Ứng dụng của nghiệm nhớt đối với toán tài chính

x, y ∈ R
x ∧ y = min(x, y), x ∨ y = max(x, y),
x
+
= max(x, 0), x
−
= max(−x, 0).
R
d
d− R = R
1
.
x = (x
1
, , x
d
), y = (y
1
, , y
d
) ∈ R
d
x.y =
d


i=1
x
i
y
i
; |x| =
√
x.x.
R
n×d
n × d. I
n
n ×n. σ = (σ
ij
)
1≤i≤n,1≤j≤d
∈ R
n×d
σ

= (σ
ji
)
1≤j≤d,1≤i≤n
σ
tr(A) =

n
i=1
a

ii
A =

a
ij

1≤i,j≤n
∈ R
n×n
R
n×d
|σ| =

tr

σσ


1
2
A
1
A
(x) =



1, x ∈ A
0, x ∈ A.
O ⊂ R

n
T = [0, T ] T = [0, ∞)
C
k
(O) k
O.
C
0
(T × O) T × O.
C
1,2
(T ×O) f T ×O
∂f
∂t
,
∂f
∂x
i
,
∂
2
f
∂x
i
∂x
j
, 1 ≤ i, j ≤ n T × O
(Ω, F, P )
P − P.
B(U) σ− U

σ(G) σ− G Ω
Q  P Q P
Q ∼ P Q P Q  P, P  Q
dQ
dP
Q  P
E
Q
(X) Q X
E(X) X P
E(X | G) X G
V ar(X) = E[(X − E(X))(X − E(X))

] X
L
p
(Ω, F, P ; R
n
) X
R
n
F− E|X|
p
< ∞, p ∈ [1, ∞)
L
p
(P ) L
p
L
∞

(Ω, F, P ; R
n
) X
R
n
F− L
∞
ν = (Ω, F, P )
X = (X
t
)
t∈T
R
d
t ∈ T.
T = [0, T ], T > 0 T = [0, ∞).
ω ∈ Ω X(ω) : t ∈ T −→ X
t
(ω)
X ω.
X ω ∈
Ω, X(ω)
Y = (Y
t
)
t∈T
X t ∈
T, X
t
= Y

t
Y = (Y
t
)
t∈T
X
P [X
t
= Y
t
, ∀t ∈ T] = 1.
ν = (Ω, F, P )
F = (F
t
)
t∈T
σ F : F
s
⊂ F
t
⊂ F
0 ≤ s ≤ t T.
F
t
t
F
¯
T
=
σ(∪

t∈T
F
t
) σ F
t
, t ∈ T
(Ω, F, F = (F
t
)
t∈T
, P )
X = (X
t
)
t∈T
X
F
X
t
= σ(X
s
, 0 ≤ s ≤ t), t ∈ T,
σ X
s
0 ≤ s ≤ t. F
X
t
X
0 t
F = (F

t
)
t∈T
F
t
+
:=

s≥t
F
s
= F
t
, ∀t ∈ T,
F
0
F
¯
T
.
F
t
t
t = 0
(X
t
)
t∈T
F ∀t ∈ T, X
t

F
t
t
F
t
X F
X
=
(F
X
t
)
t∈T
(X
t
)
t∈T
F
t ∈ T, (s, ω) −→ X
s
(ω) [0, t]×Ω
σ B([0, t]) ⊗F
t
(X
t
)
t∈T
F
(t, ω) −→ X
t

(ω) T × Ω σ
F
(X
t
)
t∈T
F
(t, ω) −→ X
t
(ω) T ×Ω σ
F
T × Ω σ B(T) ⊗ F
t
;
X
X
τ : Ω −→ [0, ∞]
F
t ∈ T,
{τ ≤ t} := {ω ∈ Ω : τ(ω) ≤ t} ∈ F
t
.
τ
(τ
n
)
n≥1
lim
n
τ

n
= τ
τ
n
< τ, ∀n {τ > 0}.
(τ
n
)
n≥1
τ.
t
τ σ τ ∧ σ
τ ∨σ τ + σ
τ
F
τ
= {B ∈ F
¯
T
: B ∩ {τ ≤ t} ∈ F
t
, ∀t ∈ T}
σ F τ F
τ
τ = σ F
τ
= F
σ
σ τ ξ
B ∈ F

σ
B ∩ {σ ≤ τ} ∈ F
τ
σ ≤ τ
F
σ
⊂ F
τ
.
{σ < τ}, {σ ≤ τ}, {σ = τ} F
σ∧τ
= F
σ
∩ F
τ
.
ξ F
τ
∀t ∈ T ξ1
τ≤t
F
t
(X
t
)
t∈T
τ
X
τ
{τ ∈ T}

X
τ
(ω) = X
τ(ω)
(ω).
X X
τ
{τ ∈ T}
τ X
τ
X
τ
t
= X
τ∧t
, t ∈ T.
(X
t
)
t∈T
τ
X
τ
1
τ∈T
F
τ
X
τ
X

O ⊂ R
d
O
σ
O
= inf{t ≥ 0 : X
t
∈ O}
inf∅ = ∞
X O
τ
O
= inf{t ≥ 0 : X
t
∈ O}
(X
t
)
t∈T
(Y
t
)
t∈T
τ X
τ
= Y
τ
{τ < ∞} X Y
T R
d

(W
t
)
t∈T
= (W
1
t
, , W
d
t
)
t∈T
W
0
= 0
0 ≤ s < t T W
t
− W
s
σ(W
u
, u ≤ s)
0
(t − s) I
d
.
d

W
i

t

t∈T
, i = 1, , d,
d d
W
t
− W
s
F
W
s
= σ (W
u
, u ≤ s) W W
d T F = (F
t
)
t∈T
F
R
d
(W
t
)
t∈T
= (W
1
t
, , W

d
t
)
t∈T
W
0
= 0
0 ≤ s < t T W
t
− W
s
F
s
0
(t − s) I
d
.
(W
t
)
t∈T
(F
t
)
t∈T
.
(−W
t
)
t∈T

λ > 0 ((1/λ) W
λ
2
t
)
t∈T
s > 0 (W
t+s
− W
s
)
t∈T
F
s
(X
t
)
t∈T
E

X
−
t

< ∞, ∀t ∈ T
E [X
t
| F
s
] ≤ X

s
, ∀0 ≤ s ≤ t T.
X −X
X
P F.
Q− Q.
ξ (Ω, F) , E[|ξ|] < ∞
X
t
= E [ξ | F
t
] , t ∈ T,
ξ
T = [0, T ] (X
t
)
t∈[0,T ]
ξ = X
T
T = [0, ∞)
X = (X
t
)
t≥0
L
1
. X
t
t → ∞.
X = (X

t
)
t≥0
(X
t
)
t≥0
X
t
L
1
t → ∞ X
∞
X
∞
X, X
t
= E [X
∞
| F
t
] , t ≥ 0.
¯
T = [0, T ] T = [0, T ]
¯
T =
[0, ∞] T = [0, ∞) .
¯
T T (X
t

)
t∈T
X
¯
T
L
1
X
t
t →
¯
T . X
¯
T
X
X
t
= E [X
¯
T
| F
t
] , t ∈ T.
(M
t
)
t∈T
σ, τ
T σ ≤ τ
E [M

τ
| F
σ
] = M
σ
, a.s.
(X
t
)
t∈T
X
τ T X
τ
∈ L
1
E [X
τ
] = X
0
.
X τ
X
τ
X = (X
t
)
t∈T
τ T
P


sup
0≤t≤τ
|X
t
| ≥ λ

≤
E |X
τ
|
λ
, ∀λ > 0,
E

sup
0≤t≤τ
|X
t
|

p
≤

p
p − 1

p
E [|X
τ
|

p
] , ∀p > 1.
X = (X
t
)
t∈T
sup
t∈T
|X
t
| < ∞
ν = (Ω, F, F, P ) .
X
X
(τ
n
)
n≥1
lim
n→∞
τ
n
= ∞
X
τ
n
n
M = (M
t
)

t∈T
E

sup
0≤s≤t
|M
s
|

< ∞, ∀t ∈ T
M
M
(M
τ
)
τ
τ
T
M M
0
∈ L
1
M
A = (A
t
)
t∈T
ω ∈ Ω, t ∈ T,
sup
n


i=1
|A
t
i
(ω) − A
t
i
−1
(ω)| < ∞
sup 0 = t
0
< t
1
< < t
n
= t
[0, t] .
A
A
A = A
+
− A
−
A
+
A
−
A
+

([0, t]) := A
+
t
A
−
([0, t]) := A
−
t
T; A = A
+
− A
−
|A| = A
+
+ A
−
|A| : |A|([0, t]) = A
+
t
+ A
−
t
A
α

t
0
|α
s
(ω)|d |A|

s
(ω) < ∞, ∀t ∈ T, ∀ω ∈ Ω,

αdA

t
0
α
s
(ω) dA
s
(ω)
t ∈ T ω ∈ Ω A
α

αdA
A A
c
A
c
t
= A
t
−

0≤s≤t
∆A
s
, ∆A
s

= A
s
− A
−
s
(∆A
0
= A
0
) .
A
c
A.
M M
0
= 0
M M
0
M = (M
t
)
t∈T
N = (N
t
)
t∈T
< M, N > 0
MN− < M, N >
M N t ∈ T
0 = t

n
0
< t
n
1
< < t
n
k
n
= t [0, t]
< M, N >
t
= lim
n→+∞
k
n

i=1

M
t
n
i
− M
t
n
i
−1

N

t
n
i
− N
t
n
i
−1

< M, N > M N
M N < M, N >= 0, MN
M = N < M, M > < M >
M
< M, N >=
1
2
(< M + N, M + N > − < M, M > − < N, N >) .
W =

W
1
, , W
d

d
< W
i
, W
j
>

t
= δ
ij
t
δ
ij
W
i
W
j
− δ
ij
t
M N
α, β
T × Ω σ B(T) ⊗ F. t ∈ T,

t
0
|α
s
||β
s
|d| < M, N >|
s
≤


t
0

α
2
s
d < M, N >
s

1
2


t
0
β
2
s
d < M, N >
s

1
2
d < M, N >
d < M > .
p > 0 c
p
C
p
M = (M
t
)
t∈T

τ
¯
T
c
p
E

< M >
p/2
τ

≤ E

sup
0≤t<τ
|M
t
|

p
≤ C
p
E

< M >
p/2
τ

.
p = 1

M E

√
< M >
t

< ∞ t ∈ T
M
M = (M
t
)
t∈T
E

|M
t
|
2

< ∞ t ∈ T; M L
2
sup
t∈T
E

|M
t
|
2


< ∞ L
2
M
¯
T
t
¯
T
H
2
c
L
2
.
M = (M
t
)
t∈T
M E [< M >
t
] <
∞ t ∈ T M
0
= 0 M
2
− < M >
E

M
2

t

= E [< M >
t
] , ∀t ∈ T.
M L
2
E [< M >
¯
T
] < ∞.
E

M
2
¯
T

= E [< M >
¯
T
] .
H
2
c
(M, N)
H
2
= E [< M, N >
¯

T
]
X
X
X = X
0
+ M + A
M 0
A 0 X
A E [A
¯
T
] < ∞ A
¯
T
= lim
t→
¯
T
A
t
X
X = X
0
+ M + A
M
A
0.
M A
X = X

0
+ M + A
< X, X >=< M, M > .
t ∈ T, 0 = t
n
0
< t
n
1
< < t
n
k
n
= t
[0, t] 0
< X, X >
t
= lim
n→∞
k
n

i=1
(X
t
n
i
− X
t
n

i−1
)
2
Q X Q−
X = (X
1
, , X
d
)
X
i
X < X, X >
< X
i
, X
j
>
X
X
A M. A A
M
< M > M
L
2
.
α
t
=
n


k=1
α
(k)
1
(t
k
,t
k+1
]
(t) ,
0 ≤ t
0
< t
1
< < t
n
T
α
(k)
F
t
k
k.
E
M L
2
M ∈ H
2
c
L

2
(M)
α E


¯
T
0
|α
t
|
2
d < M >
t

< ∞
L
2
(M)
(α, β)
L
2
(M)
= E


¯
T
0
α

t
β
t
d < M >
t

E L
2
(M) .

t
0
α
s
dM
s
=
n

k=1
α
(k)
.

M
t
k+1
∧t
− M
t

k
∧t

, t ∈ T,
H
2
c
E




¯
T
0
α
t
dM
t

2


= E


¯
T
0
|α

t
|
2
d < M >
t

.
E L
2
(M) α →

αdM
L
2
(M) H
2
c
α ∈ L
2
(M)

αdM
<

αdM, N >=

αd < M, N >, ∀N ∈ H
2
c
.

τ

t∧τ
0
α
s
dM
s
=

t
0
α
s
1
[0,τ]
dM
s
=

t
0
α
s
dM
s∧τ
, t ∈ T.
M L
2
loc

(M)
α t ∈ T

t
0
|α
s
|
2
d < M >
s
< ∞, a.s.
α ∈ L
2
loc
(M)
0 α M

αdM
<

αdM, N >=

αd < M, N >
N
A =

A
1
, , A

d

Γ dΓ =

d
i=1
d


A
i


γ =

γ
1
, , γ
d

A
i
=

γ
i
dΓ, i = 1, , d.
L
S
(A) α =


α
1
, , α
d


t
0





d

i=1
α
i
u
γ
i
u





dΓ
u

< ∞, a.s. ∀t ∈ T.

αdA =

d

i=1
α
i
γ
i
dΓ,

t
0
α
u
(ω) dA
u
(ω) = 0 ω

t
0





d


i=1
α
i
u
γ
i
u
(ω)





dΓ
u
(ω) = ∞.

αdA (γ, Γ)
L
S
(A) α =

α
1
, , α
d

i = 1, , d, α
i
∈ L

S

A
i


t
0


α
i
u


d


A
i


u
<
∞ t ∈ T

αdA =
d

i=1


α
i
dA
i
.

α

dA.
M =

M
1
, , M
d

R
d
. < M >
< M
i
, M
j
> C
d < M
i
, M
j
> dC,

C =

d
i=1
< M
i
>
c S
+
d
< M >=

cdC, tcl < M
i
, M
j
>=

c
ij
dC, i, j = 1, , d.
L
2
loc
(M) α =

α
1
, , α
d


R
d
t ∈ T

t
0
α

u
d < M >
u
α
u
:=

t
0
α

u
c
u
α
u
dC
u
< ∞, a.s.

t

0
α

u
d < M >
u
α
u
c C α ∈ L
2
loc
(M)
0,
α M

αdM
<

αdM, N >=

αd < M, N >
N
< M, N >=

< M
1
, N >, , < M
d
, N >


<

αdM,

αdM > =

α

d < M >
α
.
i = 1, , d, α
i
∈ L
2
loc
(M)

t
0


α
i
u


2
d < M
i

>
u
< ∞, ∀t ∈ T,