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Appendix: Proof of Theorem 7.13

# Appendix: Proof of Theorem 7.13

Tải bản đầy đủ - 0trang

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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Appendix: Proof of Theorem 7.13

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