1 / 74 ( ) 2019 3 8 URL: http://www.math.kyoto-u.ac.jp/ ichiro/
2 / 74 Contents 1 Pearson 2 3 Doob h- 4 (I) 5 (II) 6 (III-1) - 7 (III-2-a) 8 (III-2-b) - 9 (III-3)
Pearson
3 / 74 Pearson Definition 1 ρ Pearson : { g(x) } ρ(x) = exp f(x) dx. (1) g(x) 1 f(x) 2 Pearson 12 6
4 / 74 Pearson I 6 1 e βx2 /2 R 2 x α e βx (0, ) 3 x α (1 x) β (0, 1) 4 (1 + x 2 ) α exp{β arctan x} R 5 x α e β/x (0, ) 6 x α (1 + x) β (0, )
Pearson II α- β- γ- F- & Pareto t- 5 / 74
: α- a = 1 0F 1 β- a = x a = x 2 1F 1 γ- a = x(1 x) a = x(1 + x) a = 1 + x 2 2F 1 α- β- - γ- 6 / 74
7 / 74 1 1 : Definition 2 A = a d2 dx 2 + b d dx. (2) (2) a 2 b 1 Pearson-Kolmogorov Kolmogorov a b 1 Feller dm = ρdx Pearson Kolmogorov
A Kolmogorov a d2 dx 2 + b d dx Feller d d dm ds d dm = d ds d ds : L2 (dm) L 2 (ds) Stein ( a d dx + b) d dx a d dx + b = d dx d dx : L2 (ρ) L 2 (aρ) 8 / 74
9 / 74 Feller Stein Feller Stein Feller s pair Stein s pair d d dm ds (a d dx + b) d dx d d ds dm d dx (a d dx + b) pair 0
10 / 74 Stein Stein d dx (a d d2 + b) = a dx dx + 2 (a + b) d dx + b. a 2 b 1 Kolmogorov Stein
Doob h-
11 / 74 V = d dx d ds d = aρ d ds dx V = d dx : d ds : L2 (ρ) L 2 (1/(aρ)) d dx : L2 (ρ) L 2 (aρ).
12 / 74 U f = aρ f (3) U : L 2 (aρ) L 2 (1/(aρ)) Proposition 3 d ds = UV ( d ds ) U = V
13 / 74 L 2 (aρ) D U D L 2 (ρ) L 2 (1/(aρ))( ) L 2 (ρ) d d d = ds ds dm Figure 1: V d ds
14 / 74 Doob h- V d dm VV d d ds dm Theorem 4 VV d d ds dm VV = U 1 d ds d dm U, d ds d dm = UVV U 1. (4)
15 / 74 Kolmogorov diffusions a 2 b 1 A = a d2 dx 2 + b d dx Kolmogorov diffusions a 0 1 2 3 (I) a = 1 on (, ) (II) a = x (0, ) (III-1) a = x 2 on (0, ) - (III-2-a) a = x(1 x) on (0, 1) (III-2-b) a = x(x + 1) on (0, ) - (III-3) a = x 2 + 1 on (, )
(I)
16 / 74 a = 1 b = β R b = β R β A = d2 dx + β d (5) 2 dx ρ(x) = exp{ βdx} = e βx (6) V = d dx : L2 (ρ) L 2 (ρ) ( d dx ) = d β. (7) dx
17 / 74 I : L 2 (ρ) L 2 (dx) : J f(x) = e βx/2 f(x). (8) L 2 (ρ) J L 2 (dx) A L 2 (ρ) J d 2 dx 2 β2 4 L 2 (dx) (9)
18 / 74 σ(a) = (, β2 4 ] (10) d2 A dx 2 e (iλ β/2)x λ β 2 /4 d dx e(iλ β/2)x = (iλ β/2)e (iλ β/2)x ( d dx + β)e(iλ β/2)x = (iλ + β/2)e (iλ β/2)x V, V Stein
19 / 74 5 4 3 2 1 1 1 2 3 4 β 2 3 4 5 Figure 2: β
20 / 74 Feller Feller β y 4 3 2 1 1 1 2 3 4 fi 2 3 4 Figure 3: Feller s pair
a = 1 b = βx, β R β = 0 A = d2 dx 2 + βx d dx (11) ρ(x) = exp{ βxdx} = e βx2 /2 V = d dx : L2 (ρ) L 2 (ρ) ( d dx ) = d βx (12) dx Au = V Vu = u + βxu, Âu = VV = u + βxu + βu 21 / 74
22 / 74 Hermite Hermite H n (ξ) (n Z +, ξ R) H n (ξ) = ( 1)n e ξ2 /2 dn /2. (13) n! dξ ne ξ2
23 / 74 Ornstein-Uhlenbeck β = 1 Ornstein-Uhlenbeck ( d2 dx 2 x d dx )H n = nh n H n
24 / 74 Ornstein-Uhlenbeck β = 1 Ornstein-Uhlenbeck d ds : L2 (e βx2 ) L 2 (e βx2 ) (14) Feller
25 / 74 Ornstein-Uhlenbeck ( d dx + x)(e x2 /2 H n+1 = xe x2 /2 H n+1 + e x2 /2 H n+1 + xe x2 /2 H n+1 = e x2 /2 H n+1 = e x2 /2 H n. e x2 /2 H n (n + 1) Doob h- J : L 2 (e x2 /2 ) L 2 (e x2 /2 ) J f = ε x2 /2 f (15)
26 / 74 4 3 2 1 1 2 3 4 β Figure 4: β
27 / 74 4 3 2 1 1 2 3 4 β Figure 5: Feller s pair
28 / 74 4 3 2 1 1 2 3 4 β Figure 6: Stein s pair
29 / 74 4 3 2 1 1 2 3 4 β Figure 7: Doob s h-transformation
(II)
30 / 74 a = x, b = 1 + α, α R Bessel I = (0, ), a = x, p = x α Au = xu + (1 + α)u, Âu = xu + (2 + α)u. A and Â
31 / 74 0F 1 (c; x) = n=0 1 (c) n n! xn. (16) B(c; x) = 0 F 1 (c; x). (17)
(a) α > 1 ξ (ξ 0) B(1 + α; ξx) d [B(1 + α; ξx)] = λ dx B(2 + α; ξx). 1 + α entrance family (b) α < 0 ξ (ξ 0) x α B(1 α; ξx) d dx [x α B(1 α; ξx)] = αx α 1 B( α; ξx) exit family 32 / 74
33 / 74 Remark 1 B Bessel B(α + 1, ξx) = Γ(α + 1)(ξx) α/2 J α ( 4ξx) (18)
34 / 74 Bessel entrance family 1 ff 1 ff ff+1 ff+2 ff+3 ff+4 the spectrum of A
35 / 74 Bessel exit family 0 ff α α+1 α+2 α+3 α+4 the spectrum of A
36 / 74 a = x, b = 1 + α x, α R Kummer I = (0, ), a = x, p = x α e x. Au = xu + (1 + α x)u, Âu = xu + (2 + α x)u u. 0
37 / 74 1F 1 (a; c; x) = n=0 (a) n (c) n n! xn. (19) M(a, c; x) = 1 F 1 (a; c; x) (20) M Laguerre L (α) n (x) = (α + 1) n M( n, α + 1; x) (21) n!
38 / 74 Theorem 5 (a) α > 1 entrance family n (n = 0, 1,... ) M( n, α + 1; x) [M( n, α + 1; x)] = n M( n + 1, α + 2; x). α + 1 (b) α < 0 exit family n + α (n = 0, 1,... ) x α M( n, 1 α; x) [x α M( n, 1 α; x)] = αx α 1 M( n, α; x).
39 / 74 Kummer (Exit ) 1 0 α 1 α α+1 α+2 α+3 α+4 1 2 3 4 5..
40 / 74 Kummer (Entrance ) 5 4 3 2 1 0 α α+1 α+2 α 0 1 1 2 2 3 3 4 4 5 5
41 / 74 Kummer 5 4 3 2 1 1 2 3 4 0 α 1 2 3 4 5
42 / 74 a = x, b = 1 + α + x, α R Kummer Feller L α = x d2 d + (α + 1 + x) dx2 dx α > 1 (entrance ) (22) L α n α 1 α < 0 (exit ) L α n 1 L (α) n (x)e x. (23) L ( α) n (x)x α e x. (24)
43 / 74 1 0 1 2 3 4 5 1 2 3 4 α 0 1 2 3 4 5 α α+1 α+2 α+3 α+4 α Figure 8: Kummer Entrance
44 / 74 4 3 2 1 0 α α+1 α+2 α+3 α+4 α 0 1 1 2 2 3 3 4 4. 5. 5 Figure 9: Kummer Exit
45 / 74 5 4 3 2 1 1 2 3 4 0 α 1 2 3 4 5 Figure 10: Kummer
(III-1) -
46 / 74 (III-1) a = x 2, I = (0, ). A = x 2 d d + (αx β) dx2 dx. (25) β = 0 Black-Scholes -
47 / 74 β = 0 Black-Scholes A = x 2 d dx + αx d 2 dx. (26) σ(a) = (, 1 4 (α 1)2 ] (27)
48 / 74 Black-Scholes G = x 2 d2 dx 2 + αx d dx 16 14 12 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 55 60 2 4 6 8 10 12 14 16 α
49 / 74 β = 1 A = x 2 d d + (αx + 1) dx2 dx. (28) n = 0, 1, 2,... λ n (α) λ n (α) = n(n 1 + α). (29) σ ess (A) = (, 1 4 (α 1)2 ] σ p (A) = {λ n (α); 0 n < 1 α }. 2
50 / 74 L (1 2n α) n P (α) n (x) = xn L (1 2n α) Laguerre n (1 ). (30) x
51 / 74 β = 1 G = x 2 d2 d +(αx +1) dx2 dx 16 14 12 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 55 60 2 4 6 8 10 12 14 16 α
52 / 74 β = 1 A = x 2 d d + (αx 1) dx2 dx. (31) n = 1, 2,... ξ n (α) by ξ n (α) = n(n + 1 α) (32) σ ess (A) = (, 1 4 (α 1)2 ] σ p (A) = {ξ n (α); 1 n < α 1 }. 2
53 / 74 x α+2 e 1/x P (4 α) n 1 L (α 2n 1) n 1 (x) = xn α+1 e 1/x L (α 2n 1) n 1 Laguerre (1 ). x
54 / 74 β = 1 G = x 2 d2 d +(αx 1) dx2 dx 16 14 12 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 55 60 2 4 6 8 10 12 14 16 18 α
55 / 74 β = 0 G = x 2 d2 dx 2 + αx d dx 16 14 12 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 55 60 2 4 6 8 10 12 14 16 α
56 / 74 β = 1 G = x 2 d2 d +(αx +1) dx2 dx 16 14 12 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 50 55 60 2 4 6 8 10 12 14 16 α
(III-2-a)
57 / 74 (III-2-a) a = x(1 x), b = α + 1)(1 x) (β + 1)x I = (0, 1), a = x(1 x), ρ = x α (1 x) β Au = x(1 x)u + ((α + 1)(1 x) (β + 1)x)u, Âu = x(1 x)u + ((α + 2)(1 x) (β + 2)x)u (α + β + 2)u.
58 / 74 Gauss (a) n (b) n 2F 1 (a, b; c; x) = (c) n n! xn. (33) K(x) = K(α, β, γ; x) = 2 F 1 ( γ, α + β + γ + 1; α + 1; x) (34) n=0 Remark 2 K Jacobi P (α,β) n (x) = Γ(α + n + 1) n!γ(α + 1) 1 x K(α, β, n; ). (35) 2
59 / 74 (a) α > 1, β > 1 [entrance,entrance] family n(n + α + β + 1) (n = 0, 1,... ) K(α, β, n) γ(α + β + γ + 1) K (α, β, n) = K(α + 1, β + 1, n 1). α + 1 (b) α < 0, β > 1 [entrance,exit] family (n α)(n + β + 1) (n = 0, 1,... ) x α K( α, β, n) [x α K( α, β, n)] = αx α 1 K( α 1, β + 1, n).
60 / 74 (c) α < 0, β < 0 [exit,exit] family (n + 1)(n α β) (n = 0, 1,... ) x α (1 x) β K( α, β, γ) [x α (1 x) β K( α, β, n)] = αx α 1 (1 x) β 1 K( α 1, β 1, n + 1).
61 / 74 β = α + 3 α 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 α 10 20 30 40 50 60
62 / 74 Stein 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 α 10 20 30 40 50 60
63 / 74 x d2 d + (α + 1) dx2 dx x d2 d + (α + 1 x) dx2 dx x(1 x) d2 d + ((α + 1)(1 x) (β + 1)x) dx2 dx Bessel 0F 1 Laguerre 1 F 1 Jacobi 2F 1
(III-2-b) -
64 / 74 (III-2-b) a = x(1 + x), I = [0, ) A = x(1 + x) d2 d + ((α + 1)(1 + x) + (β + 1))x) dx2 dx. (36) Fisher Pareto -
65 / 74 α > 1 n = 0, 1, 2,... λ n (α, β) λ n (α, β) = ( n β + β )( β β n + α + 1 ) (n β)(n + α + 1), = 2 2 n(n + α + β + 1), Theorem 6 A σ ess (A) = ( (α + β + 1)2 ], 4 σ p (A) = {λ n (α, β); 0 n < [ α + β 1] } 2
66 / 74 G = x(1 + x) d2 d +((α + 1)(1 + x)+(β +1)x) dx2 dx, β = α 11 3 2 1 0 5 10 15 20 25 30 35 40 45 50 55 60 1 2 3 4 5 6 7 8 9 10 11 12 α
67 / 74 α < 0 n = 1, 2,... ξ n (α, β) ξ n (α, β) = ( n β β )( β + β n α 2 2 n(n α β 1), β 0, = (n + β)(n α 1), β 0. + 1 ) Theorem 7 A σ ess (A) = ( (α + β + 1)2 ], 4 σ p (A) = {ξ n (α, β); 1 n < [ α + β + 1] }. 2
68 / 74 3 2 1 5 10 15 20 25 30 35 40 45 50 55 60 1 2 3 4 5 6 7 8 9 10 11 12 G = x(1 + x) d2 d +((α + 1)(1 + x)+(β +1)x) dx2 dx β = α 11 α
(III-3)
69 / 74 (III-3) a = 1 + x 2, I = (, ) A = (1 + x 2 ) d2 d + (2(α + 1)x + 2β) dx2 dx. (37) t-
70 / 74 Theorem 8 A σ ess (A) = (, (α + 1 2 )2]. (38) α < 1 2 λ n (α) = n(n + 2α + 1), 0 n < α 1 2 (39) α > 1 2 ξ n (α) = n(n 2α 1), 1 n < α + 1 2. (40) 1 2 α 1 2
71 / 74 x K(α + iβ, α iβ, n, 1 ix ). 2
72 / 74 β G =(1+x 2 ) d2 d +(2(α +1)x +2β)) dx2 dx 8 7 6 5 4 3 2 1 0 5 10 15 20 25 30 35 40 45 50 55 60 1 2 3 4 5 6 7 α
73 / 74 G =(1+x 2 ) d2 d +(2(α +1)x +2β)) dx2 dx 8 7 6 5 4 3 2 1 0 5 10 15 20 25 30 35 40 45 50 55 60 1 2 3 4 5 6 7 α
74 / 74