2
П.В. Храпов
1 2
1 2
1
,
{ ( , ,..., ) | ( , ,...,
) ,
[1, ]},
S h
t
y y y y y y
S y
h
Q
Q
Q
/
ɝɞɟ
h
— ɮɢɤɫɢɪɨɜɚɧɧɨɟ ɱɢɫɥɨ;
S
— ɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ
ɪɟɲɟɬɤɢ
1
.
Q
]
ȼ
/
S,h
ɨɩɪɟɞɟɥɢɦ ɫɥɭɱɚɣɧɨɟ ɩɨɥɟ ɫɨ ɡɧɚɱɟɧɢɹɦɢ ɜ
X
= {+1,–1} ɢ
ɜɟɪɨɹɬɧɨɫɬɧɨɣ ɦɟɪɨɣ
P
. Ɇɧɨɠɟɫɬɜɨ ɬɨɱɟɤ
*/
S,h
ɧɚɡɨɜɟɦ ɨɞɧɨ-
ɫɜɹɡɧɵɦ, ɟɫɥɢ ɞɥɹ ɥɸɛɵɯ
t
',
t
''
*
ɫɭɳɟɫɬɜɭɟɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ɬɨɱɟɤ
t
i
*
, ɬɚɤɢɯ, ɱɬɨ
1
1
1
ȡ( ,
)
1,
1,...,
1,
,
.
i i
i
i
m
t t
t t
i
m t t t
t
c
cc
ɇɚɡɨɜɟɦ ɞɟɮɟɤɬɧɵɦ ɤɨɧɬɭɪɨɦ ɨɞɨɧɫɜɹɡɧɨɟ ɦɧɨɠɟɫɬɜɨ
*
ɬɚɤɨɟ,
ɱɬɨ
x
t
= –1 ɞɥɹ
t
*
ɢ
x
t
= 1 ɞɥɹ ɜɫɟɯ
t
w*
,
,
{
|
,
d
S h
t
t
w * / *
1
1
1,1
: ȡ( , ) },
t
t t
d
*
w * { w*
,
* *w*
.
ɋɤɚɠɟɦ, ɱɬɨ ɞɟɮɟɤɬɧɵɣ ɤɨɧɬɭɪ ɹɜɥɹɟɬɫɹ ɤɨɧɬɭɪɨɦ ɩɪɨɬɟɤɚɧɢɹ
ɤɨɧɮɢɝɭɪɚɰɢɢ, ɟɫɥɢ ɜ ɧɟɦ ɟɫɬɶ ɬɨɱɤɚ
t
1
= (
y
1
,
h
) ɜɟɪɯɧɟɝɨ ɨɫɧɨɜɚɧɢɹ ɢ
t
2
= (
y
2
, 1) ɧɢɠɧɟɝɨ ɨɫɧɨɜɚɧɢɹ ɰɢɥɢɧɞɪɚ
/
S,h
. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ
R
ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɤɨɧɬɭɪɨɜ ɩɪɨɬɟɤɚɧɢɹ, ɱɟɪɟɡ
H
(
S
,
h
,
P
) — ɜɟɪɨɹɬɧɨɫɬɶ
ɧɟɩɪɨɬɟɤɚɧɢɹ ɜ
/
S,h
(ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɜ ɤɨɧɮɢɝɭɪɚɰɢɢ
ɫɥɭɱɚɣɧɨɝɨ ɩɨɥɹ ɧɟɬ ɤɨɧɬɭɪɨɜ ɩɪɨɬɟɤɚɧɢɹ) ɢ ɧɚɡɨɜɟɦ ɟɟ
ɧɚɞɟɠɧɨɫɬɶɸ ɨɛɴɟɦɚ
/
S,h
.
ȼɜɟɞɟɦ ɧɟɡɚɜɢɫɢɦɨɟ ɩɨɥɟ, ɞɥɹ ɤɨɬɨɪɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɹɜɥɹɟɬɫɹ
ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɧɟɡɚɜɢɫɢɦɵɯ ɦɟɪ
v
0
ɜ ɬɨɱɤɚɯ
t
/
S,h
,
v
0
= ({
x
t
=
= –1}) =
p
, 0 <
p
<1. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɨɤɚɡɚɧɚ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ.
Ɍɟɨɪɟɦɚ 1
. ɉɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɯ
p < p
0
, ɮɢɤɫɢɪɨɜɚɧɧɨɦ
h
ɢ
ɥɸɛɨɦ
S
ɜɟɪɧɨ ɪɚɜɟɧɫɬɜɨ:
1
( , , ) exp{
(1 ȥ( , , ))}, ȥ( , , )
,
h
h
H S h p
S p p p S h
p S h K
Q
(1)
ɝɞɟ
p˾
=
p
/
q
,
q
= 1 –
p
,
\
(
p˾
,
S
,
h
) — ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɜ ɤɪɭɝɟ
ɪɚɞɢɭɫɚ
p˾
0
=
p
0
/
q
0
;
K
v
— ɤɨɧɫɬɚɧɬɚ, ɡɚɜɢɫɹɳɚɹ ɥɢɲɶ ɨɬ ɪɚɡɦɟɪɧɨɫɬɢ
v
.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɫɥɭɱɚɣɧɨɟ ɝɢɛɛɫɨɜɫɤɨɟ ɩɨɥɟ, ɨɩɪɟɞɟɥɹɟɦɨɟ ɜ
/
S,h
v
-ɦɟɪɧɨɣ ɦɨɞɟɥɶɸ ɂɡɢɧɝɚ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ
2
ȕ ı ı ,
,
1, ı , ı
{ 1}
t t
t
t
H
t t
t t
X
c
c
/
c
c
/
r
¦
]
ɜ ɧɢɡɤɨɬɟɦɩɟɪɚɬɭɪɧɨɣ ɨɛɥɚɫɬɢ (ɛɨɥɶɲɢɟ
E
) ɢ ɩɥɸɫɨɜɵɦɢ
ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. Ɉɩɪɟɞɟɥɢɦ ɞɟɮɟɤɬɧɵɣ ɤɨɧɬɭɪ, ɤɨɧɬɭɪ
ɩɪɨɬɟɤɚɧɢɹ ɢ ɜɟɪɨɹɬɧɨɫɬɶ ɧɟɩɪɨɬɟɤɚɧɢɹ, ɤɚɤ ɢ ɩɪɟɠɞɟ.
