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Appendix: Proof of Theorem 7.13

Appendix: Proof of Theorem 7.13

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210



G. Di Nunno et al.

T



+E



R0



0



g X(T ) + Dt,z g X(T )



∂θ (t)

∂θ (t)

Y (t) +

β(t) ν(dz) dt

∂x

∂u



× Dt+,z

T



=E



g X(T )

0



+



R0



∂b(t)

∂σ (t)

+ Dt g X(T )

∂x

∂x

∂θ (t)

ν(dz) Y (t) dt

∂x



Dt,z g X(T )

T



+E



∂b(t)

∂σ (t)

+ Dt g X(T )

∂u

∂u



g X(T )

0



+



R0



∂θ (t)

ν(dz) β(t) dt

∂u



Dt,z g X(T )

T



+E



g X(T ) Dt+

0

T



+E



g X(T )

0



∂σ (t)

Dt+ Y (t) dt

∂x



T



+E



g X(T ) Dt+

0

T



+E



g X(T )

0

T



+E



R0



0

T



+E



R0



0



∂σ (t)

Y (t) dt

∂x



∂σ (t)

β(t) dt

∂u



∂σ (t)

Dt+ β(t) dt

∂u



g X(T ) + Dt,z g X(T ) Dt+,z

g X(T ) + Dt,z g X(T )



∂θ (t)

Y (t)ν(dz) dt

∂x



∂θ (t)

∂θ (t)

+ Dt+,z

∂x

∂x



× Dt+,z Y (t)ν(dz) dt

T



+E



R0



0

T



+E

0



R0



g X(T ) + Dt,z g X(T ) Dt+,z

g X(T ) + Dt,z g X(T )



∂θ (t)

β(t)ν(dz) dt

∂u



∂θ (t)

∂θ (t)

+ Dt+,z

∂u

∂u



× Dt+,z β(t)ν(dz) dt

T



=E



g X(T )

0



∂b(t)

∂σ (t)

+ Dt+

+

∂x

∂x



R0



Dt+,z



∂θ (t)

ν(dz)

∂x



7 A General Maximum Principle for Anticipative Stochastic Control



∂σ (t)

∂x



+ Dt g X(T )

+



R0



∂θ (t)

∂θ (t)

+ Dt+,z

ν(dz) Y (t) dt

∂x

∂x



Dt,z g X(T )

T



+E



211



∂σ (t)

∂b(t)

+ Dt+

+

∂u

∂u



g X(T )

0



R0



Dt+,z



∂θ (t)

ν(dz)

∂u



∂σ (t)

+ Dt g X(T )

∂u

+



R0



∂θ (t)

∂θ (t)

+ Dt+,z

ν(dz) β(t) dt

∂u

∂u



Dt,z g X(T )

T



+E



g X(T )



∂σ (t)

Dt+ Y (t) dt

∂x



g X(T )



∂σ (t)

Dt+ β(t) dt

∂u



0

T



+E

0



T



+E



R0



0



g X(T ) + Dt,z g X(T )



∂θ (t)

∂θ (t)

+ Dt+,z

∂x

∂x



× Dt+,z Y (t)ν(dz) dt

T



+E



R0



0



g X(T ) + Dt,z g X(T )



∂θ (t)

∂θ (t)

+ Dt+,z

∂u

∂u



× Dt+,z β(t)ν(dz) dt .



Similarly, we have using both Fubini and duality theorems,

T



E

0





f (t)Y (t) dt

∂x

T



=E

0

t



+





f (t)

∂x



t



∂θ (s)

∂θ (s)

Y (s) +

β(s) N dz, d − s

∂x

∂u



R0



0

T



=E

0



0



∂b(s)

∂b(s)

Y (s) +

β(s) ds

∂x

∂u



∂σ (s)

∂σ (s)

Y (s) +

β(s) d − B(s)

∂x

∂u



0



+



t



t

0



dt



∂f (t) ∂b(s)

∂b(s)

Y (s) +

β(s) ds dt

∂x

∂x

∂u



212



G. Di Nunno et al.

T



+E



t



Ds

0



0

T



+E



t



0



0

T



+E



∂θ (s)

∂θ (s)

Y (s) +

β(s) ν(dz) ds dt

∂x

∂u



T



T



0



∂f (t)

dt

∂x



s

T



T



Ds

0



s

T



T



0



s



Ds,z



R0



s



T



T

R0



0



∂σ (s)

∂σ (s)

Y (s) +

β(s)

∂x

∂u



∂f (t)

dt

∂x



T



0



+E



∂b(s)

∂b(s)

Y (s) +

β(s) ds

∂x

∂u



∂f (t)

∂σ (s)

∂σ (s)

dt Ds+

Y (s) +

β(s) ds

∂x

∂x

∂u



T



+E



∂θ (s)

∂f (t) ∂θ (s)

Y (s) +

β(s) ν(dz) ds dt

∂x

∂x

∂u



∂f (t)

∂f (t)

+ Ds,z

∂x

∂x



R0



0



× Ds+,z



+E



Ds,z



t



0



+E



R0



0

T



=E



∂f (t)

∂σ (s)

∂σ (s)

Ds+

Y (s) +

β(s) ds dt

∂x

∂x

∂u



t



0



+E



∂σ (s)

∂f (t) ∂σ (s)

Y (s) +

β(s) ds dt

∂x

∂x

∂u



s



∂θ (s)

∂θ (s)

Y (s) +

β(s) ν(dz) ds

∂x

∂u



∂f (t)

dt

∂x



∂f (t)

∂f (t)

+ Ds,z

dt

∂x

∂x



∂θ (s)

∂θ (s)

Y (s) +

β(s) ν(dz) ds .

∂x

∂u



× Ds+,z



Changing the notation s → t, this becomes

T



=E



T



0



∂f (s)

ds

∂x



t

T



+E



T



Dt

0



t

T



+E



T



+E

0



Dt,z

t

T



t



0



T

R0



∂σ (t)

∂σ (t)

Y (t) +

β(t)

∂x

∂u



∂f (s)

ds

∂x



∂θ (t)

∂θ (t)

Y (t) +

β(t) ν(dz) dt

∂x

∂u



∂f (s)

∂σ (t)

∂σ (t)

ds Dt+

Y (t) +

β(t) dt

∂x

∂x

∂u



T



+E



∂f (s)

ds

∂x



T

R0



0



∂b(t)

∂b(t)

Y (t) +

β(t) dt

∂x

∂u



t



∂f (s)

∂f (s)

+ Dt,z

ds

∂x

∂x



7 A General Maximum Principle for Anticipative Stochastic Control



∂θ (t)

∂θ (t)

Y (t) +

β(t)

∂x

∂u



× Dt+,z

T



=E



T



0



t

T



+



Dt

t



+



Dt,z



R0



t

T



T



0



t



T



+



Dt

t



Dt,z

t

T



+E

0



∂f (s)

ds

∂x



0



∂f (s)

ds

∂x



t



∂f (s)

ds

∂x



∂θ (t)

ν(dz)

∂x



R0



Dt+,z



∂θ (t)

ν(dz)

∂u



∂θ (t)

∂θ (t)

+ Dt+,z

ν(dz) β(t) dt

∂u

∂u



T



∂f (s)

∂σ (t)

ds

Dt+ β(t) dt

∂x

∂u

T



R0



0



R0



∂b(t)

∂σ (t)

+ Dt+

+

∂u

∂u



∂σ (t)

∂f (s)

ds

Dt+ Y (t) dt

∂x

∂x



T



+E



Dt+,z



∂θ (t)

∂θ (t)

+ Dt+,z

ν(dz) Y (t) dt

∂x

∂x



T

t



T



+E



∂σ (t)

∂b(t)

+ Dt+

+

∂x

∂x



∂σ (t)

∂f (s)

ds

∂x

∂u



T

R0



ν(dz) dt



∂f (s)

∂σ (t)

ds

∂x

∂x



T



+E



+



∂f (s)

ds

∂x



213



t



∂f (s)

∂f (s)

+ Dt,z

ds

∂x

∂x



∂θ (t)

∂θ (t)

+ Dt+,z

∂x

∂x



∂f (s)

∂f (s)

+ Dt,z

ds

∂x

∂x



∂θ (t)

∂θ (t)

+ Dt+,z

∂u

∂u



× Dt+,z Y (t)ν(dz) dt

T



+E

0



T

R0



t



× Dt+,z β(t)ν(dz) dt .



(7.78)



Recall that

T



K(t) := g X(T ) +

t





f s, X(s), u(s) ds,

∂x



and combining (7.33)–(7.78), it follows that

T



0=E



K(t)

0



∂b(t)

∂σ (t)

+ Dt+

+

∂x

∂x



R0



Dt+,z



∂θ (t)

ν(dz)

∂x



214



G. Di Nunno et al.



+ Dt K(t)

+



R0



∂σ (t)

∂x



Dt,z K(t)

T



+E



K(t)

0



∂θ (t)

∂θ (t)

+ Dt+,z

ν(dz) Y (t) dt

∂x

∂x

∂σ (t)

∂b(t)

+ Dt+

+

∂u

∂u



R0



Dt+,z



∂θ (t)

ν(dz)

∂u



∂σ (t)

+ Dt K(t)

∂u

+



R0



Dt,z K(t)

T



+E



K(t)



∂σ (t)

Dt+ Y (t) dt

∂x



K(t)



∂σ (t)

Dt+ β(t) dt

∂u



0

T



+E

0



T



+E



∂θ (t)

∂f (t)

∂θ (t)

+ Dt+,z

ν(dz) +

β(t) dt

∂u

∂u

∂u



R0



0



K(t) + Dt,z K(t)



∂θ (t)

∂θ (t)

+ Dt+,z

∂x

∂x



× Dt+,z Y (t)ν(dz) dt

T



+E

0



R0



K(t) + Dt,z K(t)



∂θ (t)

∂θ (t)

+ Dt+,z

∂u

∂u



× Dt+,z β(t)ν(dz) dt .



(7.79)



We observe that AG contains all βα given as βα (s) := αχ[t,t+h] (s) for some t, h ∈

(0, T ), t + h ≤ T , where α = α(ω) is bounded and Gt -measurable. Then Y (βα ) (s) =

0 for 0 ≤ s ≤ t, and hence (7.79) becomes

A1 + A2 + A3 + A4 + A5 + A6 = 0,

where

T



A1 = E



K(t)

t



+



R0



∂b(s)

∂σ (s)

+ Ds+

+

∂x

∂x



Ds,z K(s)



× Y (βα ) (s) ds ,



R0



Ds+,z



∂θ (s)

ν(dz)

∂x



∂θ (s)

∂θ (s)

∂σ (s)

+ Ds+,z

ν(dz) + Ds K(s)

∂x

∂x

∂x



(7.80)



7 A General Maximum Principle for Anticipative Stochastic Control

t+h



A2 = E



K(t)

t



+



R0



Ds,z K(s)



T



A3 = E



K(s)

t



K(s)

t

T



A5 = E



∂f (s)

∂θ (s)

ν(dz) +

∂u

∂u



∂θ (s)

∂θ (s)

∂σ (s)

+ Ds+,z

ν(dz) + Ds K(s)

α ds ,

∂u

∂u

∂u



∂σ (s)

Ds+ α ds ,

∂u



K(s) + Ds,z K(s)



R0



t



R0



Ds+,z



∂σ (s)

Ds+ Y (βα ) (s) ds ,

∂x



t+h



A4 = E



∂σ (s)

∂b(s)

+ Ds+

+

∂u

∂u



215



∂θ (s)

∂θ (s)

+ Ds+,z

∂x

∂x



× ν(dz)Ds+,z Y (βα ) (s) ds ,

t+h



A6 = E



K(s) + Ds,z K(s)



R0



t



∂θ (s)

∂θ (s)

+ Ds+,z

∂u

∂u



× ν(dz)Ds+,z α ds .

Note that by the definition of Y with Y (s) = Y (βα ) (s) and s ≥ t + h, the process

Y (s) follows the dynamics

dY (s) = Y s −

+



R0



∂b

∂σ

(s) ds +

(s) d − B(s)

∂x

∂x

∂θ

(s, z)N dz, d − s

∂x



(7.81)



for s ≥ t + h with initial condition Y (t + h) at time t + h. By Itô’s formula for

forward integral, this equation can be solved explicitly, and we get

Y (s) = Y (t + h)G(t + h, s),



s ≥ t + h,



(7.82)



where, in general, for s ≥ t,

s



G(t, s) := exp

t

s



+

t



∂σ

r, X(r), u(r), ω dB − (r)

∂x



s



+

t



1 ∂σ

∂b

r, X(r), u(r), ω −

∂x

2 ∂x



R0



ln 1 +



∂θ

r, X(r), u(r), ω

∂x



2



r, X(r), u(r), ω



dr



216



G. Di Nunno et al.







∂θ

r, X(r), u(r), ω

∂x

s



+



∂θ

r, X r − , u r − , ω

∂x



ln 1 +



R0



t



ν(dz) dt

N dz, d − r



.



Note that G(t, s) does not depend on h, but Y (s) does. Defining H0 as in (7.27), it

follows that

T



A1 = E



∂H0

(s)Y (s) ds .

∂x



t



Differentiating with respect to h at h = 0, we get

d

A1

dh



t+h



d

E

dh



=

h=0



+



t



∂H0

(s)Y (s) ds

∂x



T



d

E

dh



t+h



h=0



∂H0

(s)Y (s) ds

∂x



.

h=0



Since Y (t) = 0, we see that

t+h



d

E

dh



t



∂H0

(s)Y (s) ds

∂x



= 0.

h=0



Therefore, by (7.82),

d

A1

dh



=

h=0



d

E

dh

T



=

t



T



=

t



T

t+h



∂H0

(s)Y (t + h)G(t + h, s) ds

∂x



∂H0

d

E

(s)Y (t + h)G(t + h, s)

dh

∂x

d

∂H0

E

(s)G(t, s)Y (t + h)

dh

∂x



h=0



ds

h=0



ds,

h=0



where Y (t + h) is given by

t+h



Y (t + h) =



Y r−



t



+



R0



∂θ

(r, z)N dz, d − r

∂x

t+h





t



∂b

∂σ

(r) dr +

(r) d − B(r)

∂x

∂x



∂σ

∂b

(r) dr +

(r) d − B(r) +

∂u

∂u



Therefore, by the two preceding equalities,

d

A1

dh



= A1,1 + A1,2 ,

h=0



R0



∂θ

(r, z)N dz, d − r

∂u



.



7 A General Maximum Principle for Anticipative Stochastic Control



217



where

T



A1,1 =



d

∂H0

E

(s)G(t, s)α

dh

∂x



t



+



t+h

t



∂θ

(r, z)N dz, d − r

∂u



R0



∂b

∂σ

(r) dr +

(r) d − B(r)

∂u

∂u

ds,



h=0



and

T



A1,2 =

t



+



d

∂H0

E

(s)G(t, s)

dh

∂x

R0



∂θ

(r, z)N dz, d − r

∂x



t+h



∂b

∂σ

(r) dr +

(r) d − B(r)

∂x

∂x



Y r−



t



ds.

h=0



Applying again the duality formula, we have

T



A1,1 =

t



t+h



d

E α

dh



t



+ F (t, s)Dr +

+



∂θ

(r, z)F (t, s) ν(dz) dr

∂u



T



=



E α

t



+



∂σ

(r)

∂u



∂θ

∂θ

(r, z) + Dr + ,z (r, z) Dr,z F (t, s)

∂u

∂u



R0



+ Dr + ,z



∂σ

∂b

(r)F (t, s) +

(r)Dr F (t, s)

∂u

∂u



∂b

∂σ

(t) + Dt +

(t) +

∂u

∂u



∂σ

(t)Dt F (t, s) +

∂u



× Dt,z F (t, s)ν(dz)



ds

h=0



R0



Dt + ,z



∂θ

(t, z)ν(dz) F (t, s)

∂u



∂θ

∂θ

(t, z) + Dt + ,z (t, z)

∂u

∂u



R0



ds,



where we have put

F (t, s) =



∂H0

(s)G(t, s).

∂x



Since Y (t) = 0, we see that

A1,2 = 0.

We conclude that

d

A1

dh



= A1,1 .

h=0



(7.83)



218



G. Di Nunno et al.



Moreover, we see that

d

A2

dh



=E



∂σ (t)

∂b(t)

+ Dt+

+

∂u

∂u



K(t)



h=0



R0



Dt+,z



∂θ (t, z)

ν(dz)

∂u



∂σ (t, z)

∂f (t)

+ Dt K(t)

+

∂u

∂u

∂θ (t, z)

∂θ (t, z)

Dt,z K(t)

+ Dt+,z

ν(dz) α ,

+

∂u

∂u

R0

d

A4

dh

d

A6

dh



= E K(t)

h=0



=E



R0



h=0



(7.84)



∂σ (t)

Dt+ α ,

∂u



(7.85)

∂θ (t, z)

∂θ (t, z)

+ Dt+,z

∂u

∂u



K(t) + Dt,z K(t)



× ν(dz)Dt+,z α .



(7.86)



On the other hand, by differentiating A3 with respect to h at h = 0, we get

d

A3

dh



=

h=0



d

E

dh

+



t+h



K(s)

t



d

E

dh



∂σ (s)

Ds+ Y (s) ds

∂x



T



K(s)

t+h



∂σ (s)

Ds+ Y (s) ds

∂x



h=0



.

h=0



Since Y (t) = 0, we see that

d

A3

dh



=

h=0



d

E

dh

T



=

t



T



=

t



T



K(s)

t+h



∂σ (s)

Ds+ Y (t + h)G(t + h, s) ds

∂x



∂σ (s)

d

E K(s)

Ds+ Y (t + h)G(t + h, s)

dh

∂x

d

∂σ (s)

E K(s)

Ds+ G(t + h, s) · Y (t + h)

dh

∂x



+ Ds+ Y (t + h) · G(t + h, s)



ds

h=0



T



=

t



d

∂σ (s)

E K(s)

Y (t + h)Ds+ G(t, s)

dh

∂x



+ Ds+ Y (t + h)G(t, s)



ds.

h=0



h=0



ds

h=0



7 A General Maximum Principle for Anticipative Stochastic Control



219



Using the definition of p and H given respectively by (7.36) and (7.35) in the theorem, it follows by (7.80) that



H t, X(t), u(t) Gt + E[A] = 0 a.e. in (t, ω),

∂u



E



(7.87)



where

d

A3

dh



A=



+

h=0



d

A4

dh



+

h=0



d

A5

dh



+

h=0



d

A6

dh



.



(7.88)



h=0



2. Conversely, suppose that there exists u ∈ AG such that (7.34) holds. Then

by reversing the previous arguments, we obtain that (7.80) holds for all βα (s) :=

αχ[t,t+h] (s) ∈ AG , where

T



A1 = E



∂σ (s)

∂b(s)

+ Ds+

+

∂x

∂x



K(t)

t



+



R0



Ds,z K(s)



+ Ds K(s)



K(t)

t



+



R0



Ds,z K(s)



T



A3 = E



K(s)

t



K(s)

t

T



A5 = E

t



∂θ (s)

∂θ (s)

+ Ds+,z

ν(dz)

∂x

∂x



R0



∂σ (s)

∂b(s)

+ Ds+

+

∂u

∂u



R0



Ds+,z



∂θ (s)

∂f (s)

ν(dz) +

∂u

∂u



∂θ (s)

∂θ (s)

∂σ (s)

+ Ds+,z

ν(dz) + Ds K(s)

α ds ,

∂u

∂u

∂u



∂σ (s)

Ds+ Y (βα ) (s) ds ,

∂x



t+h



A4 = E



∂θ (s)

ν(dz)

∂x



∂σ (s) (βα )

(s) ds ,

Y

∂x



t+h



A2 = E



R0



Ds+,z



∂σ (s)

Ds+ α ds ,

∂u



K(s) + Ds,z K(s)



∂θ (s)

∂θ (s)

+ Ds+,z

∂x

∂x



× ν(dz)Ds+,z Y (βα ) (s) ds ,

t+h



A6 = E

t



R0



K(s) + Ds,z K(s)



∂θ (s)

∂θ (s)

+ Ds+,z

∂u

∂u



× ν(dz)Ds+,z α ds

for some t, h ∈ (0, T ), t + h ≤ T , where α = α(ω) is bounded and Gt -measurable.

Hence, these equalities hold for all linear combinations of βα . Since all bounded



220



G. Di Nunno et al.



β ∈ AG can be approximated pointwise boundedly in (t, ω) by such linear combinations, it follows that (7.80) holds for all bounded β ∈ AG . Hence, by reversing the

remaining part of the previous proof, we conclude that

d

J1 (u + yβ)

dy



= 0 for all β,

y=0



and then u satisfies (7.33).



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