stochastic pdeâ s on networks with non local boundary...
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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
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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.
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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
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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
![Page 5: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/5.jpg)
Main motivations
(i) quantum mechanics;
(ii) electrical circuits;
(iii) traffic flow;
(iv) neurobiology;
(v) smart grid optimization;
(vi) system of interconnected banks.
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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
![Page 7: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/7.jpg)
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
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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
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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
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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
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Main aim
Write the diffusion equation as an abstract operatorial problem
Boundary conditions?
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Main aim
Write the diffusion equation as an abstract operatorial problem
Boundary conditions?
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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 .
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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 .
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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
![Page 16: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/16.jpg)
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 ,
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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 ,
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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 ,
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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 ,
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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 ,
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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)
.
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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
,
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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
,
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The abstract equation
u(t) = Au(t) , t ≥ 0 ,
u(0) = u0 ∈ X 2 .
Does A generate a C0−semigroup?
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The abstract equation
u(t) = Au(t) , t ≥ 0 ,
u(0) = u0 ∈ X 2 .
Does A generate a C0−semigroup?
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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.
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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.
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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 .
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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 .
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PropositionFor any t ≥ 0, the semigroup T (t) ∈ L2(X 2), moreover there existsM > 0 such that
|T (t)|HS ≤ Mt−14 .
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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 ,
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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 ,
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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 ,
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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 ,
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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)
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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)
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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)
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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)
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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|) .
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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|) .
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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
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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) .
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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) .
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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) .
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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) .
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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 .
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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 .
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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 ,
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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
,
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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
.
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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
.
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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 ,
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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 ,
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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?
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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?
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On the infinitesimal generator
A :=
Am 0 0C B Φ0 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
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On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
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On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
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On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
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On the infinitesimal generator
A0 :=
Am 0 0C B 00 0 Aθ
,
Aa :=
(Am 0C B
),
A0 :=
(Aa 00 Aθ
), D(A0) = D(A) ,
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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.
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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.
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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.
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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.
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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 .
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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.
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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.
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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) .
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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) .
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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) .
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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.
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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.
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The abstract equation
dX(t) = [AX(t) + F (t,X)] dt + G (t,X(t))dW (t) , t ≥ 0 ,
X(0) = X0 ∈ E2 ,
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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)
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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)
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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)
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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)
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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).
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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).
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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
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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 .
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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 .
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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 .
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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 ,
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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 ,
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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 ,
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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.
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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.
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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
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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.
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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.
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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).
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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) .
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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 .
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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 .
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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;
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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;
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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;
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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) ,
![Page 100: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/100.jpg)
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 .
![Page 101: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/101.jpg)
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 .
![Page 102: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/102.jpg)
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 .
![Page 103: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/103.jpg)
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 .
![Page 104: Stochastic PDEâ s on networks with non local boundary ...vpsms2018.org/wp-content/uploads/2017/09/Cordoni.pdf · boundary conditions and application to nance F. Cordoni, University](https://reader031.vdocument.in/reader031/viewer/2022022419/5a78d1327f8b9a4f1b8cbf3a/html5/thumbnails/104.jpg)
Thank you for your attention!