.
. . http://synset.com. [email protected] , .
v. 0.1,
4 2009 ., (printed: 26 2011 .)
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
910 14 18 22 26 30 34 38 42
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
. . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
4748 52 54 56 58 64 68 72
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
3 3.1 3.2 3.3 3.4 3.5 . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
7778 82 88 92 96
t
. . . . . . . . . . . . . .
. . . . . . . . . .
3
4
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7
101. . . . . . . . . . . . . 102
P (x0 , t0 x, t)
. . . . . . . . . . . . . . . . 104
- . . . . . . . . . . . . . 108 . . . . . . . . . . . . . . . .
. . . . . . . 110
. . . . . . . . . . . . . 114 . . . . . . . . . . . . . 116
x(t, )
. . . . . . . . . . . . . . . . . . . . . 120
5 5.1 5.2 5.3 5.4 5.5 5.6
123. . . . . . . . . . . . . . 124
. . . . . . . . . . . . . . . . . . . . . . . . . 130
. . . . . . . . . . . . . . . . . 134 . . . . . . . . . 140
. . . . . . . . . . . . . . . . . . . 142 . . . . . . . . . . .
148
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7
151. . . . . . . . . . . . . . . . 152 . . . . . . . . . . . . .
156
. . . . . . . . . . . . . . . . . . 160 . . . . . . . . . . . .
. . . . 164 . . . . . . . . . . . . . . . 168
. . . . . . . . . . . . . . . . . 172 . . . . . . . . . . . . .
176
?
7 7.1 7.2 7.3 7.4 7.5
181. . . . . . . . . . . . . . . . . . . . . 190
. . . . . . . . . . . . . . . . 182 . . . . . . . . . . . . . .
. . . . 186
. . . . . . . . . . . . . . . . . . . . . . . 194
. . . . . . . . . . . . . . . . . . . . . 198
8 8.1 8.2 8.3 8.4 8.5 8.6 8.7
203. . . . . . . . . . . . . . . . 208
. . . . . . . . . . . . . . . . . . . . . . . 204
. . . . . . . . . . . . . . . . . . . . . . . . 212 . . . . . .
. . . . . . . . . . . . . . 216 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 220 - . . . . . . . . . . . . . . . . . . . .
224 . . . . . . . . . . . . . . . . . . . . . . . 228
5
9 9.1 9.2 9.3 9.4 9.5 9.6
233
C ++
. . . . . . . . . . . . . . . . . . . . . . 234
. . . . . . . . . . . . . . . . . . . . . . . . . . . 238
. . . . . . . . . . . . . . . . . . . . . . . . 244 . . . . . .
. . . . 250 . . . . . . . . 254 . . . . . . . . . . . . . . . . . .
. . . . 258
R: I II III IV V VI VII
261. . . . . . . . . . . . . . . . 262 ,
. . . . . . . . . . . . . . . . . . . . . . . . . 266
x
n=1 . x , n = 1
. . . . . . . . 268 . . . . . . . . 274
. . . . . . . . . . 279 . . . . . . . . . . . 280 . . . . . . .
. . . . . . . . . 282
VIII . . . . . . . . . . . . . . . 290 IX . . . . . . . . . . .
. . . 292
M: I II III IV V VI VII
295. . . . . . . . . . . . . . . 304
. . . . . . . . . . . . . . . . . . . . . . 296 . . . . . . . .
. . . . . . . . . . . . . . . 300
. . . . . . . . . . . . 308 . . . . . . . . . . . . . . . . . .
. . . . 312 . . . . . . . . . . . . . . . . . . . 314
. . . . . . . . . . . . . . . . . . . . . 316 . . . . . . . . .
. . . . 318
VIII IX
. . . . . . . . . . . . . . . . . . . . 320
H: C:
323 353 369
6
Altus Assets Activities, . , . , . , , . , , . , . . :
, . . , . , . , ( Hi ),
i
. , Ci ),
(
, . ,
i.
-
, . . , . .
7
, . , 51
2.6,
. 64,
5.1,
. 124.
: synset.com/ www.altus.ua/research/. , , .
http://synset.com
8
1
. . , , , . , . , . , . , , . . . , . . 296.
9
10
1.
1.1 . , , . :
dx = x dt
=>
x(t) = x0 et .
(1.1)
x(t) > 0
, -
, . , , , .
> 0,
, -
.
x0 ,
, ,
x0 = x(0) > 0
t0 = 0 .
. , . , .
dx/x = A dt x. A(x) = x + ..., . (- ). , . , -
/
(
> 0):(1.2)
dx = x x2 dt (t ,
=>
x(t) =
. ( /x0 ) et x
(1.2) (
H1 )
) dx/dt = 0 (
x(t) = 1/y(t). = /
C1 ). , (1.2)
, , ( C2 ).
11
.
F (x) {
p = mx
:
p = F (x) x = p/m,
(1.3)
x = dx/dt, m . , F (x) = kx, x(t) = x0 cos(wt) + (p0 /m) sin(wt)
w = k/m ( H2 ). , .
x0 = x(0)
p0 = p(0).
, :
dx = a(x, t) dt,
(1.4)
x(t) = {x1 (t), ..., xn (t)}
, -
.
a(x, t) .
, , (1.4) . (1.3). (1.4)
x(t)
-
dt.
(1.4) , . ,
x = xk+1 xk , t = tk+1 tk .
(1.4) :
xk+1 = xk + a(xk , tk ) t.
(1.5)
x0 ,
x1
t.
x1
x0
x2 .
,
x(t)
t0 , t1 = t0 + t, t2 = t0 + 2t, ..
t, (1.5)
(1.4).
12
1.
, , 300 . , . , . . . , , , .
x(t) ,
. () . . , (1.2), . .
x0 et
, , . , , . , , , . , . :
. ( C3 ).
, : , , .
13
, , , :
dx = a(x, t) dt + Noise(x, t, dt).
(1.6)
( ) ()
x.
dx
-
, ,
dt
.
Noise(x, t, dt),
.
x(t),
. , , . , :
x(t)
x(t)
t ,
t
x(t)
-
, . ,
[x(t + t) x(t)]/t t 0.
, . . ,
dx
-
. , (1.6). , , . , , . , .
14
1.
1.2 , x1 , x2 , ... . x1 , x2 , ... x. . ,
xi
ni
,
n.
x
:
1 x i ni = xi p i = x = x = n i i
x P (x) dx, (1.7)
pi = ni /n
( )
xi , , n. xi , . . . -
xi .
P (x),
dx pi , x x x + dx. x [..] P (x). , :
P(x)dx = pi
P(x)i
pi =
P (x)dx = 1.
(1.8)
x x+dx
. . , .
x x < 0 .
. , .
15
, -
F (x)
x:
F (x) = F (x) = .
F (x) P (x) dx.
EF (x).
(), . , :
f (x) = f (x) , - :
f (x) + g(x) = f (x) + g(x) . 2 x = x2 . (x x)2 P (x) dx.
! , , -
:
2 = (x x)2 =
.
2
,
= Var(x).
2
x
-
, :
2 = (x x)2 = x2 2x + x2 = x2 2 x + x2 = x2 x2 . x x
, -
x. x . , P (x), 0 x = x. . ,
asym = (x x)3 / 3 ,
excess = (x x)4 / 4 3 P (x)
(1.9)
. . , .
P (x)
16
1.
, -
.
.
-
, :
0.40
P( )0.24 0.05 -2 -1 0 1 2
e 2 P () = 21
2
(1.10)
= 0, 2 2 = 1. , = 1. : N (0, 1). x = + , 2 ( C4 ), x N (, ). ,
[. (14), . 312]:
e =
e P () d = e
2
/2
.
(1.11)
(1.11) n ( H3 ). 4 : 3, , , excess = 0. (1.9) .
excess > 0, , ,
, .. (
x ).
, .
:
x F (x) =
e /2 d 22
(1.12)
,
x.
17
P (x)
x,
-
y,
y = f (x). F (y). , P (x):
x
F (y) =
( ) F y P (y) dy =
( ) F f (x) P (x) dx.
(1.13)
P (y) , P (x) y = f (x) F (...). . F (y) P (y) y . r = + , r . x
= x0 e , x0 . F (x) =
( ) d 2 F x0 e+ e /2 = 2
0
F (x) e[ln(x/x0 )]
2
/2 2
dx . x 2
.
x = x0 e+ , dx = xd. -
x
0: [ ] (ln(x/x0 ) )2 PL (x) = exp . 2 2 x 2 1
(1.14)
PL (x)
. -
x PL (x) x = + ,
P ()
(
H4 ).
, : ,
X , -
x ...
, -
x,
.
, , .
18
1.
1.3 x y . {x1 , y1 }, {x2 , y2 }, .., .
P (x, y) , x y .
-
:
F (x, y) =
F (x, y) P (x, y) dx dy.
(1.15)
y , P (x, y) -
x:(1.16)
P (x, y) dy = P (x).
x
. -
. (1.15),
F (x, y) = 1, 1 = 1. x y . , x , y , . , . ,
x
y
(, ..). ,
P (x, y)
, -
.
y = f (x). , x , y . y = f (x, ), , , . ,
y = f (x),
, -
.
19
x
y
. ,
y,
x.
-
P (x, y), P (x) (. . 299 ): P (x y) = P (x, y) . P (x) P (x, y)
e
(1.17)
P (x)
(1.10), :
P (x, y) =
e
(x2 +y 2 + 2 xy)
2
,
P (x y) =
(x2 /2+y 2 + 2 xy)
.
:
P(x,y)
P(x => y)
P (x, y) , P (x y)
.
y
x: P (x y) dy = 1.(1.18)
, (1.18) (1.16). ,
P (y|x).
-
P (x y) P (x, y),
.
P (x y),
. , , - .
20
1.
. 33-, _, :
p(_) = 0.163, p() = 0.0940, p() = 0.0696, ..., p() = 0.0002. ,
,
, **, .
P ( ) , ,
, ), ,
*
. :
* ( ,
p() = N ()/N () = 0.002,
p( ) = N ()/N () = 0.739,
N
, .
n
N () = n 1,
N () = p() n.
,
,
332 = 1089.
( ). , 14 , :
p() = 0.051.
,
. ,
.
, . , ,
p(...cba x).
...cba
x
-
, :
P (x):
-
;
P (a x): P (ba x):
-
;
;
P (cba x): . . , ..
21
xt S&P500. rt = ln(xt /xt1 ) ( C6 ). :
(...3%), [3%...1%), [1%...+1%], (+1%...+3%], (+3%...+). , :
,
(... 3%)
(+3%...).
-
rt
,
. , , -2,-1,0,1,2.
p(rt1 , rt ) , -
rt1
rt .
,
25 = 52
: {(0,0); (0,1);
(0,-1);...}. 19902007 . .
n = 4531 0.761
. -
:
( p(r) = 0.007
0.110
0.125
) 0.007 .
, ,
n. [1%... + 1%], 0.100 0.014 0.004 . 0.013 0.030
3451 = 0.76 4531 . 0.067 0.022 rt ) = 0.004 0.006 0.000 0.167
0.146 0.107 0.084 0.303 0.400 0.651 0.783 0.759 0.515 0.267 0.168
0.102 0.138 0.152
:
p(rt1
. . , ( ),
p(r). ,C5 )
. (
. - , [ . (1.18)].
22
1.
1.4 , , :
P (x, y) = P1 (x) P2 (y) . , . (1.17) ,
P (x y) = P (y) y . ,
.
y
x,
.
:
x y =
x y P (x)P (y) dxdy = x y .
cov(x, y):(1.19) C7 ).
cov(x, y) = (x x)(y y ) = xy x y . (
x y P (z). ,
z = f (x, y)
F (z),
z:
F (z) =,
) F f (x, y) P (x, y) dxdy =
(
F (z)P (z) dz.(1.20)
x y x , y , z = x + y : ( ) x2 /22 y2 /22 dxdy dz 2 2 x y F (z)
= F x+y e = F (z)ez /2 , 2x y 2 2 2 2 = x + y . z = x + y , u = x,
u (14) .
312 . ,
.
23
x y . z , z = x + y . , z = x + y . : 2 2 2 z = (z z)2 = (x x +
y y)2 = x + y + 2 (x x) (y y) , , ,
2 x = (x x)2 .
(!)
x
y
-
, ( ) :
(x x) (y y) = x x y y = 0.
:
2 2 2 z = x + y .
n
:
z = x1 + ... + xn
=>
2 2 2 z = 1 + ... + n .
(1.21)
2 , x3 , z
x1 +x2 2 2 2 2 + 3 = 1 + 2 + 3 , .. xi 0 , -
,
z = 0 n. . , (1.21)
Noise,
xi .
. -
. . .
z z(-
, , , ). , - , . ,
z = x+y
. ,
, . ( , ) . , , . (1.21) .
24
1.
x
y
y = + x. , , . :
y = + x + . . ,
(1.22)
-
= 0.
.
,
( ) :
2 = 2 = (y x)2 = min.
(1.23)
, (:
H5 )
.
=
xy x y (x x)(y y ) = . 2 x x2 x2
(1.24)
:
xx yy = (x, y) + . y x y :
(1.25)
xy = (x, y) =
cov(x, y) . x y
(1.26)
(1.19).
( = 0) x, y y = f (x) x = g(y). , z , x, y , . , - . , ( ). . (
C8 ).
25
. , . , (1.24) (1.22). ,
= 0
y = + x:
=
(x x)( (x x) + ) x =+ 2 . 2 x x
x = 0, y : 2 2 y = (y y )2 = ( (x x) + )2 = 2 x + 2 . = (x, y)y
/x , E=
:
= 1 2 (x, y). (1.27) y 2 2 = y = + x. y , y = y . , E . , .
, , -
, , ,
y
-
.
R2 = 1 E 2 = 2 .
,
||
1.
(1.22) -
. 1) ,
y, y
x
(
P (x y)). y
,
x
.
. -
x
. ,
x
-
(, ). .
x . 2) , x, y
x
. ,
x
y
.
.
26
1.
1.5
(q)
- (.
314)
x: eqx (q) dq.
(q) =
eqx P (x) dx,
P (x) =
1 2
x.
- -
,
xn : =q=0
1 dn (q) n dq n
xn P (x) dx = xn .
:
(q) = eqx . , (0) = 1. (q) q x: (q) = e =qx n xn n=0
n!
q n = 1 + x q
1 2 2 x q + ... 2
(1.28)
:
(q/) = (q) = e .qx
q q/
:
, y x y = a + b x. : ( ) qy q(a+bx) y q = e = e = eqa eqbx .,
,
q
:(1.29)
y = a + bx
=>
( ) y (q) = eqa x b q .
b = 0,
y (q) = eqa ,
,
- (. 315), -
P (y) = (y a). , y = a.
27
:
: : :
e(xx0 ) /2 P (x) = , 2 a/ P (x) = , (x x0 )2 + a2 ( )1 x 1 ex/ ,
P (x) = () 2 2
(q) = ex0 q
2 2
q /2
.
(q) = ex0 qa|q| . (q) = 1 . (1 q)
(q) . (q) P (x). (16), . 313, - . ,
(q) m q x m > 1. x, y P1 (x), P2 (y) z = x + y . P (z) z . (
): F (z) = F (x + y) P1 (x)P2 (y) dx dy = F (z) P1 (x)P2 (z x) dx
dz,
P (z)
y = z x. P (z) = P1 (x)P2 (z x) dx.
:
z (q) = e
q(x+y)
= eqx eqy = x (q) y (q), x
n
y . , xi -
:
z = x1 + ... + xn
=> xi
z (q) = 1 (q) .. n (q). z (q) = n (q). H6 ).
,
, (
28
1.
. ,
1 , ..., n
i N (0, 1),
:
1 + ... + n =
n.
(1.30)
n , N (0, 1) [ (1.21), . 23 ]. i (q)n = ( n q)
(, , ..).
(q) =
e
q 2 /2
.
n
P (x) , an bn , (1.31)
x1 + ... + xn = an + bn x,
x1 , ..., xn
x
P (x).
an = 0,
.
bn =
n.
, (1.31) , . , (1.31) , , . (1.29), :
n (q) = eiqan (bn q).
(1.32)
, . -, , . , (1.32), -:
(q) = eq[1+ sign(q) tg(/2)] |q| ,
(q) = eq|q| q ln |q| ,
sign(q) = q/|q| q , 0 < 2. , || 1, 0. , , 1. ( ), .
29
n
x1 , ..., xn
, , :
u=
x1 + ... + xn n xi = 0,
n .
,
x x x
.
u . xi x: 2 2 x1 + ... + x2 n u = = x2 = 2 . n xi
(q)
n
[ ( [ )]n ]n q 2 q2 u (q) = = 1 + .. , 2 n n
u
:
(1.29) . ,
(q/ n) q , ,
x = 0. , e = (1 + x/n)n , n . u :x
u (q) e (
2 2
q /2
. z (q) = n (q).
(1.33)
H7 )
n
(1.33) :
., , . . . : 1) ( ) 2) () . 8.
30
1.
1.6 . :
=
n i=1
Si i = Si i = (S ) .
(1.34)
, . i . , , . , . (1.34) ,
S = S n
= {1 , ..., n }
. ,
.
i j
. :
{
i j = ij =
1 i=j 0 i = j. :
, ,
T = Si Sj i j = Si Sj ij = Si Si = Si Si = (SST ) .
(1.35)
ij i = j . ( j ) , i. T Si = Si . . .
S
S1 ,
:
S S1 = S1 S = 1,
1 = ij
( ). , -
= (1 , ..., n ) =>
:
=S
= S1 , S1 .
31
= (1 , ..., n ) i N (0, 1), = (1 , ..., n ) (1.34) S . (1.35): D
= , D = S ST , :
D = D . .
b = (b1 , ..., bn ) b = b1 1 + ... + bn n ( n !): b bS b S 1 2 2
e = e = e i i1 1 ... ebi Sin n = e 2 {(bi Si1 ) +...+(bi Sin ) } .
i , -
, (1.11), . 16. :
T (bi Si1 )2 + ... + (bi Sin )2 = bi Sik bj Sjk = bi Sik Skj bj
= b S ST b. :
1 (b) = eb = e 2 bDb .
b , . , D . b . , b D b bi Dij bj , :
(b) 1 = (Dj bj + bi Di ) (b) = Di bi (b), b 2 , D = D . 2 (b) =
D (b) + Di bi Dj bj (b). b b
-
:
b=0
,
2 eb b b
D = .
b=0
= , -
:
k = D Dk + D Dk + Dk D .
,
D.
-
32
1.
1 , ..., n . 1 , ..., n : e 2 (1 +...+n ) P (1 , ..., n ) = P (1
) ... P (n ) = . (2)n/21 2 2
= S n d = d1 ...dn , :
dn = det
n d = (det S) dn .
, ,
det D = (det S)2
, :1 1
e 2 D P (1 , ..., n ) = , (2)n/2 det D
= S1 :T
1 1 1 2 = Si Si = S 1 i Si = S1 S1 = (S ST )1
T
(A B)1 = B1 A1 (. . 304). , P (1 , ..., n ) , ,
b e , n- :
eb 2 D
1
1
dn = (2)n/2
det D e 2 bDb .
1
(1.36)
:
= S = 0. , : = + S .
,
n-
:1 1
e 2 ( )D ( ) P (1 , ..., n ) = , (2)n/2 det D
P (1 , ..., n )
= S1 ( ).
33
2
n = 2.
D
1 ,
:
,
) 2 1 1 2 D= . 2 1 2 2 D 2 2 det D = 1 2 (1 2 ),
(
D
:
D1
1 = det D
(
) 2 2 1 2 . 2 1 2 1 1 , 2
:
exp{(x2 2 x1 x2 + x2 )/2(1 2 )} 1 2 P (1 , 2 ) = , 21 2 1 2 xi =
(i i )/i i i . 2 2 i : (1 1 ) = D11 = 1 , : = x1 x2 . T D = SS , S
. D , S ,
S.
, :
(
S=
) 1 cos 1 sin , 2 sin 2 cos
= sin( + ). , , . = , = 0, D , 1 = 2 = 1 . S, SST = 1, .
= 0, = sin , 1 = 2 = 1, ( ) 1 0 S= , 1 2
( D=
) 1 . 1
(1.37)
1 , 2 N (0, 1), 1 2 = 0 { 1 = 1 2 = 1 + 1 2 2
=>
1 , 2 N (0, 1) : 2 2 1 2 = , 1 = 2 = 1.
, , .
34
1.
1.7
x
. -
. . ,
x = x0 .
x
-
t = 1, 2, ...
(),
. x (1.38)
:
xt = x0 + (1 + ... + t ),
i N (0, 1)
-
.
t
,
. :
Wt = 1 + ... + t =
t.
(1.39)
t t (. 2223). , i , , 2 : = 0, = 1, .. N (0, 1). (1.38) : xt =
x0 + Wt . , .
x0 = 0,
1 , 2 ,
... -
(1- ):
xt5 1 0 4 3 2 6 7 8 9
xt
0.4
P(x,t)0
t=1
t=3 t=5
t0 1 2 3 4 5 6 7 8 9 10
t
x0
k
, -
xt = x(t) x.
(. 2- ). -
t = const
.
35
x(t), , x = x(t) P (x). , .
P (x, t),
x(t)
(t)
.
, 2-
t.
xt .
-
.
xt
.
xt
-
x 0 = 0.
3- , .
P (x, t),
-
x0 = x(t0 ) t0 . , , P (x0 , t0 x, t). t0 t , k . , xt = x(t) .
. ,
x
-
, ,
P (x).
t-
xt
i .
, -
Wt
:
P (1 , ..., t ) = P (1 ) ... P (t ), , .
i . , Wt -
1 + ... + t = t. , t 1 , 2 , ... . , , t. , P (1 , ..., t ). , :
, , , . , , , . .
36
1.
-
.
s
,
t s.
s
t (s < t):
Ws = 1 + ... + s , Wt = 1 + ... + s + s+1 + ... + t . ,
Wt Ws = s+1 + ... + t = t s = Wts .
ts
:
t s t s. , Ws Wt : Ws = a s, (1.40) Wt = a s + b t s, a , b , ,
. a s , b a t s . Ws Wt . Wt = 0, : ( ) cov(s, t) = Ws Wt = a s a s
+ b t s = s, 2 , a = 1 a b = 0. , s = min(s, t), Ws Wt . (1.25) Ws
Wt . s t, , :
, -
Wt cov(s, t) W = s + s t s t t
=>
Wt = Ws + . s
Ws , Wt Ws . , , (1.40) , ( C9 ).
s
:
= s+1 + ... + t = b t s.
, (i
Wi Wj Wk = 0,[
2 Wi Wj Wk = 2i2 + ij,
< j < k ): Wi Wj2 Wk = 3ij.C10 ).]
Wk
(
37
. x = x1 t = t1 , t = t2 x2 ? , x: ( ) (x2 x1 )2 2 exp 2(t2 t1 )
e /2 P (x1 x2 ) = = . 2 2(t2 t1 ) = 1 = x2 x1 . , x1 x2 .
xt = {x1 , x2 , x3 , ...},
x
t.
x(t)
-
, . , . . ,
P (x1 , x2 , x3 , ...)
xt+1
-
xt . C11 ): (1.41)
(
P (x1 , ..., xt xt+1 ) = P (xt xt+1 ).
xt ,
xt+1
xt
, x1 , ..., xt1 .
. ,
P (x1 , ..., xt xt+1 ) = P (xt+1 ).
. :
P (x1 , x2 , x3 ) = P (x1 ) P (x1 x2 ) P (x2 x3 ).
(1.42)
P (x1 , x2 , x3 ) = P (x1 , x2 ) P (x1 , x2 x3 )
.
P (x1 , x2 ) = P (x1 )P (x1 x2 ) : P (x1 , x2 x3 ) = P (x2 x3 ).
, x1 , x2 , x3 , , x1 . , , x2 , ..
38
1.
1.8
x1 , x2 , ....
, -
,
x(t). t , . t , . x(t) . , , ,
x(t)
( ).
:
P (x1 , x2 , x3 , ...) P (x1 , t1 ; x2 , t2 ; x3 , t3 ;
...),
(1.43)
ti
,
xi .
,
. .
xi , , P (x, t).
P (x1 , t1 ; x2 , t2 ), .. , t, x, , . ,
x0
t0 .
-
. , :
P (x0 x1 ) P (x0 , t0 x1 , t1 ) :
P (x0 x1 , x2 ) P (x0 , t0 x1 , t1 ; x2 , t2 ). . , , , . , , .
. , . , .
39
,
(1.43) , . , , :
P (..., xt2 , xt1 , xt xt+1 ) =
P (..., xt2 , xt1 , xt , xt+1 ) = P (xt xt+1 ), P (..., xt2 ,
xt1 , xt )
. , .
P (x1 , t1 x2 , t2 ),
.
(1.42). , , (1.43).
P (x0 , t0 x, t),
-
. , , . , .
x0
t0 ,
:
x(t, x0 , t0 ) = ( ):
x P (x0 , t0 x, t) dx.
(1.44)
2 (t, x0 , t0 ) =
(
)2 x x(t) P (x0 , t0 x, t) dx.
(1.45)
. ,
Noise -
x , -
. .
40
1.
x(t)
(t)
-
. :
x(t)
x(t)
t
t
( ) , . , , ( C12 ).
.
t1 < t2
,
t = t0
x0 = x(t0 ): covt0 (t1 , t2 ) = ( ) ( ) xt1 xt1 xt2 xt2 , t,
(1.46)
xt = x(t)
xti = x(ti ). t2 . x[.
- ,
t1
P (x0 , t0 x, t)
(1.44).]
xt
t,
-
x0
t0 .
:
covt0 (t1 , t2 ) =
(x1 x1 )(x2 x2 )P (x0 , t0 x1 , t1 ; x2 , t2 ) dx1 dx2 ,
(1.47)
P (x0 , t0 x1 , t1 ; x2 , t2 ) x1 x2 t1 t2 , t0 x0 = x(t0 ).
41
. ( ):
P (x0 x1 , x2 ) =
P (x0 , x1 , x2 ) . P (x0 ) P (x0 , x1 , x2 )
(1.48)
-
[. (1.42), . 37]:
P (x0 , x1 , x2 ) = P (x0 ) P (x0 x1 ) P (x1 x2 ). (1.48) ,
:
P (x0 , t0 x1 , t1 ; x2 , t2 ) = P (x0 , t0 x1 , t1 ) P (x1 , t1
x2 , t2 ).
(1.49)
x1
x2
. , ,
x1 .
(1.47) ,
. t0 , . , :
cov(t1 , t2 ) = xt1 xt2 xt1 xt2 , . , :
(1.50)
(1.46) -
t1 = t2 = t
(t) = cov(t, t).
2
:
(t1 , t2 ) =
cov(t1 , t2 ) . (t1 )(t2 )
(1.51)
,
x2 = x(t2 ),
x1 = x(t1 ). x1 , x2 .
-
.
x0 = x(t0 ),
42
1.
1.9 , , . . ,
x = x0 ,
. . ,
x0 . x = x0 + 1 + ... + n i.
,
P (1 , ..., n ) = P (1 ) ... P (n ).
,
,
n:
(1 + ... + n ) P (1 , ..., n ) d1 ...dn = 0. ,
x = x0 .
, .
. , :
x 6 5 4
7 5 3 n
x 6 5 4
8 5 2 n
x0 = 5
6 4 .. . .
{1/4, 1/2, 1/4}
:
0.25 7 + 0.5 5 + 0.25 3 = 5,
0.25 8 + 0.5 5 + 0.25 2 = 5.
. 4. , :
0.5 5 + 0.5 2 = 3.5 = 4.
, 4 6.
43
-
. , , -
. ,
F
(, F, P),
-
P
-
. .
-
, , ( ). , , :
= {1, 2, 3, 4, 5, 6}.
4=(5
F
3=(3
, . -
A =
6)
6). A B A (. 298). . F , .. , F . , , -.
B = A + B ,
P p : A P (A) : A F ,A
F
1. , . P(F), P(), , . . ,
0
p
x . F x ,
(, ).
x
,
F x
F.
,
x.
,
P
.
x(t) xt = x1 , x2 , ... -
x(t)
,
, , . ,
x = (x1 , x2 , ..., xt ).
44
1.
x(t)
F
.
t
,
Ft
Ft . -
:
Ft = ..., xt2 , xt1 , xt . , . , . , . :
ti
-
Fj = ..., xj1 , xj
x P (...; xj1 , tj1 ; xj , tj xi , ti ) dxi .
E(xi |Fj ) = xi j =
,
E(xi |Fj ) = xj ,
j
i. ti tj .
(1.52) -
,
, ,
x(t)
x(t0 )
. , :
t0 < t
-
E(x(t)|x(t0 )) x(t)x(t0 ) =
x P (x0 , t0 x, t) dx = x(t0 ) = x0 ,
P (x0 , t0 x, t).
-
. .
P = 0
x < 0, , ,
.
45
, , -
. :
E(xi |Fj ) :
xj . xj .
E(xi |Fj )
, . , , . . , , . , .
p = 1/2,
.
p = 1/2
-
, .
-
, .
x > x0 , x < x0 , | x x0 |. , . , . , , . . , , , , . , ,
. . , .
x
x0 ,
46
1.
2
. . , , , , . , . , , , . , , . . , .
47
48
2.
2.1 (. 34), , i , x 0 . n x :
x = x0 + 0 n + 0 n . 0 .
(2.1)
0 > 0, n
( ) , . .
1 + ... + n =
N (0, 1)
-
t,
n = (t t0 )/t. 2 = 0 /t, = 0 /t. x 2
t t0
, :
x(t) = x(t0 ) + (t t0 ) + t t0 .
(2.2)
-
x
t.
,
x(t)
, -
, , t t0 . dx = x(t) x(t0 ) dt = t t0 . (2.2) :
dx = dt + W, (2.3) W = dt. dx = a(x, t)dt, 1/2. , , d. , (2.3),
-
. (n
),
i
.
. 29, . ,
t(
C13 ).
49
b(x, t). , t, x:
a(x, t)
dx = a(x, t) dt + b(x, t) W
,
(2.4)
W = dt , N (0, 1). a(x, t) , b(x, t) 2 , b (x, t) . , a(x, t)
b(x, t) , , -
(
C14 ).
(2.4) -
-
xk+1 = xk + a(xk , tk ) t + b(xk , tk )
t k .
(2.5)
t x0 . 1 x1 . x1 x0 , t1 t0 + t. x0 , x1 , x2 ,... , . ,
k .
(2.5) .
dx = a(x, t) dt
xk+1 = xk + a(xk , tk ) t, , x0 = x(t0 ) t ,
t 0. k
-
!
t
,
x(t),
-
. (2.5) ,
(t)
t x(t), P (x0 , t0 x, t)
x(t).
50
2.
a(x, t)
b(x, t)
.
x
t0
x0 , t 0 : (x x0 )2 x x0 = a(x0 , t0 ), = b2 (x0 , t0 ), (2.6) t
t x0 = x(t0 ). :
(x x0 )k = t0
(x x0 )k P (x0 , t0 x, t) dx.
t
,
x0
x. t0
, (2.5) (2.6). :
x0
x x0 = a(x0 , t0 ) t + b(x0 , t0 ) t .
(2.7)
,
x
,
x0
. :
(x x0 )2 = a2 (t)2 + 2a0 b0 (t)3/2 + b2 t 2 = a2 t2 + b2 t, 0 0
0 0 2 a0 = a(x0 , t0 ), b0 = b(x0 , t0 ), , = 0, = 1. 2 t , b (x0 ,
t0 ). (2.7) x0 , . k , (x x0 ) k/2 (t) t k > 2 . , -
(2.6), . , (2.6)
x0 . , ,
, , ..
(x x0 )k /t
k>2
t 0.
P (x0 , t0 x, t).
51
. . t0 , x = x0 . x x = x(t). t > t0 x . . :
x = f (x0 , t0 , t, ),
(2.8)
x
t
,
,
,
, ,
P (x0 , t0 x, t)
-
. (2.8) ,
.
x(t) , t. , . x . , ,
.
t0
t1
t2 ,
: (2.9) (2.10)
x1 = f (x0 , t0 , t1 , 1 ) x2 = f (x0 , t0 , t2 , 2 ) = f (x1 ,
t1 , t2 , 3 ).
(2.9) t1 . -
x0
, -
. .
x1
.
1 . 2
(2.10) . , ,
, (, ,
1 ,
x1
1 ) , ,
x2 .
x1 = x(t1 ),
(2.10).
3
t1 ,
, ,
1 .
(2.10) -
x2
x1 , 3 .
,
f
(2.9), (2.10) ,
i
N (0, 1).
52
2.
2.2 , .
dt. -
b(x, t)
, ,
x
( ).
?
, -
? , :
dx = dt.
dx/dt
. :
x1 = x0 + 1 t,
x2 = x1 + 2 t = x0 + (1 + 2 )t,
...
n
, -
,
n: x = x0 + (1 + ... + n )t = x0 +
n t. t 0, n .
, ,
nt = t
,
t0 = 0. x = x0 + t t, t 0 x0 . .
, -
dt: dx = 2 dt.
:
x = x0 + (2 + ... + 2 ) t = u (nt) = u t, 1 n :
2 + ... + 2 n . u= 1 n ? 2 = 1, u = 1. ,
i n , t 0 t = n t.
53
u:(2.11)
n 2 2 2 ] 1 2 2 1 [ 4 2 2 =1+ . u = 2 i j = 2 n + (n n) n i,j=1
n n
n2 . n 4 2 2 21 , n n : 1 2 , .. ( C15 ). i i
j
, :
2 2 2 2 1 2 = 2 . , 1 2 : = 1, 4 = 3.
u
n u = 1. ! 4 . m dx = dt ( H8 ). 2 2 , (W ) = dt x(t) = t, , 2 .
P (u)
n . ,
2 u = u2 u2 = 2/n
:
(W )2
dt.
(2.12)
. -
dx = (x, t) 2 dt, t (x, t) . t n . , Noise dt . dt , . , ( C16
).
, , (2.4). ,
. .
54
2.
2.3
x(t)
. -
( ) F (t) = F x(t), t
F (x, t).
x
x(t),
. ,
:
dF = A(x, t) dt + B(x, t) W
(2.13)
x = G(F, t),
G
F
.
A
B,
,
.
F (x, t) = F (x0 + x, t0 + t) x0 x t:
F 1 2F F F (x, t) = F (x0 , t0 ) + x + t + ..., (x)2 + ... + x0
2 x2 t0 0
x0 , t0 .
-
x.
(2.7)
(x)
2
:
)2 ( (x)2 = a0 t + b0 t + ... = b2 2 t + ..., 0
t.
, -
t0
F0 = F (x0 , t0 ), , , ( C17 ): F F b2 2 F 2 F = F0 + (a0 t + b0
t) + 0 2 t + t t + ... x0 2 x0 0 (2.6)
(2.14)
t 0
:
F F0 F b2 2 F F A(x0 , t0 ) = = a0 + 0 + , t x0 2 x2 t0 0
(2.14)
F
,
= 0, 2 = 1.
, :
)2 ( (F F0 )2 F B 2 (x0 , t0 ) = = b2 . 0 t x0 t 0
. , .
55
, . ,
t 0
. , (2.13). , . (2.14) , ,
2.2.
t 2 2 . 1. F (x, t) W = dt: ) dt + b(x, t) F W . x
2
,
( dF =
F F b2 (x, t) 2 F + a(x, t) + t x 2 x2
(2.15)
. ( C18 ).
, -
F (x, t), dx = a(x, t)dt, :
x = x(t) ) dt.
F F dF = dt + dx = t x
(
F F + a(x, t) t x
(2.16)
,
b2 (x, t)
x.
, ,
dt.
, -
, , .
dx = dt + W
y = x2 ,
(2.15), :
d(x2 ) = (2x+ 2 ) dt+2 x W
=>
dy = (2 y+ 2 ) dt+2 y W.
, , . .
56
2.
2.4 , (2.4) -
W .
(2.5).
x
-
k
(
C19 ). ,
, . .
f (t)
s(t):(2.17)
dx = f (t) dt + s(t) W. W .
(2.5): ,
x1 = x0 + f0 t + s0 1 t, x2 = x1 + f1 t + s1 2 t = x0 + (f0 + f1
) t + (s0 1 + s1 2 ) t, ...,
fk = f (tk )
sk = s(tk ).
n
:
x = x0 + (f0 + ... + fn1 ) t + (s0 1 + ... + sn1 n ) t. , ,
(
sk . s2 + ... + s2 . , 0 n1H9 ):
t x(t) = x(t0 ) +t0
f ( ) d +
t
1/2 s2 ( ) d . x(t) (2.18)
t0 -
(2.18) (2.17) ,
, . . (2.18) , ,
s(t)
, -
x
t,
x(t)
(t).
57
,
a(x, t)
b(x, t)(2.19)
dx = a(x, t) dt + b(x, t) W,
(2.17), . :
( dF =
F b2 (x, t) 2 F F + a(x, t) + t x 2 x2f (t)
) dt + b(x, t)s(t)
F W. x
(2.20)
F (x, t) , W dt (2.20) s(t) f (t), : [ ] F a(x, t) 1 b(x, t)
s(t) F = , + s(t) = f (t), (2.21) x b(x, t) t b(x, t) 2 x
F/x
dt
(2.21) (2.21)
x ( H10 ). t x. , } 1 2 b(x, t) = 2 x2 x {
:
1 s(t) t
{
s(t) b(x, t)
} a(x, t) . b(x, t)
(2.22)
a(x, t)
b(x, t)
s(t),
(2.22) , (2.19) :
( ) ( ) F x(t), t = F x(t0 ), t0 + f (t)
tt0
t 1/2 f ( ) d + s2 ( ) d ,t0
(2.23)
(2.21), C20 ).
F (x, t)
(2.21) (
(2.23)
x(t)
F (x, t).
,
(2.22) . , , .
58
2.
2.5 :
dx = x dt + x W
,
(2.24)
. (2.24)
. ( (
= 0),
> 0)
(
< 0):
dx = x dt
=>
x(t) = x0 et .
, , . .
a(x, t) = x b(x, t) = x (2.22) . 57. s(t) s(t) = 0, . , s(t) , .
(2.21) F (x, t) = ln x, , , f (t) 2 /2. (t0 = 0) :2 x(t) = x0 e(
/2) t+
t
.
(2.25)
x x < 0, . (2.24).
x = 0 . x 0.
(1.11) . 16, :
x(t) = x0 et ,
x (t) = x(t)
e2 t 1.
, . , (2.24)
x,
:
dx/x = d ln x.
(2.15)
d(ln x) = ( 2 /2) dt + W .
-
, , . 57, (2.25).
59
:
dx = xW .
, ,
x = 0,
, .
x,
. :
dx = 0.05 x (dt + W ).
-
.3 500 2 400 300 1 200 100 0 0
, (), ().
Wt = W (t) = t,2
-
:
x(t) = e(,
/2)t+ Wt
.:
x(t) = F (t, W )
x x 2x 2 = ( /2) x, = x, = 2 x. 2 t W W Wt a = 0 b = 1. (2.15) :
( ) x 1 2 x x + dt + W = x dt + x W. dx = t 2 W 2 W x W , F x. x =
F (t, Wt ), x0 = F (0, 0), . F (t, Wt ) Wt , Wt = G(t, x), G F . x0
, . (R38 ) (R43 ) (. 276). , , , .
60
2.
- :
dx = (x ) dt + W ,
(2.26)
x
, -
.
.
x , . x x(t) . > 0 . (2.22) (2.21)
s(t) = s(t).
F (x, t),
, (2.23), ,
x
:
s(t) = et ,
F (x, t) = xet ,
f (t) = et . = 0):(2.27)
(t0
( ) 1 e2t . x(t) = + x0 et + 2 ,
x(t)
-
, .
> 0, . / 2 . x(t) x0 . (2.26) x(t) ,
/ 2 .
- -
.
x(t)
,
, . ,
/ 2 ,
,
. , , , . , , x(t) , .
61
- . ,
(
C21 ),
- . - . 2
= 0.1, = 0.1.
= 1, = 0.5.
.2
1
1
0
0
, ,
Wt ,
,
W t = t.
, , .
,
Wt ,
Wt / t.
, (2.27) , (2.26).
x
-
:
x > , x
) x dx = x ln 1 dt + x W. , x < H11 ).
(
(2.28) .
x = 0.
(
-. ,
x
(2.26), ,
y = ex
(2.28). (2.28) (2.26), . :
dx = (x ) dt + (x ) W.
(2.29)
x = ,
, .
x=
(
H12 ).
62
2.
, -
:
dx = a(x) dt + b(x) W. :
( a ) s(t) 1 = bb b = , s(t) 2 b
(2.30)
x,
,
. ,
x,
, -
.
,
:
( 2 ) b dx a= + b b , 4 b -
.
b(x) = = const
(2.26), . 60.
b(x) = x b(x) = x
(2.28), .
:
2 a(x) = + x + 2x. 4 (x0
= x(0), > 0):
]2 [ ) ( t 2t t x0 e + e 1 . x(t) = e 1 + 2 8
a(x)/b(x) = const,
a(x) = 0,
(2.30) :
b = . 2 b :
b , -
x=
db , + 4 ln b
.
63
. ,
x,
t: . < T ),(2.31)
dx =
x dt + W T t
T
(t
. :
s(t) =
, T t
F (x, t) =
x , T t
f (t) =
. (T t)2 (x0 = x(t0 )):
(2.32)
x(t) = + (x0 )
T t + T t0
(t t0 )(T t) . T t0
. . , x(t) x(T ) = : 2 2.5 2
tT
1
1.5
1
0
0.5
= 1. = 0.1, = 0.05. x0 = x(0) x(T ) = -
. , , :
) dx = (t) x (t) dt + (t) W.t
(
:
s(t) s(t) , f (t) = (t)(t) . (t) (t) (t) = /(T t), (t) = , (t) =
, , , T , (t0 = 0): [ ( )]1/2 x0 (T t) (T t)21 x(t) = + (T t) + 1 .
T 2 1 T 21 (t) s(t) = (t)e t0 , F (x, t) = x .
(t)dt
64
2.
2.6
x0 = x(t0 ) : x(t) = f (x0 , t0 , t, ). ,
-
P (x0 , t0 x, t). f
-
.
,
x(t)
. . , :
x(t) = x0 +
t t0 . x(t)
(2.33)
, , , , . ,
.
,
(x0 , t0 )
-
. , :
x1 = f (x0 , t0 , t1 , 1 ) x2 = f (x1 , t1 , t2 , 2 ) x3 = f (x2
, t2 , t3 , 3 ), ...,
x11 2
x33
x0 x2
ti ti+1
. -
(xi , ti ) (xi+1 , ti+1 ) , 1 , 2 , 3 ,.. -
. , , . , :
x2 = f (f (x0 , t0 , t1 , 1 ), t1 , t2 , 2 ). , , :
xt = x 0 +
t k=1
k .
xt
, ,
, ,
.
65
. -
t = [0..T ]:(2.34)
sin(k t/T ) t k x(t) = x0 + 0 + 2T , k T k=1
k N (0, 1)
. , (2.33). ,
2 x
(
x = x0 ):(2.35)
sin2 (k t/T ) 2 t2 2 x = x0 + + 2T = x2 + t, 0 2k2 T k=1
i j = 0, i = j 2 2 x = x0 + t 2 f (t) = t t /T t = [0..T ] ( H16
). 2 i = 1. , (2.33). (2.33) (2.34) ,
0 ,1 ,... , , , t.
- , ,
T.
, , -
k = N.
0 ,...,N ,
-
. :
N = 10, 20, 100.
0 , 1 ,..
:
N=10
N=20
N=100
, ,
N
.
66
2.
, -
. , .
. ,
P (x0 , t0 x, t) -
. , ..
P (x0 , t0 x, t),
, . , , . - (. 82).
.
, .
t .
, , .
. W (t) . ,
x(t) = x0 exp{( 2 /2)t + W (t)} W (t) x(t). -
. ( ). ,
( ) W (t) , x t, W (t) , x(t). , ( ) x t, W (t) . , - W (t)
,
. , -
W (t).
67
, . . , , , , , . .
-
. .
x(t)
,
,
.
, . , , , . .
x = f (x0 , t0 , t, ) , f
-
P (x0 , t0 x, t). , F (x) , , x ( x0 , t0 ):
F (x) =
F (x) P (x, t) dx =
( ) F f (, t) P () d,
P () . x = f (t, ), , , , t : { } 1 2 1 g(x, t) exp g (x, t) , P
(x0 , t0 x, t) = 2 2 x (2.36)
g(x, t)
x = f (t, )
, ..
= g(x, t).
. . , , .
68
2.
2.7 (. 40) , .
t1 < t2
,
t = t0 x0 = x(t0 ): ( ) ( ) covt0 (t1 , t2 ) = xt1 xt1 xt2 xt2 ,
(2.37) t,
xt = x(t)
xti = x(ti ).
,
-
. , ,
x0 = x(t0 ): x(t) = x0 + (t t0 ) + t t0 .
t0 = 0, t1 = t t2 = t + . , x, xt+ = x(t + ), xt = x(t). [0...t]
[t...t + ]. xt = 0 xt+ : (2.38) xt+ = xt + + .
xt
t, xt = xt = 0, : xt+ xt = x2 + xt . t
:
xt = x0 + t,
2 xt xt 2 = 2 t,
:
cov(t, t + ) = xt+ xt xt+ xt = 2 t.
t0 = 0 t .
(. 36). . ( H13 ) ( H14 ).
69
- :
x(t) = + x0 e
(
)
(tt0 )
+ 1 e2(tt0 ) 2 = 0): ] 2 [ 1 e2 t e . 2 ,
(2.39)
( H15 ). (t0
cov(t, t + ) = 2 (t) e =
(2.40)
t,
(2.40) ,
= t2 t1 : 2 cov(t, t + ) e . 2(2.41)
, . ,
x(t) = const, (t) = const,
-
cov(t1 , t2 ) = cov(t2 t1 ).
. , ,
t1 .
-
, .
t t0
-
. -
t .
x0 ,
,
(
). , ,
x0
. -
. ,
x = f (x0 , t t0 , ).
-
. , , .
70
2.
x(t) k k (t), x(t) = x(t) + k
:
k
. . ,
k (t)
x(t)
.
. ,
i j = ij
cov(t1 , t2 ) =
k
i ,
k (t1 )k (t2 ),
2 (t) =
k
k (t):
2 (t). k
k (t)
. -
[T /2..T /2] k = 2k/T .
(. 314) :
x(t) = x +
k=0
{k ak cos(k t) + k bk sin(k t)} ,
k , k
-
. :
cov(t1 , t2 ) =
{ k=0
} a2 cos(k t1 ) cos(k t2 ) + b2 sin(k t1 ) sin(k t2 ) . k k
= t2 t1 .
,
a2 = b2 : k k k=0
cov(t1 , t2 ) = cov( ) = :
a2 cos(k ), k
2 a2 = k T
T /2
cov( ) cos(k ) d.T /2
a2 k k .
, .
71
S() = a2 / = a2 T /2 k k
T
. , -
, : :
cov(t1 , t2 ) = cov(t2 , t1 ), cov( ) = cov( ). :
1 S() =
1 cov( ) cos( ) d =
cov( ) ei d.
. , , . ,
cov( ) = cov(t2 t1 ): - :
2 S() = 2
e
i | |
2 / . d = 2 + 2 = 0. -
, -
( ). , , .
, -
. ,
P (x0 ).
x0
, . ,
x0 = x0 = 0.:
, -
(x(t) x)2 = (x0 x0 + t t0 )2 = (x0 x0 )2 + 2 (t t0 )
2 2 x = x0 + 2 (t t0 ).
-
.
72
2.
2.8
W
Wt .
:
Wt
W .
,
xt = f (t, Wt ),
. ,
W ,
. :
{
dx = f (t) W dy = g(t) W.
(2.42)
( (2.18) . 56). ,
W , x = x0 + y = y0 + t = x0 + F (t) gj1 j t = y0 + G(t) , fi1 i
t f 2 ( ) d,t0
:
:
t F 2 (t) = G2 (t) =
g 2 ( ) dt0
,
k . fi gi , : F (t) G(t) =
i,j=1
fi1 gj1 i j t =
i=1
t fi1 gi1 t =t0 :
f ( )g( ) d,
i j
i = j.
1 = (t) = F (t) G(t),
t f ( )g( ) d = 1.t0(2.43)
.
73
-
-:
dx = (x ) dt + W.
dy = et W
=>
y(t) = F (t, x) = et (x): 2t y(t) = y0 + e 1 , 2
N (0, 1), y0 = x0 . x ( > 0): t 1 e2t , x(t) = + (x0 )e +
2
, . ,
Wt = t, (2.43)
Wt , : 2 1 et = = , t 1 + et f (t) = 1
g(t) = et .
, :
= 1 , :
= 1 + 1 2 2 .
2 2 = = 1,
= ,
2 2 = 1 + 22 ,
.. , -
x,
x: Wt xt
( ) t 1 e2t = 1 e t . = 2
, :
xt+ = + (xt )e + 1 e2 , 2 Wt xt+ = Wt xt e = (1 e ), [t...t +
]
:
t.
74
2.
-
W :
{
dx = W dy = f (x, t) W.
x0 = x(0) = 0, Wt ,
x(t) = Wt , y W ,
.
, , :
x i = x0 +
i j=1
j
t t.
yn = y0 +
n1 i=0
f (xi , ti ) i+1
yn xi i . i+1 , yn = y0 . :
(yn y0 )
2
=
n1 i,j=0
f (xi , ti )f (xj , tj ) i+1 j+1 t. i
, , :
j,
i,j
=
ij
+
i=j
.
,
f (x1 , t1 )f (x2 , t2 )2 3 . 3 ,
3 = 0. 2 2 2 2 f (x1 , t1 )2 = f (x1 , t1 ) 2 . :
2 (t) = (y(t) y0 )2 =
tt0
2 f (x0 + , ) d, x.
(2.44)
,
, . , , .
75
.
{
x0 = x(0)
y0 = y(0):(2.45)
dx = W dy = x W.
:
dy = x dx
x2 x2 0 y y0 = . 2
(2.46)
, -
y = y(x).
! ,
,
W
,
dx, dy x dx =
- , . ,
d(x2 )/2
(
C22 ). .
(2.45) :
{
x = x0 + W y = y0 + x0 W + 1 (W 2 t). 2 y = F (t, W ),
,
W,
.
(
a = 0,
dy =
y 1 2 y + t 2 W 2
)
dW = W , b = 1:
dt +
y W = (x0 + W ) W = x W, W
(2.45). ( ( H17 ) (2.45) H18 ) (2.44).
, ,
dx
x(t).
-
. 2:
2xdx = d(x ).
,
. . ,
t 0,
. -
, !
76
2.
3
x(t)
. , , . , . . , . : . .
77
78
3.
3.1
x(t)
-
, , , , . , .
x(t + dt) = x + a(x, t) dt + b(x, t)
t
t + dt:(3.1)
dt.
x = x(t) x(t + dt). -
. (3.1) (3.1)
P (x0 , t0 x + dt, t + dt). P (x0 , t0 x, t) P (), P () . x = 0,
(3.1) , :
x(t + dt) = x(t) + a(x(t), t) dt.
x(t)
dt,
-
:
d x = x = a(x, t) . dt
(3.2)
a(x, t) = (t) + (t) x,
(3.2) , -
:
x = (t) + (t) x .
b(x, t)
x
. -
! ,
F = F (x, t),
(2.15), . 55, :
d F (x, t) = dt
F F b2 (x, t) 2 F + a(x, t) + t x 2 x2 F (x, t),
.(3.3)
-
.
79
-:
dx = (x ) dt + W, . :
x, -
( ) x = x x0 .
=>
( ) x = + x0 et . ,
t0 = 0
,
t0 = 0
x =n
x = x0 , n x0 . ,
, - :
P (x0 , t0 x, t0 ) = (x x0 ). t0 = 0.
,
n x
x
F = x , : x2 = 2 x2 + 2 x + 2 .
2
, :
= / 2 .
2 [ ( ) ]2 ( ) x = + x0 et + 2 1 e2t , :
x (t) =
1 e2t .
, . . , - ,
xn = 0
=>
xn = xn1 + (n 1) 2 xn2 . 0 : x = 1 = 1, 3 x = 3 + 3 2 ,
F = xn ,
:
-
:
x = ,
2 x = 2 + 2 ,
4 x = 4 + 62 2 + 3 4 .
, , (. 60):
x = + . ,
2n+1
= 0,
x = 1 3 5 .. (2n 1).2n( H19 ).
-
:
dx = ( + x) dt + ( + x) W
80
3.
(3.3) , -
P (x)
.
F (x), , F (x) . :
[ ] b2 (x) 2 F F + dx = 0. P (x) a(x) x 2 x2
, , ,
P (x)
, :
] [ (a P ) 1 2 (b2 P ) + F (x) dx = 0. x 2 x2 F (x) , , -
. - :
] 1 2 [ 2 ] [ a(x) P = b (x) P x 2 x2 :
a(x)P =
] 1 [ 2 b (x) P . 2 x xm .
, ,
x ,
, -
. , - :
a(x) b (x) 1 P (x) = 2 , 2 P (x) b (x) b(x)
(3.4)
x.
:
{ } C a(x) P (x) = 2 . exp 2 dx b (x) b2 (x)
(3.5)
C
. -
. , (. 58)
a(x) = x
b(x) = x
P (x) x2+2/
2
.
.
81
- -:
dx = (x ) dt + W. (3.5) :
1 P (x) = P (x)
{ } (x )2 exp 2 , /
. :
x = + , 2
N (0, 1)
H20 )
. , (
dx = (x ) dt + x W .
:
dx =
2 + x2 W.
x = x0 .
a = 0,
:
x2 = 2 (2 + x2 )
=>
2 2 x = (2 + x2 ) e t 2 . 0 ( 2 ) t e 1
2 x (t)
= ( +
2
x2 ) 0
t
. , -
- :
P (x) =
/ , x2 + 2
. . ,
n x
n > 1.
82
3.
3.2 :
dx = (x ) dt + x W.
(3.6)
> 0, > 0
-
[(2.27), . 60], .
x
x(t)
, . , (
) x > 0. (3.6) . ,
x
.
P (x0 , t0 x, t)
(. 270). (3.2):
x
x = x +
=>
x = + (x0 )et . t0 = 0
x0 = x(0).
- .
F = x2
(3.3) :
x2 = 2 x2 + (2 + 2 ) x . (
x
, -
H21 )
2 x (t) = x2 x2 :
[ ]2 [ ] 2 x (t) = 1 et + 2x0 1 et et ,
= 2 /2 .
, ,
x
( ),
.
(t ).
- ,
x0 ,
-
= /2
2
t .
,
.
83
t
, -
- (. 80):
{ } { ( ) dx } C C a(x) P (x) = 2 exp 2 dx = 2 exp 1 . b (x) b2
(x) x x
, -:
( )1 1 x P (x) = ex/ , (3.7) () = / x > 0. - () (. . 313).
:
P(x)
x = ,
xmax = ( 1)
x xmax x, (
xn = ( + 1) .. ( + n 1) n .
P (0) = 0
> 1
2 < 2 .
H22 )
,
F (x) = xn .
:
dn (t, p) (t, p) = e , x = . p=0 dpn px F = e : ( ) 1 d p x e =
p ep x + p2 p xep x , dtpx n
= (t, p): ( ) 1 = p + p2 p . t p
(
H23 ) (. 316):
[ ( )]/ (t, p) = 1 p 1 et exp
{
x0 p e ( ) 1 p 1 et
t
}
(3.8)
(0, p) = ep x0 . p p, , - P (x0 , 0 x, t).
84
3.
,
.
.
u:
x(t) = x0 e
t
+
( ) 2x0 e t (1 e t ) + 1 e t u,
= 2 /2 ,
u
:
ek + p u =
{ 2 } 1 k /2 . exp 1p (1 p)/ } { 2 2 1 p f2 /2 + p f1 exp 1 pf3
(1 pf3 )/2 f2 = 2f1 f3 ,
(3.9)
, , :
px p (f1 +f2 +f3 u) e = e =
(3.8),
f1 (t), f3 (t)
:
f1 (t) = x0 et ,
( ) f3 (t) = 1 e t . ,
u -. t , x(t) u u. ek + p u k p u: 2 = 1, 4 = 3, u = , u2 =
(1+), u3 = (1+)(2+), :
u = 0,
2 u = 1 + ,
u2 = 0,
2 u2 = 2 + 3 + 2 ,
= / .
.
x = f1 + f2 + f3 u, , : 2 2 2 2 2 2 2 x = f1 + f2 2 + f3 u2 +
2f1 f3 u = f1 + f2 + (1 + ) f3 + 2f1 f3 ,2 x (t).
85
u. p p, k k -:
P (, u) =
ek p uk /2(1p) dp dk . (1 p) (2)22
k
(14), . 312:
e /2 P (, u) = 22
ep (u /2) dp . (1 p) 22
p
- -
(. 27) c
x = u 2 /2
0.
:
( P (, u) =
2 u 2
)3/2
eu , ( 1/2) 2
u
2 . 2
(3.10)
P (, u) = 0. , , x(t) ( , ).
u < 2 /2,
(, u)
(), ():
P (, u) u, 2
:
e /2 P () = , 2 (3.9).
P (u) =
u1 eu , ()
.. -. ,
86
3.
, .
x(t) = f (x0 , tt0 , , u), :
xt = f (x0 , t, 1 , u1 ),
xt+ = f (xt , , , u), 1 , u1 ,, ,
, u
xt .
,
xt+
xt
, :
xt+ xt = x2 e + (1 e ) xt , t ,
3/2 3/2 xt = x t = 0, = 0, u = = / .
, :
2 cov(t, t + ) = xt+ xt xt+ xt = x (t) e .
t
,
-, .
(. 71)
2 | | cov( ) = e , 2
2 / S() = 2 . + 2 .
-
,
, . , . , , . :
x 2 x, ( dx =
x.
-
x x 2
) dt + W,
= 2 2 /2. ,
.
87
, x: dx = a dt + x W. (3.11)
a = const,
0,
,
(3.11) :
x(t) = x0 + :
2t x0 t + u, 2
(3.12)
a
u,
-
ek + p u =
1 exp (1 p)2a/2
{
k 2 /2 1p
} .
2a/ 2 .
:
x(t) = x0 + a t,
a t2 2 x (t) = 2 x0 t + 2
(
) .
, . , . ,
a < 0 ,
x = 0
,
. . , . (3.10) , , ,
x(t) > 1/2,
a > /4.
2
, -
, . . , .
,
x0
,
x = .
x 2 . , > /4.
88
3.
3.3 (. 10).
x0 = x(0):
dx = (x x2 ) dt + x W. , , : :
t t/, x x/ .
dx = x (1 x) dt +
2 x W,
= 2 /2. , W = dt. , . , :
t t,
x
x,
x0
x0 .
(
= 0) =>
:
dx = a(x) = x (1 x) dt
x(t) =
1 , 1 (1 1/x0 ) et
t , x0 , x = 1. x0 = 1, . . a(x ) = 0 x = 0 x = 1. a(x) , :
dx = a(x) a (x ) (x x ) + .. dt a (x ) > 0, . , x > x
dx/dt , x , x . , a (x ) < 0. x = 1. , , .
89
. -
(3.3),
t . 78, F = ln x F = x:
ln x = 1 x x = x x2 . , :
x = 1 ,
2 x = x ,
2 x = (1 ) .
(3.13)
, ,
1.
,
< 1. -
- -:
1 P (x) = ()
( )1 x ex/ ,
= (1)/ . xmax = (1)/ - . , , .
asym = 2/ excess = 6/ . P (x) (. . 83),
x,
xmax . x1 = (2 1) x1 + 1, [ F = 1/x:(3.14)
:
1 x =
] 1 1 1 + e(21) t . x0 2 1 2 1,
(3.15)
(1/x
x(t).
= 1/x), y(t) = 1/x(t), ,
:
[ ] dy = 1 + (2 1)y dt 2 y W. (3.15),
= 1/2
. , :
x
1
=
x1 + t. 0 = 1/2.
2 1. 1/2
(3.14),
90
3.
.
, (. 49), .
x (t)
P (x0 , t0 x, t). C ++
x,
, .
x0 = 1.
), :
( -
1
1/80.8 0.6
1
20.8 0.6 0.4
1/4 1/2
1/2 1/4 1/8
1
0.4 0.2 0 0 1 2 3 4 5 6 7 8
1 29 10
0.2 0 0 1 2 3 4 5 6 7 8 9 10
< 1,
1 .
x =
1
, .
,
x = 0.
-
,
x = 1.
.
x = 1.
. , , ,
x = 0. x
, , , ,
dx = 0.
x=0
, -
,
x < 0.
, ,
x = 0. x = 1 .
x0 = 1 ,
-
,
91
x = 1, dx = x(1 x) dt -
. . ,
dx = a(x) dt + b(x) W
a(x) x , a(x ) = 0, b(x) dx = a (x ) (x x ) dt + b(x ) W,
:
x. t :
a (x ) < 0,
, -
,
x(t) x + b / 2a .
b(x ) 2a (x )
, x
(3.16)
-
x = 1,
a (x ) = 1,
b(x ) =
2, t
(3.16) :
x(t) 1 +
,
(3.17)
.
1,
.
(3.13),
, (3.17)
.
, , - -. ,
.
, - , , . , , .
92
3.
3.4 t
t.
, .
:
dx = a(x, t) dt + b(x, t) W x0 = x(t0 ): x = x0 + a(x0 , t0 ) (t
t0 ) + b(x0 , t0 ) t t0 . = 0 2 = 1, , t t0 , : x = x0 + a(x0 , t0
) (t t0 ) + ... 2 [ ] x = x2 + 2x0 a(x0 , t0 ) + b2 (x0 , t0 ) (t
t0 ) + ... 0 ,
2 x (t) = b2 (x0 , t0 ) (t t0 ) + ... , .
:
dx = x (1 x) dt + :
2 x W.
x = x0 + x0 (1 x0 ) t + f t2 + ... 2 [ ] x = x2 + 2 x2 (1 x0 ) +
x2 t + ... 0 0 0 :
f.
2 x = x x , t:
[ ] x0 (1 x0 ) + 2 f t + ... = x0 (1 x0 ) + x0 1 (3 + 2)x0 + 2x2
t + ..., 0:
2 f = 1 (3 + 2)x0 + 2x2 . 0 x0 .
93
n . (3.3), . 78, F (x) = x , :
xn = (n + n (n 1)) xn n xn+1 ,
:
[ ] n [ ] x = xn 1 + fn,1 t + fn,2 t2 + ... = xn 1 + fn,k tk . 0
0k=1
t, (fn,0
k = 1, 2, ... -
= 1):
k fn,k = n (1 + (n 1)) fn,k1 nx0 fn+1,k1 . Matematica Wolfram
Research, Inc.
t5
:
f[n_, 0] := 1; f[n_,k_] := (n/k)*((1+(n-1)*g)*f[n,k-1] -
x0*f[n+1,k-1]); av = x0; Do[ av += x0*f[1, k]*tk, {k, 1, 5}];
Collect[av, t, Simplify]
f.
Do
t.
, . , :
tn ,
f[n_, 0] := 1; num = 5; Do[ Do[ f[n,k] =
(n/k)*((1+(n-1)*g)*f[n,k-1]-x0*f[n+1,k-1]), {n, 1, num-k+1}], {k,
1, num}] av = x0; Do[ av += x0*f[1, k]*tk, {k, 1, num}];
Collect[av, t, Simplify] f[n_,
k
n
-
fn,k . , : k_]:=f[n,k]=(n/k)...
94 :
3.
[ ] [ ] 2 2 t = 1 + 1 x0 t + 1 (3 + 2)x0 + 2x0 2! [ ] t3 + 1 (7
+ 10 + 4 2 ) x0 + (12 + 16) x2 6x3 + ... 0 0 3! 2 2 2 x : x (t) = x
x x02 [ ] t2 [ ] t3 x (t) + 12 + 12 + 4 2 (48 + 46)x0 + 38x2 + ...
= t + 4 + 2 6x0 0 2x2 2! 3! 0 . ,
x0 = 1.
= 0
. : :
x0 x = 1 .
-
x 1 t2 t3 t4 = + (3 2) (7 38 + 4 2 ) 2 2! 3! 4! 5 t + (15 334 +
284 2 8 3 ) 5! t6 (31 2146 + 7012 1848 + 16 ) + ... 6! ( = 1/2) ( k
= 12 3 40.5
k = 10)0 1
() () :0.8
0.9
0
0.5
. , , .
95
t.
,
-
. . n,0 (t), n n x , , n,0 (0) = x0 . , . ,
n x = n,1 (t).
.
n,k (0) = xn . 0 t
n,0 (t),
-
, . :
xn = n (1 + (n 1) ) xn n xn+1 . :
n,0 (t) = xn . 0
-
n,1 = n (1 + (n 1) ) xn nxn+1 , 0 0:
n,1 = xn + xn [1 + n(1 x0 ) + n(n 1) ] t, 0 0 ..
t,
x
n
n
.
:
n,0 = xn ent . 0
( ) n,1 = xn + xn [1 + (n 1) ] 1 ent 0 0
( ) n xn+1 1 e2nt . n+1 0
.
.
96
3.
3.5 :
dx = a(x, t)dt + b(x, t) W
b(x, t)
-
.
.
c(t)
:
c = a(c, t).
(3.18)
:
z=
x c(t) .
:
dz =
1 [a(c + z, t) a(c, t)] dt + b(c + z, t) W, F = zn:
c
(3.18).
(3.3), . 78,
n(n 1) n2 2 z b (c + z, t) . z n = n z n1 [a(c + z, t) a(c, t)]
+ 2
2
a
b2 : Dk (t) (z)k .
a(c + z, t) =
k=0
Ak (t) (z) ,
k
b (c + z, t) =
k=0
c(t) Ak = Ak (t), Dk = Dk (t). A0 = a(c(t), t), A0 , :
] [ k+n2 k n+k n(n 1) zn = Dk z . nAk+1 z + 2k=0
(3.19)
:
n n zi (t) i z = i=0
(3.20)
1 = 1,
zin , n , ! , 0 zi0 = 0 i > 0 z0 = 1.
97
(3.20) (3.19). :
i=0
zin (t) i
=
[
n Ak+1 zin+k k < i .
k,i=0
] n(n 1) k+n2 Dk zi k+i . + 2
k = k.
i = i k ,
i > 0,
, :
zin (t)
=
i { k=0
n+k n Ak+1 zik
} n(n 1) k+n2 + Dk zik . 2
(3.21)
:
1 z0 (t) = A1 z0 1 2 z0 (t) = 2A1 z0 + D0 2 3 1 z0 (t) = 3A1 z0
+ 3 D0 z0 3 4 2 z0 (t) = 4A1 z0 + 6 D0 z0 4 ... 1 2 z1 (t) = A1 z1
+ A2 z0 1 2 3 1 z1 (t) = 2A1 z1 + 2A2 z0 + D1 z0 2 3 4 1 2 z1 (t) =
3A1 z1 + 3A2 z0 + 3D0 z1 + 3D1 z0 3 ... 1 2 3 z2 (t) = A1 z2 + A2
z1 + A3 z0 1 2 3 4 1 2 z2 (t) = 2A1 z2 + 2A2 z1 + 2A3 z0 + D1 z1 +
D2 z0 2 ... 1 2 3 4 z3 (t) = A1 z3 + A2 z2 + A3 z1 + A4 z0 , ...
1
x0 =
c(t0 ),
z(t) z(t0 ) = 0. n z t = t0 . (3.21)
, , .
t
,
zin
= 0,
, ,
. :
Ak = Ak (), Dk = Dk (),
A2 D0 z = ... 2A2 1 2 D0 D0 2 2 2 + z = 4 (D0 (5A2 3A1 A3 ) 3D1
A1 A2 + D2 A1 ) + ... 2A1 4A1 , - .
98
3.
:
dx = x dt + x W. (. 58), :
x = x0 et ,
] [ 2 4 t2 2 x = x2 e2t+ t = x2 e2t 1 + 2 t + + ... . 0 0 2
x,
c(t)
-
. :
A1 = ,
D0 = x2 e2t , 0
D1 = 2x0 et ,
D2 = 1.
, :
] n(n 1) [ 2 2t n2 t n1 n x0 e zi + 2x0 e zi1 + zi2 . = + 2 (n =
1) i- 1 1 zi = zi . z(0) = 0, zi = 0, , , 1 x = c(t) = x0 et . :
zin n zin2 z0 = 2 z0 + x2 e2t 2 0 2 2 z1 = 2 z 1 2 2 z 2 = 2 z2 + z
0 2
=> => =>
2 z0 = x2 e2t t 0 2 z1 = 0 2 z0 = x2 e2t t2 /2, ... 0 .
-
:
dx = x (1 x) dt + x W. (. . 10):
[ ]1 c(t) = 1 et ,
= 1x1 . 0 A2 = 1, D0 = c2 (t), D1 = 2 c(t), D2 = 1.
:
A1 = 1 2 c(t),
t
,
2 c(t) z, z -
(3.13), c.89.
99
(3.21) :
z0 (t) 1
[ ] 1 = 1 2c(t) z0 (t)
=>
1 z0 (t)
z0 et = . (1 et )2
z(0) = 0, , , z0 , , , z , , 1 3 2 1 z0 (t) = 0. z1 (t) = z1 (t)
= z2 (t) = 0. 2 z [ ] 2 z0 (t) = 2 1 2c(t) z0 (t) + c2 (t), 2
2 z0 (0) = 0
:
2 z0 (t) =
4 z
1 4et + (22 t + 4 1)e2t . 2(1 et )4
2 z0 :
( 2 )2 4 z0 (t) = 3 z0 (t) ., :
1 z1 (t)
1 2(1 + (t 1))et + (1 2)e2t = . 2(1 et )3
. -
= 0, .. x = 1. 4 x : ,
( )2 2 ( )( ) 4 x = 1 1 et + et 2 3et 2t 3 + 4et e2t . 2 4 :
2 ( ) 2 x = 1 1 4et + 3e2t + ... 2 , . .
,
- .
100
3.
4
P (x0 , t0 x, t),
.
. , -. - .
x(t) = f (t, ) .
f (t, )
-
.
101
102
4.
4.1 .
= 0 x(t)
: x = x0 + t t0
= (x x0 )/ t t0 ,
, (. . 67), :
1 (x x0 )2 P (x0 , t0 x, t) = exp 2 t t0 2 (t t0 ) 1
{
} .(4.1)
t t0 ,
,
t t0
- :
P (x0 , t0 x, t) = (x x0 )
t t0 .
(4.2)
x = x0
, ,
x
x0
(. , .
315).
t = t0 . x x0 .
,
t0
-
- :
t t0 (t t0 )/ . (x x0 )2 + (t t0 )2
P (x0 , t0 x, t) =
(4.3)
x
,
x0 .
. . . . ,
t t0 .
x0 .
, , , .
103
-
.
t1 < t2 < t3 ,
x(t)
x1 , x2
x3 .
x1
x3
: (4.4)
P (x1 , x3 ) =
P (x1 , x2 , x3 )dx2 ,
ti . (4.4)
x2 .
-
.
P (x1 , x3 ) = P (x1 ) P (x1 x3 )
, , , (. (1.42), . 37):
P (x1 , x2 , x3 ) = P (x1 ) P (x1 x2 )P (x2 x3 ). , :
P (x1 , t1 x3 , t3 ) =
P (x1 , t1 x2 , t2 ) P (x2 , t2 x3 , t3 ) dx2 .
(4.5)
-. ( H24 ) , H25 ) -
(4.1). (
- ,
P (x0 , t0 x, t) = P (x x0 , t t0 ),
(4.3). - . , . (4.5)
t1 , t 2
t3
. .
P (x0 , t0 x, t) {x0 , t0 }
-
. , , ,
{x, t}. (4.5) {x0 , t0 }, . -,
{x, t}. -
. .
104
4.
4.2 P (x0, t0 x, t) x0 , t0 . -. t0 , t0 t0 + t.
t1 = t0 , t2 = t0 + t
t3 = t:
P (x0 , t0 x, t) =
P (x0 , t0 y, t0 + t) P (y, t0 + t x, t) dy. (yx0 )
y , t0 + t, x0 t0 . y x0 , y = x0 , , , :
t
1 2P P 2 (y x0 ) + P (y, t0 + t x, t) = P + 2 (y x0 ) + ..., x0
2 x0
P = P (x0 , t0 + t x, t).
,
y,
. , :
P (x0 , t0 x, t) = P (x0 , t0 + t x, t) P (x0 , t0 y, t0 + t) dy
P + (y x0 ) P (x0 , t0 y, t0 + t) dy x0 1 2P + (y x0 )2 P (x0 , t0
y, t0 + t) dy 2 2 x0 + ... ( ), . ,
P.
P (x0 , t0 x, t) t 0 :
t.
P (x0 , t0 + t x, t) P (x0 , t0 x, t) P (x0 , t0 x, t) , t
t0
t0 .
105
y
:
P (x0 , t0 x, t) 1 2 P (x x0 )2 P (x x0 ) + + = 0. t0 x0 t 2 x2
t 0 ,
(x x0 )3
, ..
(. . 50).
t 0
, , (t
0): (x x0 )m P (x0 , t0 x, t0 + t) dx.
(x x0 )m =
t 0 , , t t, t 0. , , . , , -
:
P P 1 2 2P + a(x0 , t0 ) + b (x0 , t0 ) =0 t0 x0 2 x2 0
,
(4.6)
P = P (x0 , t0 x, t).
,
x
t,
x0
t0 .
x0 = x(t0 )
, -
:
P (x0 , t0 x, t) = (x x0 )
t t0 .
(4.7)
, .
x x0
t > t0 .
106
4.
P (x0 , t0 x, t) x, t. t t x. t y : y = x + a t + b t, (4.8) a =
a(x, t t), b = b(x, t t). x P (x, t t) = P (x0 , t0 x, t t). c P
(). y t . P (y, t) = P (x0 , t0 y, t), (. . 22):
F (y) =
F (x + at + b t ) P (x, t t) P () dx d
F (y)
P (x,)
(4.9)
, c
F (y)
t. , , (4.8)
x, y
,
, (4.9) .
t
o,
F (..)
,
t:
) 1 2F 2 2 F ( F (x + at + b t) = F (x) + a t + b t + b t + ...
x 2 x2
x
, (4.8)
t. t t.
t,
t.
-
t
-
. ,
a = a(x, t), b = b(x, t). t: P (x, t) t + ... t
P (x, t t) = P (x, t)
, .
107
(4.9), :
t.
= 0, 2 = 1,
F (y) =
[ F (x)P (x, t)dx t
] F 1 2F 2 P aP b P dx. F t x 2 x2
F = F (x), P = P (x, t). t ( x y ). . ( C23 ),
F (x),
:
] 1 2 [ 2 ] P [ + a(x, t) P b (x, t) P = 0 , t x 2 x2 (
(4.10)
F (x)).
- ,
P = P (x0 , t0 x, t).
- . , . , , . , (4.7), . ,
t0
x
x0 ,
-
. , .
:
P (x0 , t0 x, t) dx = 1, . (4.10) (4.11)
P,
P
-
. (4.11).
108
4.
4.3 - -
a(x, t) = 0 P 2 2P = . t 2 x2
b(x, t) = :(4.12)
.
2 . P (x, t) (-
) - (. , . 314):
P (x, t) =
(k, t) eikx
dk . 2
(4.13)
(4.12),
(s, t) 2k2 = . t 2
: (4.14)
, :
P (x, t0 ) = P (x0 , t0 x, t0 ) = (x x0 ) = -
ei(xx0 )k t = t0
dk . 2
(k, t0 ) = e
ix0 k
. (4.14) :
(k, t) = e
2 2
k (tt0 )/2+ix0 k
.
(4.13) (14), . 312, :
P =,
e
2 2
k (tt0 )/2ik(xx0 )
{ } 1 1 (x x0 )2 dk = exp 2 . 2 2 (t t0 ) 2(t t0 ) x0 .
, -
t t0 .
x0
, -
. ,
x
x0 .( H26 ), ,
-, :
dx = f (t)dt + s(t)W
(. 316), - :
dx = (x ) dt + W
(
H28 ).
109
2.7,
. 71,
. , ,
x0 = x(t0 ).
-
. - , - , - :
P0 (x0 ). dk . 2
P0 (x0 ) =
0 (k) eikx0
:
(k, t) = 0 (k) e
2 2
k (tt0 )/2
. x, -
(4.13). , :
1 2 2 2 2 P0 (x0 ) = e(x0 a) /2b => 0 (k) = eiakb k /2 , b 2
a - , b - ( x0 ). P (x, t), , :
2 (t t0 ) b2 + 2 (t t0 ). ,
x
-
b
-
(t t0 ).2
P0 (x0 ). t .
,
t0 = 0,
N (0, 1)
:
x = x0 +
x0
2 x
= (x x)
, :
2
( )2 = (x0 x0 )2 + 2 t, = x 0 x0 t
x0 = x0 = 0.
110
4.
4.4 (2.24), . 58,
x , -
x > 0.
. . .
dx
. , , , . , , . . () , . .
-
.
x
,
. .
[..]
, -
. , .
, -
x =
x
-
x = ,
-
. , .
,
, ,
=0
= 2
.
. ,
x
. -
.
111
- (4.10), . 107,
P = P (x0 , t0 x, t):
] 1 2 [ 2 ] P [ + a(x, t) P b (x, t) P = 0. t x 2 x2 :
J P + = 0, t x
[ ] b2 (x, t) P 1 J(x, t) = a(x, t) P . 2 x
(4.15)
J(x, t)
.
x
[..].
. (4.15)
x:
dp(t) = J(, t) J(, t), dt
p(t) =
P (x0 , t0 x, t) dx.
(4.16)
x < x <
J
. (4.15)
, (4.16) . . ,
[..]
,
( , ).
n(x, t).
N , P (x, t), n(x, t) = N P (x, t).
J = v n(x, t). J dS,S
:
n + J = 0, t
d dt
n(x, t)dV = V
J = v n, v S
,
- .
V,
J = Jx /x + Jy /y + Jz /z dS S , .
, , , .
112
4.
p
x
[..]
:
dp(t) =0 dt
=>
J(, t) = J(, t). x =
x = , . , ( ). . :
, , , .
P (x, t), = P (x0 , t0 x, t)):
.
, (P (x, t)
:
: :
J(, t) = 0 P (, t) = 0 J(, t) = J(, t), P (, t) = P (, t).
(, , ). ,
P (, t) = 0.
. , . , , . , . . - . .
113
,
x
, -
, ( ):
x
dx = dt + W.
x = 0
. -
- .
x0 > 0, -
. . . . , -
P/t = 0: P (x) = 2 2x/2 e . 2
2 P (x) = 0 P (x) 2
=>
, . , . .
, [..]. (4.15) : 2 P (x) P (x) = J0 2
=>
P (x) =
J0 2 + P0 e2x/ . J0 P/t = 0.
,
x
-
,
J() = J()
J(x) = J0 = const. P () = P () P0 = 0. P (x) J0 /, . , P (x) =
1/( ). .
x = .
x = ,
. ,
. , . , .
114
4.
4.5
-
[..].
,
t0
= 0 < x0 < . p(x0 , t) , t [..], : p(x0 , t) =
P (x0 , 0 x, t) dx =
P (x0 , t x, 0) dx.
(4.17)
, . . , t0
= t,
t = 0. t p = p(x0 , t) : (4.18)
(4.17) (4.6), . 105.
b2 (x0 ) 2 p p p + = a(x0 ) . x0 2 x2 t 0
-
P (x0 , 0 x, 0) = (x x0 ). (4.17) : p(x0 , 0) = 1 ( < x0 <
). , x0 , [..] , : p(, t) = p(, t) = 0.
T
. ,
T-
p(x0 , t)
,
T
t
( ). ,
T < t,
1 p(x0 , t).
t
-
:
[..].
, ,
T =0 ,
) ( t 1 p(x0 , t) dt = t
p(x0 , t) dt.0
p(x0 , ) = 0, .. [..] . n- n T Tn (x0 ) = T , Tn (x0 ).
115
T n ,
:
Tn (x0 ) = T n = 0
p(x0 , t) tn dt = n t ntn1
tn1 p(x0 , t) dt.0(4.19)
(4.18)
dt:
a(x0 ) Tn (x0 )
b2 (x0 ) + Tn (x0 ) = nTn1 (x0 ). 2 1 = 1
T0 (x0 ) = 1.
, . ,
T (x0 ) = T1 (x0 ): b2 (x0 ) T (x0 ) = 1 a(x0 ) T (x0 ) + 2
T () = T () = 0
(
x0
, ).
,
= 0
:
2 T = 1 2
=>
2 x2 T = 0 + Ax0 + B, 2 2
A
B
.
:
x = 0, L. T (0) = T (L) = 0 x0 (L x0 ) . 2 -
T = T (x0 ) =
T = L2 /4 2
,
x0 = L/2.
( )
.
x0
x = 0,
L
. H27 ) -
(
L .
116
4.
4.6
-
a(x)
D(x) = b2 (x):
] 1 2 [ ] [ P + a(x) P D(x) P = 0. t x 2 x2
P = u (x) et . u(x) ( - x): [ ] 1 [ ] a(x)u (x) D(x)u (x) = u
(x). (4.20) 2 (. 111) [...] :
1 , 2 , ...
( )
u (x).
, -
-.
a(x) = 0 D = 2 . (4.20) : 2/ . u (x) + 2 u (x) = 0, :
w=
u (x) = A sin(x) + B cos(x).
0 x = L u (L) = 0.
[0..L] . x = : u (0) =
:
un (x) =
2 sin(n x), L
n =
n L n =
n = 1, 2, ...
,
2 2 n /2.
2/L
-
, :
L un (x)um (x)dx =0
2 L
L sin(n x) sin(m x)dx = nm ,0(4.21)
nm , n = m , m = n. .
117
, :
P (x0 , 0 x, t) =
n=0
An un (x) en t . un (x) -
.
P (x0 , 0 x, 0) = (x x0 )
(4.21), :
L An =0 :
L P (x0 , 0 x, 0) un (x)dx =0
(x x0 ) un (x)dx = un (x0 ).
2 P (x0 , 0 x, t) = sin(n x0 ) sin(n x)en t . L n=0
[0..L]
,
. .
x=0
x=L
(4.15), . 111:
2 et 2 P (x, t) J(x, t) = = u (x) 2 x 2 , , , :
u (0) = u (L) = 0. : n 1 2 un = cos(n x), n = , u0 (x) = , L L L
, -
n = 1, 2, ...
. :
1 2 P (x0 , 0 x, t) = + cos(n x0 ) cos(n x)en t . L L n=0
t
P (x0 , 0 x, t) 1/L,
L.
-.
118
4.
, A (2 2 , A = d /dx ), : Au(x) = (x) u(x), (4.22)
(x) . (x) (x) : (x)A(x) dx =
(x)A (x) dx,
(4.23)
A
. (
) .
un (x), um (x) (4.22), n m . (4.22), :
u (x)Aun (x) dx = n m
u (x)un (x)(x) dx, m
un (x)A u (x) dx = m m
u (x)un (x)(x) dx, m
(4.22)
(x). = u , = un ). m :
A
,
(
(n ) m
u (x)un (x) (x) dx = 0. m
n = m,
, , -
, (n
= n ).
n = m
, -
(x).
A
, ,
. ,
u (x)un (x) (x) dx = nm m
(x).
119
:
F (x) =
fn un (x), fn => fn =
F (x)u (x) (x) dx, n
.
A
(4.20) . -
(4.20)
= (x) ,
(4.23). :
{
} } { ( ) ( ) 1 1 ( ) dx = ) a D dx + I, a D 2 2 (
I
:(4.24)
I = a
1 (D ) 2
+
1 ( ) D . 2
. ,
( ). , :
2a = D D
=>
(x) = exp
D (x) 2a(x) dx. D(x)
(4.25)
, , (I (4.15) :
= 0).
J = a
1 (D ) , 2
J = a
1 (D ) . 2 (x),
(4.25) (4.24) :
I = (x)((x)J (x) (x)J (x))
= 0.
, , -
4.4,
. 110
I.
, , (4.20),
(x) (4.25), . n
- :
P (x, t) =
an un (x)en t , an
an =
P (x, 0) u (x) (x) dx, n P (x, 0).
120
4.
4.7 x(t, )
x = f (t, )
t
.
,
f (t, )
. , . ,
f
= g(x, t).
-
f
g . : dx = f d + t f dt, dx , :
d = x g dx + t g dt,
x g = g/x,
..
(
f x g = 1,
t g = x g t f,
2 x g
= x
1 f
)
2 f x g = . ( f )2
(4.26)
t
,
g(x, t). x, 1 .
t + dt
-
( ) 2 = g x + dx, t + dt , 1 = g(x, t). ( ) k k 2 k - 2 = g x +
dx, t + dt dt, dx: k 2 k 1k1
2 :
2 [ ] k2 2 k1 (dx) = + kg (g dx + gdt) + k(k 1)g g + kg g + ..,
2 x, . dx dx = adt + b dt, k 1 . 2 = k , = 0, 2 = 1 k k + 1, : 1 (
) D k k1 D 2 g g a + g + g + kg g = 0, 2 2
D = b2
. -
F (g) g a + g +
(
Fk
D g 2
)
k = 0, 1, ...: D = 0, + F (g) g 2 2
F (g) = F0 + F1 g + F2 g 2 + .. 1 = g P (1 ). g (x, t) , x = f
(1 , t) g (x, t).
121
:
D F (1 )g 2 P (1 ) d1 = 2
[ ] 2 D F (1 ) g P (1 ) d1 . 1 2
:
[ ] [ ] [ ] D D x D 1 g 2 = g 2 = g 2 , 1 2 x 2 1 x 2 g ,
x/1 = f = 1/g
(. (4.26)). :
(1 ) = P (1 )/P (1 ),
[ ] ) D 2 D 2 D 1 (1 ) g = 0. F (g) g a + g + g + g 2 2 x 2 g
(
F
-
,
1 = g(x, t)
:
g=
] D(x, t) [ 1 D(x, t) g a(x, t)g (g) g 2 g . 2 x 2
(4.27)
(4.26),
f (t, ):
[ ] D (f, t) D(f, t) () f f = a(f, t) + + 2 , 2 2 f f
(4.28)
D = D/f
1 . = 0),
(D
, (4.28) ,
f = a(f, t).
-
x(t0 , ) = x0 .
() = .
. 1 P () e () = ( 1)/.H42 ) , -
(
(4.27) (4.28) -.
122
4.
5
, , . . , . ,
dt,
W .
, . -
dt,
.
W .
, -
, .
123
124
5.
5.1 n 1 , ..., n , :
i N (0, 1),:
-
,
t = t/n
Wn = W (tn ) = (1 + ... + n ) t = nt = t. 1 , ..., n
(5.1)
n
.
:
xk+1
= a(xk , tk )t + b(xk , tk ) k t,
,
Wt . W = dt Wt . ,
Wt = W (t), t:
-
Wt
tSt
St =0
W d.
(5.2)
, , :
St =
n k=1
Wk1 t = [1 + (1 + 2 ) + ... + (1 + ... + n1 )] (t)3/2 ,
[0..t]
(5.3)
n
t.
k
-
k
.
1 , ..., n , t
St
.
St
Wt ,
.
Wt = W (t), , St
.
125
] [ (n 1) 1 + ... + 1 n1 (t)3/2 = 1 12 + 22 + ... + (n 1)2
(t)3/2 . ,
(5.3) :
1 N (0, 1). 2 2 . 1 + ... + (n 1) (n 1)n(2n 1)/6. n , t 0, nt =
t, :
t St =0 ,
t3/2 W d = 1 . 3
St , 3/2 3 t , .. St N (0, t /3). . 1 Wt . , Wt k , St : [ ] Wt
= 1 + 2 + ... + n1 + n (t)1/2 = 2 t [ ] t3/2 St = (n 1) 1 + (n 2) 2
+ ... + 1 n1 (t)3/2 = 1 . 3
t
2
.
, .
1
2
N (0, 1). , :
t2 (n 1)n t2 2 2 Wt St = 1 2 = (1 + 2 + ... + n 1)(t) = (t) . 2
2 3 3/2 1 2 = (. . 33)
, : 1 =
3 1 + , 2 2
2 = .
:
Wt = t,
Wt t3/2 t3/2 t+ . St = ( 3 + ) = 2 2 3 2 3
(5.4)
, ,
2 2 Wt St = 5 t4 /6.
126
5.
St
-
.
t0
t,
W0 = W (t0 ) Wt = W (t). (5.4) Wt Wt W0 W0 (tt0 ): W0 + Wt (t t0
)3/2 St = (t t0 ) + . 2 2 3
Wt W0 t-t0
W0
Wt
,
(W0 + Wt )(t t0 )/2. ,
,
W0
Wt .
,
S = f (W0 , Wt ).
,
-
(t t0 )
3
.
t
n + 1 W0 , W1 , ..., Wn , t t0 = n t, n
:
) Wn W0 t + W1 + ... + Wn1 + t + t t0 , Sn = 2 2 2 3 , (1 + ...
+ n ) t = nt = t t0 . t 0 (
.
[0...t]
t + : St+
[t...t + ]. t
Wt + Wt+ 3/2 = St + + . 2 2 3 Wt+ t + = t + , = Wt + W
(. . 68). :
St+ = St + Wt + S ,
(5.5)
S
Wt
W
St .
(
H40 ) (5.3).
127
Wt .
-
, ,
St
,
, .
Wt
St
, :
Wt2 = 2 t = t, S
t3 t3 St2 = 2 = . 3 3 St
(5.5)
t
t + .
St
Wt ,
:
,
t3 t2 St St+ = St2 + St Wt = + , 3 2 Wt St = t2 /2.
St
St+ ,
St St+ 1 + 3T /2 3 (St , St+ ) = = 1 T 2 + ..., (1 + T )3/2 8 2
St2 St+ Wt Wt+ 1 1 T + ... = (Wt , Wt+ ) = 1+T 2 W2 Wt t+ Wt St
.
S:
T = /t.
W:
:1.0
T
-
T (St,St+ ) (Wt,Wt+ )
1
Wt St
0.7
.
Wt
St . , St
, . , . , .
128
5.
f (t) N (0, 1) : t f (s)Ws ds = (t) ,0
(t) =0 s
2
t [ t f ( ) d
]2 ds.(5.6)
Wt = t,
:
=
1 = (t) t
t [ t0 s
] f ( ) d ds.
,
f (t) = tn : = 6 + 7n + 2n2 . 2(2 + n) t.
t0
n+3/2 2t , sn Ws ds = 6 + 7n + 2n2
= +
1 2 1 . 2 2
1 :
.
,
, :
t It =t0 ,
f (W ) d
W,
:
ft (Wt ) =
f (t, Wt ),
, , (5.6).
f
. , , . . ,
Wt
-
. .
129
. , , :
gt (Wt )
ft (Wt ). -
gt (Wt )
t f (W ) d0
[ ] = gt (1 + ... + n ) f1 (1 ) + f2 (1 + 2 ) + ... t , t k k -
g ,
.
.
,
gt (1 +...+k +k+1 +...+n )fk (1 +...+k ) = gt (a k+a n k)fk (a
k).
nk
:
a
