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





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?

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 ,

⟨(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

,

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.

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 ,


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) ,


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 ,


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) ,


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 ,


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)


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|) .

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) .

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

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 ,


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) ,

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)

,

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 ,


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.


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

−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.


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 +

∫ 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)


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).

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

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)] 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 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∑

j=1

(Πi,j(t))T uj(t) + X i (t)− ui (t) .


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


)+ 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 .

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;


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


)+ 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 .

Thank you for your attention!

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

Documents