = d
,
The abstract PDE
L :(H1([0, 1])
)m → Rn is the boundary evaluation operator
Lu(t, x) :=(d1(t), . . . , dn(t)
)T, dα(t) := uj(t, vα) .
C : D(A)→ Rn is the feedback operator
Cu(t, x) :=
− m∑j=1
φj1u′j (t, v1), . . . ,−
m∑j=1
φjnu′j (t, vn)
T
.
The abstract PDE
L :(H1([0, 1])
)m → Rn is the boundary evaluation operator
Lu(t, x) :=(d1(t), . . . , dn(t)
)T, dα(t) := uj(t, vα) .
C : D(A)→ Rn is the feedback operator
Cu(t, x) :=
− m∑j=1
φj1u′j (t, v1), . . . ,−
m∑j=1
φjnu′j (t, vn)
T
.
The abstract PDE
Φ : H1([−r , 0];Rn)→ Rn is the delay operator
Φdt =
∫ 0
−rdα(t + θ)µ(dθ) .
Aθ : D(Aθ) ⊂ Z 2 → Z 2
Aθη :=∂
∂θη(θ) , D(Aθ) = η ∈ H1([−r , 0];Rn) : η(0) = d0 ,
The abstract PDE
Φ : H1([−r , 0];Rn)→ Rn is the delay operator
Φdt =
∫ 0
−rdα(t + θ)µ(dθ) .
Aθ : D(Aθ) ⊂ Z 2 → Z 2
Aθη :=∂
∂θη(θ) , D(Aθ) = η ∈ H1([−r , 0];Rn) : η(0) = d0 ,
The abstract equation
u(t) = Au(t) , t ∈ [0,T ] ,
u(0) = u0 ∈ E2 ,
A is defined as
A :=
Am 0 0C B Φ0 0 Aθ
,
with domain D(A) := D(Am)× D(Aθ).
Does A generate a C0−semigroup?
The abstract equation
u(t) = Au(t) , t ∈ [0,T ] ,
u(0) = u0 ∈ E2 ,
A is defined as
A :=
Am 0 0C B Φ0 0 Aθ
,
with domain D(A) := D(Am)× D(Aθ).
Does A generate a C0−semigroup?
On the infinitesimal generator
A :=
Am 0 0C B Φ0 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
TheoremThe matrix operator (A0,D(A0)) generates a C0−semigroup given by
T0(t) =
Ta(t) 0
00 Tt T0(t)
,
Ta is the C0−semigroup generated by (Aa,D(Aa))T0(t) is the nilpotent left-shift semigroup
(T0(t)η) (θ) :=
η(t + θ) t + θ ≤ 0 ,
0 t + θ > 0 ,, η ∈ Z 2 ,
Tt : Rn → Z 2 is defined by
(Ttd) (θ) :=
e(t+θ)Bd −t < θ ≤ 0 ,
0 −r ≤ θ ≤ −t ,, d ∈ Rn ,
e(t+θ)B being the semigroup generated by the finite dimensional n × nmatrix B.
TheoremThe matrix operator (A0,D(A0)) generates a C0−semigroup given by
T0(t) =
Ta(t) 0
00 Tt T0(t)
,
Ta is the C0−semigroup generated by (Aa,D(Aa))
T0(t) is the nilpotent left-shift semigroup
(T0(t)η) (θ) :=
η(t + θ) t + θ ≤ 0 ,
0 t + θ > 0 ,, η ∈ Z 2 ,
Tt : Rn → Z 2 is defined by
(Ttd) (θ) :=
e(t+θ)Bd −t < θ ≤ 0 ,
0 −r ≤ θ ≤ −t ,, d ∈ Rn ,
e(t+θ)B being the semigroup generated by the finite dimensional n × nmatrix B.
TheoremThe matrix operator (A0,D(A0)) generates a C0−semigroup given by
T0(t) =
Ta(t) 0
00 Tt T0(t)
,
Ta is the C0−semigroup generated by (Aa,D(Aa))T0(t) is the nilpotent left-shift semigroup
(T0(t)η) (θ) :=
η(t + θ) t + θ ≤ 0 ,
0 t + θ > 0 ,, η ∈ Z 2 ,
Tt : Rn → Z 2 is defined by
(Ttd) (θ) :=
e(t+θ)Bd −t < θ ≤ 0 ,
0 −r ≤ θ ≤ −t ,, d ∈ Rn ,
e(t+θ)B being the semigroup generated by the finite dimensional n × nmatrix B.
TheoremThe matrix operator (A0,D(A0)) generates a C0−semigroup given by
T0(t) =
Ta(t) 0
00 Tt T0(t)
,
Ta is the C0−semigroup generated by (Aa,D(Aa))T0(t) is the nilpotent left-shift semigroup
(T0(t)η) (θ) :=
η(t + θ) t + θ ≤ 0 ,
0 t + θ > 0 ,, η ∈ Z 2 ,
Tt : Rn → Z 2 is defined by
(Ttd) (θ) :=
e(t+θ)Bd −t < θ ≤ 0 ,
0 −r ≤ θ ≤ −t ,, d ∈ Rn ,
e(t+θ)B being the semigroup generated by the finite dimensional n × nmatrix B.
The Miyadera-Voigt perturbation theorem
TheoremLet (G ,D(G )) be the generator of a C0 semigroup (S(t))t≥0. Assumethat there exist constants t0 > 0 and 0 ≤ q ≤ 1, such that∫ t0
0
‖KS(t)x‖dt ≤ q‖x‖ , ∀ x ∈ D(G ) .
Then (G + K ,D(G )) generates a strongly continuous semigroup(U(t))t≥0 on X , which satisfies
U(t)x = S(t)x +
∫ t
0
S(t − s)KU(s)xds ,
and ∫ t0
0
‖KU(t)x‖dt ≤ q
1− q‖x‖ , ∀ x ∈ D(G ) , t ≥ 0 .
A1 :=
0 0 00 0 Φ0 0 0
,
A = A0 +A1 .
TheoremThe operator (A,D(A)) generates a strongly continuous semigroup.
Proof.Apply Miyadera-Voigt perturbation theorem.
A1 :=
0 0 00 0 Φ0 0 0
,
A = A0 +A1 .
TheoremThe operator (A,D(A)) generates a strongly continuous semigroup.
Proof.Apply Miyadera-Voigt perturbation theorem.
The perturbed non–linear stochastic problem
uj(t, x) =(cju′j
)′(t, x)+fj(t, x , uj(t, x))+
+gj(t, x , uj(t, x))W 1j (t, x) ,
uj(t, vα) = ul(t, vα) =: dα(t) ,
dα(t) = −∑m
j=1 φjαu′j (t, vα) + bαd
α(t) +∫ 0
−r dα(t + θ)µ(dθ)+
+gα(t, dα(t), dαt )W 2α(t, vα) ,
uj(0, x) = u0j (x) ,
dα(0) = d0α ,
dα(θ) = η0α(θ) .
The perturbed non–linear stochastic problem
uj(t, x) =(cju′j
)′(t, x) + fj(t, x , uj(t, x))+
+gj(t, x , uj(t, x))W 1j (t, x) ,
uj(t, vα) = ul(t, vα) =: dα(t) ,
dα(t) = −∑m
j=1 φjαu′j (t, vα) + bαd
α(t) +∫ 0
−r dα(t + θ)µ(dθ)+
+gα(t, dα(t), dαt )W 2α(t, vα) ,
uj(0, x) = u0j (x) ,
dα(0) = d0α ,
dα(θ) = η0α(θ) .
The perturbed non–linear stochastic problem
uj(t, x) =(cju′j
)′(t, x) + fj(t, x , uj(t, x))+
+gj(t, x , uj(t, x))W 1j (t, x) ,
uj(t, vα) = ul(t, vα) =: dα(t) ,
dα(t) = −∑m
j=1 φjαu′j (t, vα) + bαd
α(t) +∫ 0
−r dα(t + θ)µ(dθ)+
+gα(t, dα(t), dαt )W 2α(t, vα) ,
uj(0, x) = u0j (x) ,
dα(0) = d0α ,
dα(θ) = η0α(θ) .
Assumption
|gj(t, x , y1)| ≤ Cj , |gj(t, x , y1)− gj(t, x , y2)| ≤ Kj |y1 − y2| ;
|gα(t, x , η)| ≤ Cα , |gα(t, x , η)− gα(t, y , ζ)| ≤ Kα(|x − y |n + |η− ζ|Z 2 ) .
|fj(t, x , y1)| ≤ Cj , |fj(t, x , y1)− fj(t, x , y2)| ≤ Kj |y1 − y2| .
Remarkfj(t, x , y) can be assumed also to be non–Lipschitz of polynomial growth.
Assumption
|gj(t, x , y1)| ≤ Cj , |gj(t, x , y1)− gj(t, x , y2)| ≤ Kj |y1 − y2| ;
|gα(t, x , η)| ≤ Cα , |gα(t, x , η)− gα(t, y , ζ)| ≤ Kα(|x − y |n + |η− ζ|Z 2 ) .
|fj(t, x , y1)| ≤ Cj , |fj(t, x , y1)− fj(t, x , y2)| ≤ Kj |y1 − y2| .
Remarkfj(t, x , y) can be assumed also to be non–Lipschitz of polynomial growth.
The abstract equation
dX(t) = [AX(t) + F (t,X)] dt + G (t,X(t))dW (t) , t ≥ 0 ,
X(0) = X0 ∈ E2 ,
TheoremThere exists a unique mild solution in the sense that
X(t) = T (t)X0 +
∫ t
0T (t − s)F (s,X(s))ds +
∫ t
0T (t − s)G(s,X(s))dW (s) .
Proof.Main difficulty: treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
Standard assumption G (s,u(s)) ∈ L2(X 2)
TheoremThere exists a unique mild solution in the sense that
X(t) = T (t)X0 +
∫ t
0T (t − s)F (s,X(s))ds +
∫ t
0T (t − s)G(s,X(s))dW (s) .
Proof.Main difficulty: treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
Standard assumption G (s,u(s)) ∈ L2(X 2)
TheoremThere exists a unique mild solution in the sense that
X(t) = T (t)X0 +
∫ t
0T (t − s)F (s,X(s))ds +
∫ t
0T (t − s)G(s,X(s))dW (s) .
Proof.Main difficulty: treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
Standard assumption G (s,u(s)) ∈ L2(X 2)
TheoremThere exists a unique mild solution in the sense that
X(t) = T (t)X0 +
∫ t
0T (t − s)F (s,X(s))ds +
∫ t
0T (t − s)G(s,X(s))dW (s) .
Proof.Main difficulty to treat the stochastic convolution∫ t
0
T (t − s)G (s,u(s))dW (s) .
We only require G (s,u(s)) ∈ L(X 2)
proof continued...
The matrix operator A contains Aθ := ∂∂θ
A does not generate an analytic C0−semigroup
PropositionT (t)G (s,X) ∈ L2(X 2; E2) such that
|T (t)G (s,X)|HS ≤ Mt−14 (1 + |X|E2 )
Proof.Technical computations exploiting the explicit form for T (t).
proof continued...
The matrix operator A contains Aθ := ∂∂θ
A does not generate an analytic C0−semigroup
PropositionT (t)G (s,X) ∈ L2(X 2; E2) such that
|T (t)G (s,X)|HS ≤ Mt−14 (1 + |X|E2 )
Proof.Technical computations exploiting the explicit form for T (t).
Main motivations
Notation
Non-local Kirchhoff conditionThe perturbed non–linear stochastic problem
Time–Delayed Kirchhoff conditionThe perturbed non–linear stochastic problem
Application to optimal control
Financial applications
Application to optimal control
dXz(t) = [AXz(t) + F (t,Xz) + G (t,Xz(t))R(t,Xz(t), z(t))] dt+
+G (t,Xz(t))dW (t) ,
Xz(t0) = X0 ∈ E2 ,
J(t0,X0, z) = E∫ T
t0
l (t,Xz(t), z(t)) dt + Eϕ(Xz(T ))→ min .
Admissible Control System (acs)
(Ω,F , (Ft)t≥0 ,P, (W (t))t≥0 , z
)I(
Ω,F , (Ft)t≥0 ,P)
is a complete probability space, where the
filtration (Ft)t≥0 satisfies the usual assumptions;
I (W (t))t≥0 is a Ft−adapted Wiener process taking values in E2;
I z is a process taking values in the space Z , predictable with respectto the filtration (Ft)t≥0, and such that z(t) ∈ Z P−a.s., for almostany t ∈ [t0,T ], being Z a suitable domain of Z .
Assumption
|R(t,X, z)− R(t,X, z)|E2 ≤ CR(1 + |X|E2 + |Y|E2 )m|X− Y|E2 ,
|R(t,X, z)|E2 ≤ CR ;
|l(t,X, z)− l(t,X, z)| ≤ Cl(1 + |X|E2 + |Y|E2 )m|X− Y|E2 ,
|l(t, 0, z)|E2 ≥ −C ,infz∈Z
l(t, 0, z) ≤ Cl ;
|ϕ(X)− ϕ(Y)| ≤ Cϕ(1 + |X|E2 + |Y|E2 )m|X− Y|E2 .
Girsanov theorem
W ζ(t) := W (t)−∫ t∧T
t0∧tR(s,X(s), ζ)ds ,
dXz(t) = [AXz(t) + F (t,Xz) + G (t,Xz(t))R(t,Xz(t), z(t))] dt+
+G (t,Xz(t))dW (t) ,
Xz(t0) = X0 ∈ E2 ,
Girsanov theorem
W ζ(t) := W (t)−∫ t∧T
t0∧tR(s,X(s), ζ)ds ,
dXz(t) = [AXz(t) + F (t,Xz) + G (t,Xz(t))R(t,Xz(t), z(t))] dt+
+G (t,Xz(t))dW (t) ,
Xz(t0) = X0 ∈ E2 ,
Girsanov theorem
W ζ(t) := W (t)−∫ t∧T
t0∧tR(s,X(s), ζ)ds ,
dXz(t) = [AXz(t) + F (t,Xz)] dt + G (t,Xz(t))dW ζ(t) ,
Xz(t0) = X0 ∈ E2 ,
The HJB equation
ψ(t,X,Y) := − infz∈Zl(t,X, z) + YR(t,X, z) ,
Γ(t,X,Y) := z ∈ Z : ψ(t,X,Y) + l(t,X, z) + vR(t,X, z) = 0 ,∂w(t,X)∂t + Ltw(t,X) = ψ(t,X,∇Gw(t,X)) ,
w(T ,X) = ϕ(X) ,
∇G being the generalized directional gradient.
TheoremLet w be a mild solution to the HJB equation, and chose ρ to be anelement of the generalized directional gradient ∇Gw . Then, for all ACS,we have that J(t0,X0, z) ≥ w(t0,X0), and the equality holds if and onlyif the following feedback law is verified by z and uz
z(t) = Γ (t,Xz(t),G (t, ρ(t,Xz(t))) , P− a.s. for a.a. t ∈ [t0,T ] .
Moreover, if there exists a measurable function γ : [0,T ]× E2 × E2 → Zwith
γ(t,X,Y) ∈ Γ(t,X,Y) , t ∈ [0,T ] , X , Y ∈ X 2 ,
then there also exists, at least one ACS such that
z(t))γ(t,Xz(t), ρ(t,Xz(t))) , P− a.s. for a.a. t ∈ [t0,T ] .
Eventually, we have that Xz is a mild solution.
Main motivations
Notation
Non-local Kirchhoff conditionThe perturbed non–linear stochastic problem
Time–Delayed Kirchhoff conditionThe perturbed non–linear stochastic problem
Application to optimal control
Financial applications
System of interconnected banks
Works in progress with L. Di Persio (UniVr), L. Prezioso (UniVr-UniTn),A. Bressan (Penn State University)and Y. Jiang (Penn State University).
I Multiple defualts of banks;
I Optimal control with terminal probability constraints;
I Stackelberg equilibrium;
I Stochastic impulse control.
System of interconnected banks
Works in progress with L. Di Persio (UniVr), L. Prezioso (UniVr-UniTn),A. Bressan (Penn State University)and Y. Jiang (Penn State University).
I Multiple defualts of banks;
I Optimal control with terminal probability constraints;
I Stackelberg equilibrium;
I Stochastic impulse control.
System interconnected banks: the setting
I value of the i−th bank associated to the vertex vi , i = 1, . . . , n;
I liabilities matrix L(t) = (Li,j(t))n×nI ui (t) the payment made at time t ∈ [0,T ] by vi ;
I ui (t) =∑n
j=1 Li,j(t) the total nominal obligation of the node itowards all other nodes;
I relative liabilities matrix Π(t) = (πi,j(t)) defined as
πi,j(t) =
Li,j (t)ui (t) ui (t) > 0 ,
0 otherwise .
I the cash inflow of the node i is given by∑n
j=1 (Πi,j(t))T uj(t).
System interconnected banks: the setting
total value of node vi at time t ∈ [0,T ]
V i (t) =n∑
j=1
(Πi,j(t))T uj(t) + X i (t)− ui (t) .
System interconnected banks: the setting
I liabilities evolve according to
d
dtLi,j(t) = µijLi,j(t) ,
I exogenous asset X i (t) evolves according to
dX i (t) = X i (t)(µidt + σidW i (t)
), i = 1, . . . , n .
I continuous (deterministic) default boundaries for bank i
X i (t) ≤ v i (t) :=
R i(ui (t)−
∑nj=1 (Πi,j(t))T uj(t)
)t < T ,
ui (t)−∑n
j=1 (Πi,j(t))T uj(t) t = T ,
I R i , i = 1, . . . , n, representing the recovery rate of the bank i .
System interconnected banks: the optimal control problem
I financial supervisor, (lender of last resort, (LOLR)), aims at savingthe network from default;
I LOLR minimizes the quadratic cost whilst maximizing the distanceof each bank from the respective default boundary
J(x , α) := E
[∫ τ
0
(−L (X(t)) +
1
2‖α(t)‖2
)dt − G (X(τn))
],
I τ random terminal time of default;
I controlled process
dX i (t) = X i (t)(µidt + σidW i (t)
)+ αi (t)dt , i = 1, . . . , n .
System interconnected banks: the optimal control problem
I after first default wehave a new system i = 1, . . . , n
dX i1(t) = X i
1(t)(µi
1dt + σi1dW
i (t))
+ αi1(t)dt , i = 1, . . . , n − 1 .
I LOLR minimizes the quadratic cost whilst maximizing the distanceof each bank from the respective default boundary
J1(x , α) := E
[∫ τ 1
τ
(−L1 (X1(t)) +
1
2‖α1(t)‖2
)dt − G1
(X1(τ 1)
)],
I τ 1 random terminal time of default;
I and so on until no nodes are left in the system;
System interconnected banks: the optimal control problem
I after first default wehave a new system i = 1, . . . , n
dX i1(t) = X i
1(t)(µi
1dt + σi1dW
i (t))
+ αi1(t)dt , i = 1, . . . , n − 1 .
I LOLR minimizes the quadratic cost whilst maximizing the distanceof each bank from the respective default boundary
J1(x , α) := E
[∫ τ 1
τ
(−L1 (X1(t)) +
1
2‖α1(t)‖2
)dt − G1
(X1(τ 1)
)],
I τ 1 random terminal time of default;
I and so on until no nodes are left in the system;
System interconnected banks: the optimal control problem
I multiple optimal control problems with random terminal time;
I stochastic maximum principle for global multiple stochastic optimalcontrol problem;
The maximum principle
Theorem[Maximum Principle]
∂aH(t, X(t), α(t), Y (t), Z (t)) (α(t)− α) ≤ 0 ,
where each (Y πk
(t),Zπk
(t)) solves the following BSDE’s
−dY πn−1
(t) = ∂xHπn−1
(t,Xπn−1
(t), απn−1
(t),Y πn−1
(t),Zπn−1
(t))dt − Zπn−1
dW (t) ,
Y πn−1
(τn) = ∂xGπn−1
(τn,Xπn−1
(τn)) ,
−dY πk
(t) = ∂xHπk(t,Xπ
k(t), απ
k(t),Y π
k(t),Zπ
k(t))dt − Zπ
kdW (t) ,
Y πk(τk+1) = ∂xGπ
k(τk+1,Xπ
k(τk+1)) + Y k+1(τk+1) ,
−dY 0(t) = ∂xH0(t,X0(t), α0(t),Y 0(t),Z0(t))dt − Z0dW (t) ,
Y 0(τ) = ∂xG0(τ1,X0(τ1)) + Y 1(τ1) ,
The optimal control with constrained probability of success
I LOLR minimizes amount of money lent
J(x , α) =1
2
∫ T
0
‖α(s)‖2ds ;
under fixed probability of default
P(X i (T ) ≥ v i (T )
)≥ qi , i = 1, . . . , n ,
I controlled process
dX i (t) = X i (t)(µidt + σidW i (t)
)+ αi (t)dt , i = 1, . . . , n .
The optimal control with constrained probability of success
I two regions for the optimal solution:
I Region I: the probability constraints is satisfied;
I optimal solution α(t) ≡ 0;
I Region II: the probability constraints is not satisfied;
I we guess αi (t) = ψ(t)X i (t)
I optimal solution
ψi =ln v i (T )− ln x0
t1−(√
2Erf −1(1− 2qi
))σi 1√
T+
(σ)2
2− µi .
The optimal control with constrained probability of success
I two regions for the optimal solution:
I Region I: the probability constraints is satisfied;
I optimal solution α(t) ≡ 0;
I Region II: the probability constraints is not satisfied;
I we guess αi (t) = ψ(t)X i (t)
I optimal solution
ψi =ln v i (T )− ln x0
t1−(√
2Erf −1(1− 2qi
))σi 1√
T+
(σ)2
2− µi .
The optimal control with constrained probability of success
I two regions for the optimal solution:
I Region I: the probability constraints is satisfied;
I optimal solution α(t) ≡ 0;
I Region II: the probability constraints is not satisfied;
I we guess αi (t) = ψ(t)X i (t)
I optimal solution
ψi =ln v i (T )− ln x0
t1−(√
2Erf −1(1− 2qi
))σi 1√
T+
(σ)2
2− µi .
Thank you for your attention!