確率微分方程式の数値シミュレーション

(1)

2009SE322

"!$#%$&

1

')(+*-, .0/2123240506+798;:2<+=>[email protected]/2A+BDC@E$F GIHKJMLON+PRQ CS45$6+TUWV@X

1942

Y 8SZ[+\O$]^ _2`acbedgfih2CIjIk$lWm>nK=>o0p$q2rD#$sR? PIL@t0u9v w>x N

[3]

XRy0=z0{0|z07S80}$~D$~0=122M=9 WMCIWF+TKUVWWKWF G+ ?$C LSt hKV9XR =;F7>d

,

=FI?Ih tMKu ? C L t hRV

.

=9KRTD./W13W456+h tOW@t h Q$¡ ¢Q X

2

£¥¤¥¦¥§¨ª©¬«g¢®-¯¬°±¬³²§ +M+´+Fµ7$8

1827

Y 80¶·+=0¸R¹Kp+qº+M+

(Robert Brown)

G0» 1¼T2½=I¾¿ºÀÁÂº?OÃÄ SN XÅÆ7O8IÇÈ´¹=SFº h Q Ä4 U LDN+P+Q T UWV G bÉWÊn7I8Ëy$= PRQ ? uÌ$t h NÍ+G U VlÎ8

1860

YÏ ?7I82½M=93$=9ÐÑM? P2LStKÒ ÔÓÕ w@x V h QÔuÌ 4 GIÖ×Ø CÕÙÚ8

1905

Y 8+z2{MIÛWÜ Ý {

(Albert Einstein)

? POLt 80h J$Þ V$; F= A GDßà w9x N X wá ?O8Rz${DÛWÜ Ý {I= A+78

1908

Y ?â2´ã=>¹ p$qäRW

(Jean Perrin)

? PIL@t 8>WRB2?åæ w>x N X>çWè7S8 RuSéWê

[5]

ë á =ìíRTUWVîb

[2]

ïð lñX

0

0.5 -6

-4

-2

0

2

4

6 t = 0.5

t = 1

t = 2

t = 10

x

ò

1

ó o

1

2 √

πt

exp

−

x

2 4t

ä+I=9K ë á 821=

t

ôõ

(t > 0)

=

(

UKV 4+ö? óS÷ V

)

ø$ù 8 ÷ C J+ú 8

t

ôõ9û = ù+ü = = ùMü$ë á =9ý7O8Oþ$ÿ G

0

803 G

t

=o=W3 b ò

1

ð l+? Q G Äm w@x N XI=+ ? @8WRWïWF=Û OÜWÛIDÀSçK= P+Q ? Q X K= N 8 õ

1

T uÌ VX

∆t

À $3$! w h o $ 8

t

n

= n∆t (n = 0, 1, ... )

!#" X wá ?D8

ξ

n

(n = 0, 1, ... )

À%$&'$3

(

þ0ÿ

0,

3(

1

=)$3

)

? Q+* o 2 8

W

0 = 0, W

n+1

= W

n

+

√

∆t ξ

n

(1)

? P ÙÚ8

(t = 0

T

)

,-Rë á m@Ã ÷ V

“

ï$

”

= ô.

t = t

n

? !/ V ù+ü À% VXS= ´80$3 ='1B23M? P ÙÚ8

√

∆t ξ

n

7I8$þ2ÿ

0,

3'

∆t

=4 $3 ?4 RhI8 wRá ?S8

W

n

79þDÿ

0

8@3(

n∆t = t

n

= 3 ? Q ë á 8è65=WFM=23ËÀ 7(89Û:OÜ%ÛI@TId ÷ VK G TM0V9X

3

;=<?>#@ ACBØ±¥¤Ø±D E?8$2$ $FR=$ÛF@ÜGH2Û:@SÀ Q N ? 7I8($'&3 ? Q+* oÀ´ÇI ÷ V)J'K G UWVSX¢ x ?7I8$h " ë =94L G UºÙÚ8%$'&B$CS4LM=0MWK?I8 NOP ãªfRQWS

(Box-Muller)

L TU x V>4'L G U V9XT7I8I=94LM?DWh t0V jV9X NWOP ã)fXQLM= , 7 = PRQ C =0TUV9X

X, Y

À'Y õ

(0, 1)

èK=6M'Z3 ? QC[\ CI./ ø o ÷ V >8

U =

√

−2 log X cos(2πY )

V =

√

−2 log X sin(2πY )

(2)

7@8@y x]@x 8^$6&D23 ? Q_[6\ [email protected]/ ø o C$V>X `E082o

u, v

?a >t 8

R

2 = [0, 1] × [0, 1]

b =c d

S

1 (u)

8

S

2 (v)

À

S

1 (u) = {(x, y) ∈ R

2 : (−2 log x)

1/2

cos(2πy) ≤ u}

S

2 (v) = {(x, y) ∈ R

2 : (−2 log x)

1/2

sin(2πy) ≤ v}

= PRQ ?'e ÷ V 8

area[S

1 (u)] = Φ(u), area[S

2 (v)] = Φ(v)

(3)

area

h

S

1 (u)

\

S

2 (v)

i

= area[S

1 (u)] area[S

2 (v)]

(4)

G I¥Ù \ 0X N'f+ 8

area[ · ]

74c(dM=)g'h8

Φ

7)$(&' 3 =93 ó o

Φ(u) =

√

1 2π

Z

u

−∞

e

−

ξ

2 /2

dξ

À%i ÷ X

4

jlk¥¤± Anmpoq@lr¢±ts 1345$6=9Æuv:wx

dx

dt

= f (t, x) (t ≥ t

0 ), x(t

0 ) = x

0 (5)

À uÌ V9X(R= N' 8(y'z ø o

x

7ov ó o8 ÷ C Jú 8D1345$676 [ 45$6 ÷ V9X 13456Ë=SÆuv w0x={ 1BC L7O8|K{ 64LKTWUV@X

x(t

n

)

=)}'~'v

x

n

÷ V ´8S|W{WS LM7I8

x

n+1

− x

n

∆t

= f (t

n

, x

n

)

(6)

i w@x V9X

(2)

W

n

(1)

x V * o b ò

2

ð l ÷ V >XËI= 13

W

n+1

− W

n

∆t

=

ξ

n

√

∆t

(7)

7O8?S{HD{' bo?0¬Ù =Ch O{'Dl=9o

p`| uÌ VK G TM0Vîb ò

3

l X

-4

-2

0

2

4

0

5

10

15

20 t

ò

2

FM=

(

Ä'1

)

-150

-100

-50

0

50

100

150

0

5

10

15

20 t

ò

3 “

W9{qMfO{(

”

wá ?O8

g(t, x)

À

t, x

= ó o $@t 8|K{R(0L

(6)

?GO{(= HKJLDN

x

n+1

− x

n

∆t

= f (t

n

, x

n

) + g(t

n

, x

n

)

W

n+1

− W

n

∆t

(8)

=68 ÷ C Jú 8

x

n+1

= x

n

+ f (t

n

, x

n

) ∆t + g(t

n

, x

n

)

√

∆t ξ

n

(9)

À uWÌ VSX|K{R(0L

(6)

? P V)}~v b À x( T C$h f+ =Sl7

∆t → 0

= >8$1345$6

(5)

= ? } t" XZ?98

(9)

∆t → 0

= ´8 á ë =4$526 = ?4} #" G u w@x V9Xµy$= PRQ CSÁ -Rë á u v w@x N 45$6 G 80.$/1345$6+TUºÙÚ8

dX(t) = f t, X(t)dt + g t, X(t)dW (t)

(10)

= PKQ C5RTi w>x VX$@TO8

W (t)

7R0%)5 %TU x VFM=op`| b .$/5lTU V9X|{KLM=4~

(9)

À82.$/1345$6=4}~ 6 ¡¡È>t 8S|{K f£¢¤IãF¥(¦ %T§ b¨ ÌU 8

[4]

l X

5

dX(t) = [X(1 − X

2 ) + a cos(ωt)] + σdW (t)

(11)

ø

X(t)

X = −1

uM=9}®W8

X = 1

G õ ¯ uM=9}®M?a° ÷ V9X ± ² Ý

a

8

ω

À

a = 0.11, ω = 2π/6000

?(³0 8

σ

À

σ = 0.1, 0.2, 0.3, 0.4

= PRQ ? ø´ w)µ N =Z$ G·¶ = PRQ ? ø J V ë À8

t

=4¸:¹RÀ

0 ≤ t ≤ 30000

? SL9t 8Oov$À L N Xº»µÀ ò ¼ ?½ ÷ XO='= ò À@ÄËV

σ = 0.3

= O?uMB$C®¾ ø´ G Ä áx N X ôWõ =Y õ =93'¿ o * oÀ ø Ì$t ZCº'» G Ä áx N X

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5000

10000

15000

20000

25000

30000

σ=0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5000

10000

15000

20000

25000

30000

σ=0.2

-1.5

-1

-0.5

0

0.5

1

1.5

0 5000

10000

15000

20000

25000

30000

σ=0.3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 5000

10000

15000

20000

25000

30000

σ=0.4

ò

4

ov7(8º'»

6

ÀtÁÃÂ, =9KËÀ>: @t ./W13456 GÄ¶ = PQ C = ë À !!P y)ÅÆ ÷ VK G TM N XGÇM?9.$/ÈFÉ G! DV h Q+Ê >J ë ÙOÊ;3M=ËÌ$ GËDL ë ÙÎÍIRT N =0T + =+?7 ÷WÏ "_ÐÑÈ>t hKV9X ÒlÓÔÖÕ

[1] R. Benzi, G. Parisei, A. Sutera, A. Vulpiani, A

theory of stochastic resonance in climate change,

SIAM J. Appl. Math. 43 (1983), pp. 565–578.

[2]

×Ø=Ù ÛÚ f x G , À6Ü N ëÝ 8(Þ'ßàW8áâ8

1976.

[3]

r#Ws8

Markoff

W5µÀ) V1W345$6K8)ãM·6ä èSopåæ'çè

1077 (1942), pp. 1352–1400.

[4]

Z(éêë28S!2#%0&W84ì$#'í î ïÚ 123$4$526M? P V 7(8ð$pñWò Ý 8È \ m)óW8áâ8

2004.

[5]

ô(Øõ6ö -÷Ú 22¡$F Ý 8 ¹ p

One Point 27

8 È \ m)óW8áâ8

確率微分方程式の数値シミュレーション

2009SE322

1

1942

[3]

,

.

2

1827

(Robert Brown)

1860

1905

(Albert Einstein)

1908

(Jean Perrin)

[5]

[2]

0

0.5

-6

-4

-2

0

2

4

6

t = 0.5

t = 1

t = 2

t = 10

x

1

1

2

√

πt

exp



−

x

2

4t



t

(t > 0)

(

)

t

0

t

1

1

∆t

t

n

= n∆t (n = 0, 1, ... )

ξ

n

(n = 0, 1, ... )

(

0,

1

)

W

0

= 0, W

n+1

= W

n

+

√

∆t ξ

n

(1)

(t = 0

)

“

”

t = t

n