stochastic pdeâ s on networks with non local boundary...

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

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

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

Notation

Incidence matrix

I = (ιve) , ιve =

1 · e−−−−→ v ,

−1 ve−−−−→ ·

0 otherwise ;

Adjacency matrix

A = (avw ) , avw =

e−−−−→ w ,

1 we−−−−→ v ,

0 otherwise ;

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 ;

Consider a diffusion equation on thegraph G

uj(t, x) = ∆uj(t, x) ,

uj(t, x) , on the edge 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

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) +

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

∑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) +

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

∑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) +

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 ,

)⟩X 2

:=m∑j=1

uj(x)vj(x)dx +n∑

The differential operator

(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 0C B1

D(A) =

)∈ D(A)× Rn : ui (vα) = du

α , ∀ i ∈ Γ(vα), α = 1, . . . , n

C : D(C ) := D(A)→ Rn the feedback operator

− m∑i,j=1

n∑β=1

δ1iβju′j (v1), . . . ,−

m∑i,j=1

n∑β=1

δn0iβj u′j (vn0 ), 0, . . . , 0

Operator matrix

(A 0C B1

D(A) =

)∈ D(A)× Rn : ui (vα) = du

α , ∀ i ∈ Γ(vα), α = 1, . . . , n

C : D(C ) := D(A)→ Rn the feedback operator

− m∑i,j=1

n∑β=1

δ1iβju′j (v1), . . . ,−

m∑i,j=1

n∑β=1

δn0iβj u′j (vn0 ), 0, . . . , 0

The abstract equation

u(t) = Au(t) , t ≥ 0 ,

u(0) = u0 ∈ X 2 .

Does A generate a C0−semigroup?

u(t) = Au(t) , t ≥ 0 ,

u(0) = u0 ∈ 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.

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) =

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) =

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

)′(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

uj (t, x) =∑m

)′(t, x) +

∑mi=1 pijui (t, x)+

+fj (t, x , uj (t, x))+gj (t, x , uj (t, x))W 1j (t, x) ,

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

uj (t, x) =∑m

)′(t, x) +

∑mi=1 pijui (t, x)+

+fj (t, x , uj (t, x)) + gj (t, x , uj (t, x))W 1j (t, x) ,

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

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 +

0T (t − s)F (s, u(s))ds +

0T (t − s)G(s, u(s))dW (s) .

Proof.Main difficulty: treat the stochastic convolution∫ t

T (t − s)G (s,u(s))dW (s) .

Standard assumption G (s,u(s)) ∈ L2(X 2)

u(t) = T (t)u0 +

0T (t − s)F (s, u(s))ds +

0T (t − s)G(s, u(s))dW (s) .

T (t − s)G (s,u(s))dW (s) .

u(t) = T (t)u0 +

0T (t − s)F (s, u(s))ds +

0T (t − s)G(s, u(s))dW (s) .

T (t − s)G (s,u(s))dW (s) .

u(t) = T (t)u0 +

0T (t − s)F (s, u(s))ds +

0T (t − s)G(s, u(s))dW (s) .

Proof.Main difficulty to treat the stochastic convolution∫ t

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

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) ,

dα(t) = −∑m

α(t) +∫ 0

uj(0, x) = u0j (x) ,

dα(0) = d0α ,

dα(θ) = η0α(θ) .

Dynamic time–delayed Kirchhoff condition

uj(t, x) =(cju′j

)′(t, x) ,

dα(t) = −∑m

α(t) +∫ 0

uj(0, x) = u0j (x) ,

dα(0) = d0α ,

dα(θ) = η0α(θ) .

Dynamic time–delayed Kirchhoff condition

uj(t, x) =(cju′j

)′(t, x) ,

dα(t) = −∑m

α(t) +∫ 0

uj(0, x) = u0j (x) ,

dα(0) = d0α ,

dα(θ) = η0α(θ) .

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 .

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 ∂∂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)

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)

The abstract PDE

Φ : H1([−r , 0];Rn)→ Rn is the delay operator

Φdt =

−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 =

−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 ,

u(t) = Au(t) , t ∈ [0,T ] ,

u(0) = u0 ∈ E2 ,

A is defined as

Am 0 0C B Φ0 0 Aθ

with domain D(A) := D(Am)× D(Aθ).

u(t) = Au(t) , t ∈ [0,T ] ,

u(0) = u0 ∈ E2 ,

A is defined as

Am 0 0C B Φ0 0 Aθ

with domain D(A) := D(Am)× D(Aθ).

On the infinitesimal generator

Am 0 0C B Φ0 0 Aθ

(Am 0C B

(Aa 00 Aθ

), D(A0) = D(A) ,

Am 0 0C B 00 0 Aθ

(Am 0C B

(Aa 00 Aθ

), D(A0) = D(A) ,

Am 0 0C B 00 0 Aθ

(Am 0C B

(Aa 00 Aθ

), D(A0) = D(A) ,

Am 0 0C B 00 0 Aθ

(Am 0C B

(Aa 00 Aθ

), D(A0) = D(A) ,

Am 0 0C B 00 0 Aθ

(Am 0C B

(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.

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 ,

(Ttd) (θ) :=

e(t+θ)Bd −t < θ ≤ 0 ,

0 −r ≤ θ ≤ −t ,, d ∈ Rn ,

T0(t) =

Ta(t) 0

00 Tt T0(t)

(T0(t)η) (θ) :=

η(t + θ) t + θ ≤ 0 ,

0 t + θ > 0 ,, η ∈ Z 2 ,

(Ttd) (θ) :=

e(t+θ)Bd −t < θ ≤ 0 ,

0 −r ≤ θ ≤ −t ,, d ∈ Rn ,

T0(t) =

Ta(t) 0

00 Tt T0(t)

(T0(t)η) (θ) :=

η(t + θ) t + θ ≤ 0 ,

0 t + θ > 0 ,, η ∈ Z 2 ,

(Ttd) (θ) :=

e(t+θ)Bd −t < θ ≤ 0 ,

0 −r ≤ θ ≤ −t ,, d ∈ Rn ,

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

‖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 +

S(t − s)KU(s)xds ,

and ∫ t0

‖KU(t)x‖dt ≤ q

1− q‖x‖ , ∀ x ∈ D(G ) , t ≥ 0 .

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.

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.

uj(t, x) =(cju′j

)′(t, x)+fj(t, x , uj(t, x))+

+gj(t, x , uj(t, x))W 1j (t, x) ,

dα(t) = −∑m

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) .

uj(t, x) =(cju′j

)′(t, x) + fj(t, x , uj(t, x))+

+gj(t, x , uj(t, x))W 1j (t, x) ,

dα(t) = −∑m

α(t) +∫ 0

uj(0, x) = u0j (x) ,

dα(0) = d0α ,

dα(θ) = η0α(θ) .

uj(t, x) =(cju′j

)′(t, x) + fj(t, x , uj(t, x))+

+gj(t, x , uj(t, x))W 1j (t, x) ,

dα(t) = −∑m

α(t) +∫ 0

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.

dX(t) = [AX(t) + F (t,X)] dt + G (t,X(t))dW (t) , t ≥ 0 ,

X(0) = X0 ∈ E2 ,

X(t) = T (t)X0 +

0T (t − s)F (s,X(s))ds +

0T (t − s)G(s,X(s))dW (s) .

T (t − s)G (s,u(s))dW (s) .

X(t) = T (t)X0 +

0T (t − s)F (s,X(s))ds +

0T (t − s)G(s,X(s))dW (s) .

T (t − s)G (s,u(s))dW (s) .

X(t) = T (t)X0 +

0T (t − s)F (s,X(s))ds +

0T (t − s)G(s,X(s))dW (s) .

T (t − s)G (s,u(s))dW (s) .

X(t) = T (t)X0 +

0T (t − s)F (s,X(s))ds +

0T (t − s)G(s,X(s))dW (s) .

Proof.Main difficulty to treat the stochastic convolution∫ t

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

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

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

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 ,

Xz(t0) = X0 ∈ E2 ,

Girsanov theorem

W ζ(t) := W (t)−∫ t∧T

Xz(t0) = X0 ∈ E2 ,

Girsanov theorem

W ζ(t) := W (t)−∫ t∧T

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

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).

total value of node vi at time t ∈ [0,T ]

V i (t) =n∑

(Πi,j(t))T uj(t) + X i (t)− ui (t) .

I liabilities evolve according to

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

(−L (X(t)) +

2‖α(t)‖2

)dt − G (X(τn))

I τ random terminal time of default;

I controlled process

)+ αi (t)dt , i = 1, . . . , n .

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 .

J1(x , α) := E

[∫ τ 1

(−L1 (X1(t)) +

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;

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 .

J1(x , α) := E

[∫ τ 1

(−L1 (X1(t)) +

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;

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

‖α(s)‖2ds ;

under fixed probability of default

P(X i (T ) ≥ v i (T )

)≥ qi , i = 1, . . . , n ,

I controlled process

)+ αi (t)dt , i = 1, . . . , n .

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√

2− µi .

I optimal solution

t1−(√

2Erf −1(1− 2qi

))σi 1√

2− µi .

I optimal solution

t1−(√

2Erf −1(1− 2qi

))σi 1√

2− µi .

Thank you for your attention!

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