G-expectations in infinite dimensions and related PDEITN2012/files/talk/Ibragimov.pdfOutline 1 Short overview 2 G-function 3 Main notions of the G-expectation theory 4 Viscosity solutions

G-expectations in infinite dimensionsand related PDE

Anton Ibragimov

PhD advisers:prof. Gianmario Tessitore

prof. Marco Fuhrman

Universita degli Studi Milano-Bicocca

July 5, 2012, Iasi, Romania

A. Ibragimov (Milano-Bicocca Universita) G-expectations in infinite dimensions July 5, 2012, Iasi 1 / 46

Outline

1 Short overview

2 G-function

3 Main notions of the G-expectation theory

4 Viscosity solutions

5 Stochastic integral

6 Brief conclusion of the solving PDE (P)


Outline

1 Short overview

2 G-function






Short overview

Consider the next PDE:∂tu + 〈Ax ,Dxu〉+ G (D2

xxu) = 0 ;

u(T , x) = f (x) .(P)

where (t, x) ∈ [0,T ]× H , H is a Hilbert space.

Assume:

A : D(A)→ H is a generator of C0-semigroup(etA).

G – is monotone, sublinear, L(H)-continuous.


Short overview


xxu) = 0 ;

u(T , x) = f (x) .(P)


Assume:




Short overview


xxu) = 0 ;

u(T , x) = f (x) .(P)


Assume:




Short overview


xxu) = 0 ;

u(T , x) = f (x) .(P)


Assume:




Short overview

(P) corresponds to the next associated SDE in H:dXτ = AXτ + dBG

τ , τ ∈ (t,T ] ⊂ [0,T ] ;

Xt = x .(S)

where BGt is so called G -B.m. in H;

The solution of (S) is said to be the next process:

Xτ := X t,xτ = e(τ−t)Ax +

τ∫t

e(τ−σ)AdBGσ . (1)


Short overview


τ , τ ∈ (t,T ] ⊂ [0,T ] ;

Xt = x .(S)




τ∫t



Short overview


τ , τ ∈ (t,T ] ⊂ [0,T ] ;

Xt = x .(S)




τ∫t



Short overview

And, roughly speaking, under the certain condition the functionu(t, x) := EG f (X t,x

T ) represents the mild type of solutions of (P), socalled “viscosity solution”.

Where EG is a certain sublinear functional, so called “G -expectation”,which is associated to the G -B.m. BG

t .

Also, we need to understand in this case how to treat the integral

Iτ :=

τ∫0

e(τ−σ)AdBGσ .


Short overview




t .


Iτ :=

τ∫0

e(τ−σ)AdBGσ .


Short overview




t .


Iτ :=

τ∫0

e(τ−σ)AdBGσ .


Outline

1 Short overview

2 G-function






G-function

Let G : D ⊂ LS(H)→ R

Definition 2.1

Such G is said to be G0-function if:1)A ≥ A ⇒ G (A) ≥ G (A).2)G (A + A) ≤ G (A) + G (A);3)G (λA) = λG (A), λ ≥ 0;4)L(H)-continuous.


G-function

Let G : D ⊂ LS(H)→ R

Definition 2.1

Such G is said to be G0-function if:1)A ≥ A ⇒ G (A) ≥ G (A).2)G (A + A) ≤ G (A) + G (A);3)G (λA) = λG (A), λ ≥ 0;4)L(H)-continuous.


G-function

Theorem 2.1

Let X is a linear space.F : X→ R is a sublinear functional, i.e.

1) F (x + y) ≤ F (x) + F (y);2) F (λx) = λF (x) , λ ≥ 0.

Then ∃

fθ : X→ R – linear functional , θ ∈ Θ

, such that:

F (x) = supθ∈Θ

fθ(x), x ∈ X. (2)

Moreover,(a) If F is continuous ⇒ fθ in (2) are continuous.(b) If F is a monotone, sublinear functional, such that F (c) = c ∈ R(such functional is called sublinear expectation)⇒ fθ in (2) are linear expectations.


G-function

Theorem 2.1


1) F (x + y) ≤ F (x) + F (y);2) F (λx) = λF (x) , λ ≥ 0.

Then ∃


, such that:

F (x) = supθ∈Θ

fθ(x), x ∈ X. (2)



G-function

Theorem 2.1


1) F (x + y) ≤ F (x) + F (y);2) F (λx) = λF (x) , λ ≥ 0.

Then ∃


, such that:

F (x) = supθ∈Θ

fθ(x), x ∈ X. (2)



G-function

We see that G 0-function satisfies the conditions of this theorem.

And, naturally, we could suppose that

fθ(A) := Tr[ABθ] – scalar product in Hilbert space.

Then such G 0-function admits the representation

G (A) = supθ∈Θ

Tr[A · Bθ]. (3)

For the reason of consistency we will require that Bθ will be trace-classoperator.

But does it possible to find such Bθ that fθ(A) be a linear continuousfunctional?


G-function





G (A) = supθ∈Θ

Tr[A · Bθ]. (3)




G-function





G (A) = supθ∈Θ

Tr[A · Bθ]. (3)




G-function





G (A) = supθ∈Θ

Tr[A · Bθ]. (3)




G-function

The situation clarifies the next obtained result:

Theorem 2.2

Let G : KS(H)→ R is G-function.Then ∃ the set Σ:

1)Σ ⊂ C1(H) – set of trace-class operators;2)∀B ∈ Σ ⇒ B = B∗, B ≥ 0;3)Σ is convex;4)Σ is closed subspace of C1(H);

5) G (A) =1

2supB∈Σ

Tr[A · B], ∀A ∈ KS(H).

Here we settled the isomorphism between set of trace-class operatorsC1(H) and the dual space of compact operators

(K (H)

)∗.


G-function


Theorem 2.2



5) G (A) =1

2supB∈Σ

Tr[A · B], ∀A ∈ KS(H).


(K (H)

)∗.


G-function


Theorem 2.2



5) G (A) =1

2supB∈Σ

Tr[A · B], ∀A ∈ KS(H).


(K (H)

)∗.


G-function

Interesting, if we consider f (B) :=

0, B ∈ Σ ;

∞, B ∈ C1(H) r Σ ..

Then G (A) =1

2supB∈Σ

Tr[AB]

=1

2sup

B∈C1(H)

Tr[AB]− f (B)

=

1

2f ∗(B),

where f ∗ is the Legendre transform of f .

⇒ 2G = f ∗ ⇒ 2G ∗ = f ∗∗ = f by Fenchel-Moreau theorem.

⇒ Σ =

B ∈ Σ∣∣ G ∗(B) = 0

= kerG ∗.

So, we can set the mutual correspondence: G ↔ Σ.


G-function


0, B ∈ Σ ;

∞, B ∈ C1(H) r Σ ..

Then G (A) =1

2supB∈Σ

Tr[AB]

=1

2sup

B∈C1(H)

Tr[AB]− f (B)

=

1

2f ∗(B),



⇒ Σ =

B ∈ Σ∣∣ G ∗(B) = 0

= kerG ∗.



G-function


0, B ∈ Σ ;

∞, B ∈ C1(H) r Σ ..

Then G (A) =1

2supB∈Σ

Tr[AB]

=1

2sup

B∈C1(H)

Tr[AB]− f (B)

=

1

2f ∗(B),



⇒ Σ =

B ∈ Σ∣∣ G ∗(B) = 0

= kerG ∗.



G-function


0, B ∈ Σ ;

∞, B ∈ C1(H) r Σ ..

Then G (A) =1

2supB∈Σ

Tr[AB]

=1

2sup

B∈C1(H)

Tr[AB]− f (B)

=

1

2f ∗(B),



⇒ Σ =

B ∈ Σ∣∣ G ∗(B) = 0

= kerG ∗.



G-function

One can define the extension of G to LS(H) defining:

G (A) = G Σ(A) :=1

2supB∈Σ

Tr[A · B] , A ∈ LS(H). (4)

And such G will be called G -function.It satisfies all conditions of the G 0-function,it is defined on LS(H),and it is represented in the form (4).


G-function

One can define the extension of G to LS(H) defining:

G (A) = G Σ(A) :=1

2supB∈Σ

Tr[A · B] , A ∈ LS(H). (4)

And such G will be called G -function.It satisfies all conditions of the G 0-function,it is defined on LS(H),and it is represented in the form (4).


Outline

1 Short overview

2 G-function






Sublinear expectation

Let (X, ‖ · ‖X) is a normed space, and Ω is a fixed set.

Define the next class:

C p.Lip (X) =ϕ : X→ R

∣∣∣ |ϕ(x)−ϕ(y)| ≤ C ·(1+‖x‖mX +‖y‖mX

)·‖x−y‖X

Definition 3.1

We define the class H0 to be a such class that:1) If c ∈ R ⇒ c ∈ H0;2) If ξ : Ω→ R is a random variable on a (Ω,F ,P) ⇒ ξ ∈ H0;3) If ξ1, ξ2, . . . , ξn ∈ H0 ⇒ ϕ(ξ1, ξ2, . . . , ξn) ∈ H0 ∀ ϕ ∈ C p.Lip(Rn).

Definition 3.2

SetH :=

X : Ω→ X – r.v. on a (Ω,F ,P) | ψ(X ) ∈ H0 ∀ψ ∈ C p.Lip(X)

.






∣∣∣ |ϕ(x)−ϕ(y)| ≤ C ·(1+‖x‖mX +‖y‖mX

)·‖x−y‖X

Definition 3.1


Definition 3.2

SetH :=


.






∣∣∣ |ϕ(x)−ϕ(y)| ≤ C ·(1+‖x‖mX +‖y‖mX

)·‖x−y‖X

Definition 3.1


Definition 3.2

SetH :=


.






∣∣∣ |ϕ(x)−ϕ(y)| ≤ C ·(1+‖x‖mX +‖y‖mX

)·‖x−y‖X

Definition 3.1


Definition 3.2

SetH :=


.



Definition 3.3

Functional E : H0 → R is called a sublinear expectation if it satisfies thenext conditions:1) X ≥ Y ⇒ E[X ] ≥ E[Y ] – monotonicity;2) c ∈ R ⇒ E[c] = c – constant preserving;3) E[X + Y ] ≤ E[X ] + E[Y ] – sub-additivity;4) λ ≥ 0 ⇒ E[λX ] = λE[X ] – positive homogeneity.

The triple (Ω,H,E) we will call the sublinear expectation space.

It means that ∀ξ ∈ H Fξ[φ] := Eφ(ξ) is defined, φ ∈ C p.Lip(X).



Definition 3.3






Definition 3.3





Distribution

Definition 3.4

X ∼ Y (identical distributed) if:

E[ϕ(X )] = E[ϕ(Y )], ∀ϕ ∈ C p.Lip(H).

Y ⊥⊥ X (Y is independent from X ) if:

E[ϕ(X ,Y )] = E[E[ϕ(x ,Y )]x=X

], ∀ϕ ∈ C p.Lip(H×H).

Definition 3.5

Random variable X on the (Ω,H,E) is said to be G-normal distributedif ∀ X ind. copy of X :

aX + bX ∼√

a2 + b2 X , ∀ a, b > 0. (5)


Distribution

Definition 3.4


E[ϕ(X )] = E[ϕ(Y )], ∀ϕ ∈ C p.Lip(H).


E[ϕ(X ,Y )] = E[E[ϕ(x ,Y )]x=X

], ∀ϕ ∈ C p.Lip(H×H).

Definition 3.5


aX + bX ∼√

a2 + b2 X , ∀ a, b > 0. (5)


Distribution

Definition 3.4


E[ϕ(X )] = E[ϕ(Y )], ∀ϕ ∈ C p.Lip(H).


E[ϕ(X ,Y )] = E[E[ϕ(x ,Y )]x=X

], ∀ϕ ∈ C p.Lip(H×H).

Definition 3.5


aX + bX ∼√

a2 + b2 X , ∀ a, b > 0. (5)


G-distribution

For such fact that X is G-normal distributed it is convenient to have thenext notation:

X ∼ NG (0,Σ).

where G is G -function defined in the next way:

G (A) :=1

2E[〈AX ,X 〉

](6)

Clear that the sublinear expectation provides the fulfillment of allG 0-function’s properties.

But in the same time such G defines the set Σ which we will call thecovariation set.

Because ∀h, k ∈ H:

E[〈X , h〉〈X , k〉

]= sup

Q∈Σ〈Qh, k〉. (7)


G-distribution


X ∼ NG (0,Σ).


G (A) :=1

2E[〈AX ,X 〉

](6)




E[〈X , h〉〈X , k〉

]= sup

Q∈Σ〈Qh, k〉. (7)


G-distribution


X ∼ NG (0,Σ).


G (A) :=1

2E[〈AX ,X 〉

](6)




E[〈X , h〉〈X , k〉

]= sup

Q∈Σ〈Qh, k〉. (7)


G-distribution


X ∼ NG (0,Σ).


G (A) :=1

2E[〈AX ,X 〉

](6)




E[〈X , h〉〈X , k〉

]= sup

Q∈Σ〈Qh, k〉. (7)


G-distribution

The correctness of the notation (6) clarifies the next theorem:

Theorem 3.1

Let G (A) =1

2supB∈Σ

Tr[A · B

]is a given G -function.

Then: ∃X ∼ NG (0,Σ).

Lemma 3.1

X ∼ NG (0,Σ).Then∃cm ∈ R : cm · sup

Q∈ΣTr[Qm]≤ E‖X‖2m

H ≤ cm · supQ∈Σ

(Tr[Q])m

, m ≥ 1.


G-distribution


Theorem 3.1

Let G (A) =1

2supB∈Σ

Tr[A · B



Lemma 3.1




(Tr[Q])m

, m ≥ 1.


G-distribution


Theorem 3.1

Let G (A) =1

2supB∈Σ

Tr[A · B



Lemma 3.1




(Tr[Q])m

, m ≥ 1.


G-Brownian motion

Definition 3.6

Consider stochastic process Xt : Ω×R→ H on (Ω,H,E).It is called as G-Brownian motion if:

1) B0 = 0;2) ∀t, s ≥ 0 (Bt+s − Bt) ∼ NG (0, sΣ);3) ∀t, s ≥ 0 (Bt+s − Bt) is indep. from (Bt1 , ..,Btn)

∀n ∈ N, 0 ≤ t1 ≤ . . . ≤ tn ≤ t.


G-expectation

Proposition 3.1

If Ω =ωt : R+→ U | ω0 = 0, ωt–continious

;

Bt(ωs) := ωs

∣∣s=t

– canonical process;Lip (Ω) :=ϕ (Bt1 , ..,Btn) | n ∈ N, t1, .., tn ∈ R+

, ϕ ∈ C p.Lip (U×Rn → R)

;

Then ∃ subl. expectation E : Lip (Ω)→ R, such that Bt is G-Br.m. on(Ω, Lip (Ω) , E).This sublinear expectation E will be called G-expectation.

We can easy show that correspondent G -function for Bt is:

G (A) =1

2tE[〈ABt ,Bt〉

](8)


G-expectation

Proposition 3.1

If Ω =ωt : R+→ U | ω0 = 0, ωt–continious

;

Bt(ωs) := ωs

∣∣s=t

– canonical process;Lip (Ω) :=ϕ (Bt1 , ..,Btn) | n ∈ N, t1, .., tn ∈ R+

, ϕ ∈ C p.Lip (U×Rn → R)

;

Then ∃ subl. expectation E : Lip (Ω)→ R, such that Bt is G-Br.m. on(Ω, Lip (Ω) , E).This sublinear expectation E will be called G-expectation.

We can easy show that correspondent G -function for Bt is:

G (A) =1

2tE[〈ABt ,Bt〉

](8)


Outline

1 Short overview

2 G-function






B-continuity

Consider the next parabolic PDE:∂tu + 〈Ax ,Dxu〉+ G (D2

xxu) = 0 ;

u(T , x) = f (x) .(P)

u : [0,T ]× H→ R;

f ∈ C p.Lip(H→ R);

G : LS(H)→ R – G -function;



B-continuity

We also impose the next condition on the operator A:

Condition

∃B ∈ LS(H) such that:1)B > 0;2)A∗B ∈ L(H);2)−A∗B + c0B ≥ I , for some c0 > 0.

Remark 4.1

If A is self-adjoint, maximal dissipative operator ⇒ we can takeB := (I − A)−1 with c0 := 1 which satisfy settled above condition.Usually, in applications A = ∆, so such condition for finding thecorresponding B is not too much strict.


B-continuity

We also impose the next condition on the operator A:

Condition

∃B ∈ LS(H) such that:1)B > 0;2)A∗B ∈ L(H);2)−A∗B + c0B ≥ I , for some c0 > 0.

Remark 4.1

If A is self-adjoint, maximal dissipative operator ⇒ we can takeB := (I − A)−1 with c0 := 1 which satisfy settled above condition.Usually, in applications A = ∆, so such condition for finding thecorresponding B is not too much strict.


B-continuity

Later we will need the space H−1 which defines as the completion of H

under the norm ‖x‖2−1 := 〈Bx , x〉 = 〈B

12 x ,B

12 x〉 = ‖B

12 x‖2

H.

Let ej , j ≥ 1 will be a basis of H−1 made of elements of H.

(Clear that in such case B12 ej , j ≥ 1 is a basis of H).

Define HN := spane1, .., eN.And PN will be an orthonormal projection H−1 onto

H : ∀x ∈ H−1 PNx :=N∑j=1〈x , ej〉−1.

QN := I − PN .


B-continuity

Definition 4.1

Let u, v : [0,T ]× H→ R.u is said to be B-l.s.c. (B-lower semicontinuous)

if u(t, x) ≤ limn→∞

u(tn, xn);

And v is said to be B-u.s.c. (B-upper semicontinuous)if u(t, x) ≥ lim

n→∞u(tn, xn),

whenever xnw−→ x , tn → t , Bxn

s−→ Bx.

Definition 4.2

The function which is B-l.s.c. and B-u.s.c. simultaneouslyis called B-continuous.


B-continuity

Remark 4.2

Note that B-continuity means that function u(t, x) is continuous on thebounded sets of [0,T ]× H for the [0,T ]× H−1-topology.

Definition 4.3

The function u(t, x) is locally uniformly B-continuous if it is uniformlycontinuous on the bounded sets of [0,T ]×H for the [0,T ]×H−1-topology.


Test functions and viscosity solutions

Definition 4.4

The function ψ : (0,T )× H→ R is said to be test functionif it admits the representation ψ = ϕ+ χ, such that:

1) ϕ ∈ C 1, 2(

(0,T )× H→ R)

;

ϕ is B-continuous;∂tϕ, A∗Dxϕ, Dxϕ, D2

xxϕ

are loc. uniform. continuous on

(0,T )× H;2) χ : (0,T )× H→ R, χ(t, x) = ξ(t) · η(x);

ξ ∈ C 1(

(0,T )→ (0, +∞))

;

η ∈ C 2(H), η ↑;η(x) = η(y) whenever |x | = |y |;

Dη, D2η

are loc. uniform. continuous on (0,T )× H and have

polynomial growth;


Test functions and viscosity solutions

Definition 4.5

Let u, v : [0,T ]× H→ R.u is said to be a viscosity subsolution of (P) at the point (t0, x0) if:

1) u is B-u.s.c. on [0,T ]× H;2) ∀test function ψ:

u ≤ ψ;u(t0, x0) = ψ(t0, x0);[∂tψ+ < x ,A∗Dxϕ > +G (D2

xxψ)](t0, x0) ≥ 0;

u(T , x) ≤ f (x);


Comparison principle

Using the next comparison principle described in Kelome [Ph.D -thesis:Viscosity solutions of second order equations in a separable Hilbertspace and applications to stochastic optimal control.] we can derivethe uniqueness of the given G-PDE.


Comparison principle

Theorem 4.1 (Comparison principle)

Let u and v – are sub- and super- viscosity solutions of the (P)correspondingly, such that:

1) ∃M > 0 : u ≤ M,v ≥ −M;

2) f is bounded, loc. uniform. B-continuous;3) G satisfies the next contiditions:

(i) A1 ≥ A2 ⇒ G (A1) ≥ G (A2);(ii) ∃ radial, increasing, linearly growing function µ ∈ C 2(H→ R)

with bounded first and second derivatives, such that ∀α > 0:∣∣∣G(A + αD2µ(x))− G

(A)∣∣∣ ≤ C

(1 + |x |) · α;

(iii) sup ∣∣∣G(A + λBQN

)− G

(A)∣∣∣ : ‖A‖, |λ| < p,

A = P∗NAPN , p ∈ R−−−−→N→∞

0.

Then u ≤ v.


Outline

1 Short overview

2 G-function






Basic space constructions

Let U,H are Hilbert spaces.

ei and fj are ONB in U and H correspondingly.

Bt ∈ U, Bt ∼ NG

(0, t ·Σ

)– G -B.m., Σ is fixed.

Now we would like to define a Banach space of operators Φ with values inH such that D(Φ) ⊂ U, endowed with the norm

‖Φ‖2LΣ

2:= sup

Q∈ΣTr[ΦQΦ∗

]= sup

Q∈Σ‖ΦQ

12 ‖2

L2(U,H).

It easy to see that ‖ · ‖2LΣ

2is a norm.





Bt ∈ U, Bt ∼ NG

(0, t ·Σ



‖Φ‖2LΣ

2:= sup

Q∈ΣTr[ΦQΦ∗

]= sup

Q∈Σ‖ΦQ

12 ‖2

L2(U,H).


2is a norm.





Bt ∈ U, Bt ∼ NG

(0, t ·Σ



‖Φ‖2LΣ

2:= sup

Q∈ΣTr[ΦQΦ∗

]= sup

Q∈Σ‖ΦQ

12 ‖2

L2(U,H).


2is a norm.



UΣ := spanimQ

12 (U), Q ∈ Σ

=

u =m∑i=1

αiui

∣∣αi ∈ R, ui ∈ imQ12i (U), Qi ∈ Σ, m ≥ i ≥ 1

.

Let u ∈ UΣ

‖u‖UΣ:= inf

m∑i=1‖ui‖0,i

∣∣ u =m∑i=1

ui , ui ∈ imQ12i (U), Qi ∈ Σ

.

∀Qi ∈ Σ U0,i := imQ12i (U), ‖ui‖0,i := ‖Q−

12

i ui‖U, ui ∈ U0,i .

We know that(U0,i , ‖ · ‖U0,i

)is a Banach space

[Da Prato, Zabczyk.

Stochastic Equations in Infinite Dimensions. (1992)].



UΣ := spanimQ

12 (U), Q ∈ Σ

=

u =m∑i=1

αiui


.

Let u ∈ UΣ

‖u‖UΣ:= inf

m∑i=1‖ui‖0,i

∣∣ u =m∑i=1


.

∀Qi ∈ Σ U0,i := imQ12i (U), ‖ui‖0,i := ‖Q−

12



)is a Banach space

[Da Prato, Zabczyk.




UΣ := spanimQ

12 (U), Q ∈ Σ

=

u =m∑i=1

αiui


.

Let u ∈ UΣ

‖u‖UΣ:= inf

m∑i=1‖ui‖0,i

∣∣ u =m∑i=1


.

∀Qi ∈ Σ U0,i := imQ12i (U), ‖ui‖0,i := ‖Q−

12



)is a Banach space

[Da Prato, Zabczyk.




UΣ := spanimQ

12 (U), Q ∈ Σ

=

u =m∑i=1

αiui


.

Let u ∈ UΣ

‖u‖UΣ:= inf

m∑i=1‖ui‖0,i

∣∣ u =m∑i=1


.

∀Qi ∈ Σ U0,i := imQ12i (U), ‖ui‖0,i := ‖Q−

12



)is a Banach space

[Da Prato, Zabczyk.




Proposition 5.1(UΣ, ‖ · ‖UΣ

)is a Banach space.

LΣ2 :=

Φ : UΣ → H

∣∣ ΦQ12 ∈ L2(U,H) and ‖Φ‖LΣ

2<∞

.

Proposition 5.2(LΣ

2 , ‖ · ‖LΣ2

)is a Banach space.

Proposition 5.3

LΣ2 ≡

Φ ∈ L(UΣ,H)

∣∣ ∀Q ∈ Σ ΦQ12 ∈ L2(U,H) , sup

Q∈ΣTr[ΦQΦ∗

]<∞

.




)is a Banach space.

LΣ2 :=

Φ : UΣ → H


2<∞

.

Proposition 5.2(LΣ

2 , ‖ · ‖LΣ2

)is a Banach space.

Proposition 5.3

LΣ2 ≡

Φ ∈ L(UΣ,H)

∣∣ ∀Q ∈ Σ ΦQ12 ∈ L2(U,H) , sup

Q∈ΣTr[ΦQΦ∗

]<∞

.




)is a Banach space.

LΣ2 :=

Φ : UΣ → H


2<∞

.

Proposition 5.2(LΣ

2 , ‖ · ‖LΣ2

)is a Banach space.

Proposition 5.3

LΣ2 ≡

Φ ∈ L(UΣ,H)

∣∣ ∀Q ∈ Σ ΦQ12 ∈ L2(U,H) , sup

Q∈ΣTr[ΦQΦ∗

]<∞

.




)is a Banach space.

LΣ2 :=

Φ : UΣ → H


2<∞

.

Proposition 5.2(LΣ

2 , ‖ · ‖LΣ2

)is a Banach space.

Proposition 5.3

LΣ2 ≡

Φ ∈ L(UΣ,H)

∣∣ ∀Q ∈ Σ ΦQ12 ∈ L2(U,H) , sup

Q∈ΣTr[ΦQΦ∗

]<∞

.


Construction of the stochastic integral

Let Lip(ΩT ) :=ϕ(Bt1∧T , ..,Btn∧T ) | n ∈ N, t1, .., tn ∈ R+

, ϕ ∈ C p.Lip (Un → R)

.

HL2,0G

(ΩT

):= m∑

i=1ai (ω)φi , ai ∈ Lip

(ΩT

), φi ∈ LΣ

2 , m ≥ i ≥ 1

.

HL2G

(ΩT

)is completion of HL2,0

G

(ΩT

)under the

‖Φ‖Σ =(E[‖Φ‖2

LΣ2

]) 12

=(E[

supQ∈Σ

Tr[ΦQΦ∗

]]) 12.





.

HL2,0G

(ΩT

):= m∑


(ΩT

), φi ∈ LΣ

2 , m ≥ i ≥ 1

.

HL2G

(ΩT


G

(ΩT

)under the

‖Φ‖Σ =(E[‖Φ‖2

LΣ2

]) 12

=(E[

supQ∈Σ

Tr[ΦQΦ∗

]]) 12.





.

HL2,0G

(ΩT

):= m∑


(ΩT

), φi ∈ LΣ

2 , m ≥ i ≥ 1

.

HL2G

(ΩT


G

(ΩT

)under the

‖Φ‖Σ =(E[‖Φ‖2

LΣ2

]) 12

=(E[

supQ∈Σ

Tr[ΦQΦ∗

]]) 12.



HM2,0G

(0,T

):=

Φ(t) =N−1∑k=0

Φk(ω)1[tk ,tk+1)(t)∣∣∣

Φk(ω) ∈ HL2G (Ωtk ), 0 = t0 < t1 . . . < tN = T

.

HM2G

(0,T

)is completion of HM2,0

G (0,T ) under

|||Φ|||T :=(E[ T∫

0

‖Φ(t)‖2LΣ

2dt]) 1

2.

For Φ ∈ HM2,0G

(0,T

): I (Φ) =

T∫0

Φ(t)dBt :=N−1∑k=0

Φk(Btk+1− Btk ) .



HM2,0G

(0,T

):=

Φ(t) =N−1∑k=0

Φk(ω)1[tk ,tk+1)(t)∣∣∣

Φk(ω) ∈ HL2G (Ωtk ), 0 = t0 < t1 . . . < tN = T

.

HM2G

(0,T


G (0,T ) under

|||Φ|||T :=(E[ T∫

0

‖Φ(t)‖2LΣ

2dt]) 1

2.

For Φ ∈ HM2,0G

(0,T

): I (Φ) =

T∫0





HM2,0G

(0,T

):=

Φ(t) =N−1∑k=0

Φk(ω)1[tk ,tk+1)(t)∣∣∣

Φk(ω) ∈ HL2G (Ωtk ), 0 = t0 < t1 . . . < tN = T

.

HM2G

(0,T


G (0,T ) under

|||Φ|||T :=(E[ T∫

0

‖Φ(t)‖2LΣ

2dt]) 1

2.

For Φ ∈ HM2,0G

(0,T

): I (Φ) =

T∫0





HK 2,0G

(0,T

):=

Φ(ω) =N−1∑k=0

Φk(ω)(Btk+1− Btk )

∣∣∣Φk(ω) ∈ HL2

G (Ωtk ), 0 = t0 < t1 . . . < tN = T

.

HK 2G

(0,T

)is completion of HK 2,0

G (0,T ) under ‖I‖ΩT:=(E‖I‖2

H

) 12.



HK 2,0G

(0,T

):=

Φ(ω) =N−1∑k=0

Φk(ω)(Btk+1− Btk )

∣∣∣Φk(ω) ∈ HL2

G (Ωtk ), 0 = t0 < t1 . . . < tN = T

.

HK 2G

(0,T

)is completion of HK 2,0

G (0,T ) under ‖I‖ΩT:=(E‖I‖2

H

) 12.


Ito’s isometry and BDG

For such integral we have the next obtained result:

Theorem 5.1 (Ito’s isometry inequality)

For stochastic integral I : HM2,0G

(0,T

)→ HK 2,0

G

(0,T

)holds the next

inequality

E[‖

T∫0

Φ(t) dBt‖2H

]≤ E

[ T∫0

‖Φ(t)‖2LΣ

2dt]. (9)

or‖I (Φ)‖ΩT

≤ |||Φ|||T . (10)






(0,T

)→ HK 2,0

G

(0,T

)holds the next

inequality

E[‖

T∫0

Φ(t) dBt‖2H

]≤ E

[ T∫0

‖Φ(t)‖2LΣ

2dt]. (9)

or‖I (Φ)‖ΩT

≤ |||Φ|||T . (10)






(0,T

)→ HK 2,0

G

(0,T

)holds the next

inequality

E[‖

T∫0

Φ(t) dBt‖2H

]≤ E

[ T∫0

‖Φ(t)‖2LΣ

2dt]. (9)

or‖I (Φ)‖ΩT

≤ |||Φ|||T . (10)



Owning to Ito’s isometry inequality one can extent integralto the M2

G

(0,T

):

I : HM2G

(0,T

)→ HK 2

G

(0,T

)Theorem 5.2 (BDG inequality)

E[‖

T∫0

Φ(t) dBt‖pH]≤ Cp · E

( T∫0

‖Φ(t)‖2LΣ

2dt) p

2. (11)




G

(0,T

):

I : HM2G

(0,T

)→ HK 2

G

(0,T


E[‖

T∫0


( T∫0

‖Φ(t)‖2LΣ

2dt) p

2. (11)




G

(0,T

):

I : HM2G

(0,T

)→ HK 2

G

(0,T


E[‖

T∫0


( T∫0

‖Φ(t)‖2LΣ

2dt) p

2. (11)


Outline

1 Short overview

2 G-function






The solving PDE

Turn back to the PDE (P):∂tu + 〈Ax ,Dxu〉+ G (D2

xxu) = 0 ;

u(T , x) = f (x) .(P)

It solves in the viscosity sense by using the notion of B-continuity.


The solving PDE

Turn back to the PDE (P):∂tu + 〈Ax ,Dxu〉+ G (D2

xxu) = 0 ;

u(T , x) = f (x) .(P)

It solves in the viscosity sense by using the notion of B-continuity.


The solving PDE

Summarize all of that we arrive to the such final result:

Theorem 6.1 ∂tu + 〈Ax ,Dxu〉+ G (D2

xxu) = 0 ;

u(T , x) = f (x) .(P)

u : [0,T ]× H→ R;f ∈ C p.Lip(H→ R);G : LS(H)→ R is G -function;A : D(A)→ H is a generator of C0-semigroup

(etA), such that

∃ positive B ∈ LS(H) : A∗B ∈ L(H) , −A∗B + c0B ≥ I , for somec0 > 0.


The solving PDE



xxu) = 0 ;

u(T , x) = f (x) .(P)


(etA), such that



The solving PDE



xxu) = 0 ;

u(T , x) = f (x) .(P)


(etA), such that



The solving PDE



xxu) = 0 ;

u(T , x) = f (x) .(P)


(etA), such that



The solving PDE



xxu) = 0 ;

u(T , x) = f (x) .(P)


(etA), such that



The solving PDE



xxu) = 0 ;

u(T , x) = f (x) .(P)


(etA), such that



The solving PDE



xxu) = 0 ;

u(T , x) = f (x) .(P)


(etA), such that



The solving PDE

Theorem

Then ∃Bt – G -B.m. with correspondent G -expectation E such thatu(t, x) := E[f (X t,x

T )] is unique C p.Lip-class viscosity solution of (P),

where X t,xτ = e(τ−t)Ax +

τ∫t

e(τ−σ)AdBσ is solution of (S):dXτ = AXτ + dBτ , τ ∈ (t,T ] ⊂ [0,T ] ;

Xt = x .(S)


The solving PDE

Theorem




τ∫t


Xt = x .(S)


The solving PDE

Theorem




τ∫t


Xt = x .(S)


The solving PDE

Theorem




τ∫t


Xt = x .(S)


References

Peng, S. (2007) Lecture Notes: G-Brownian motion and dynamic riskmeasure under volatility uncertainty, in arXiv:0711.2834v1 [math.PR].

Peng, S. (2010) Nonlinear Expectations and Stochastic Calculus underUncertainty, in arXiv:1002.4546v1 [math.PR].

Da Prato, G. (2001) An Introduction to infinite dimensional analysis.

Da Prato, G., Zabczyk, J. (1992) Stochastic Equations in InfiniteDimensions.

Kelome, D. (2002). Ph.D -thesis: Viscosity solutions of second orderequations in a separable Hilbert space and applications to stochasticoptimal control.


The end

Thank you for your attention!


Documents

G-expectations in infinite dimensions and related PDEITN2012/files/talk/Ibragimov.pdfOutline 1 Short overview 2 G-function 3 Main notions of the G-expectation theory 4 Viscosity solutions