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1 1 / 74 ( ) URL: ichiro/

2 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)

3 Pearson

4 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

5 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, )

6 Pearson II α- β- γ- F- & Pareto t- 5 / 74

7 : 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

8

9 7 / : 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

10 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

11 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

12 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

13 Doob h-

14 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).

15 12 / 74 U f = aρ f (3) U : L 2 (aρ) L 2 (1/(aρ)) Proposition 3 d ds = UV ( d ds ) U = V

16 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

17 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)

18 15 / 74 Kolmogorov diffusions a 2 b 1 A = a d2 dx 2 + b d dx Kolmogorov diffusions a (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 on (, )

19 (I)

20 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

21 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)

22 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

23 19 / β Figure 2: β

24 20 / 74 Feller Feller β y fi Figure 3: Feller s pair

25 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

26 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

27 23 / 74 Ornstein-Uhlenbeck β = 1 Ornstein-Uhlenbeck ( d2 dx 2 x d dx )H n = nh n H n

28 24 / 74 Ornstein-Uhlenbeck β = 1 Ornstein-Uhlenbeck d ds : L2 (e βx2 ) L 2 (e βx2 ) (14) Feller

29 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)

30 26 / β Figure 4: β

31 27 / β Figure 5: Feller s pair

32 28 / β Figure 6: Stein s pair

33 29 / β Figure 7: Doob s h-transformation

34 (II)

35 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

36 31 / 74 0F 1 (c; x) = n=0 1 (c) n n! xn. (16) B(c; x) = 0 F 1 (c; x). (17)

37 (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

38 33 / 74 Remark 1 B Bessel B(α + 1, ξx) = Γ(α + 1)(ξx) α/2 J α ( 4ξx) (18)

39 34 / 74 Bessel entrance family 1 ff 1 ff ff+1 ff+2 ff+3 ff+4 the spectrum of A

40 35 / 74 Bessel exit family 0 ff α α+1 α+2 α+3 α+4 the spectrum of A

41 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

42 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!

43 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).

44 39 / 74 Kummer (Exit ) 1 0 α 1 α α+1 α+2 α+3 α

45 40 / 74 Kummer (Entrance ) α α+1 α+2 α

46 41 / 74 Kummer α

47 42 / 74 a = x, b = 1 + α + x, α R Kummer Feller L α = x d2 d + (α 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)

48 43 / α α α+1 α+2 α+3 α+4 α Figure 8: Kummer Entrance

49 44 / α α+1 α+2 α+3 α+4 α Figure 9: Kummer Exit

50 45 / α Figure 10: Kummer

51 (III-1) -

52 46 / 74 (III-1) a = x 2, I = (0, ). A = x 2 d d + (αx β) dx2 dx. (25) β = 0 Black-Scholes -

53 47 / 74 β = 0 Black-Scholes A = x 2 d dx + αx d 2 dx. (26) σ(a) = (, 1 4 (α 1)2 ] (27)

54 48 / 74 Black-Scholes G = x 2 d2 dx 2 + αx d dx α

55 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

56 50 / 74 L (1 2n α) n P (α) n (x) = xn L (1 2n α) Laguerre n (1 ). (30) x

57 51 / 74 β = 1 G = x 2 d2 d +(αx +1) dx2 dx α

58 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

59 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

60 54 / 74 β = 1 G = x 2 d2 d +(αx 1) dx2 dx α

61 55 / 74 β = 0 G = x 2 d2 dx 2 + αx d dx α

62 56 / 74 β = 1 G = x 2 d2 d +(αx +1) dx2 dx α

63 (III-2-a)

64 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.

65 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

66 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).

67 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).

68 61 / 74 β = α + 3 α α

69 62 / 74 Stein α

70 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

71 (III-2-b) -

72 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 -

73 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

74 66 / 74 G = x(1 + x) d2 d +((α + 1)(1 + x)+(β +1)x) dx2 dx, β = α α

75 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 ) Theorem 7 A σ ess (A) = ( (α + β + 1)2 ], 4 σ p (A) = {ξ n (α, β); 1 n < [ α + β + 1] }. 2

76 68 / G = x(1 + x) d2 d +((α + 1)(1 + x)+(β +1)x) dx2 dx β = α 11 α

77 (III-3)

78 69 / 74 (III-3) a = 1 + x 2, I = (, ) A = (1 + x 2 ) d2 d + (2(α + 1)x + 2β) dx2 dx. (37) t-

79 70 / 74 Theorem 8 A σ ess (A) = (, (α )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 (40) 1 2 α 1 2

80 71 / 74 x K(α + iβ, α iβ, n, 1 ix ). 2

81 72 / 74 β G =(1+x 2 ) d2 d +(2(α +1)x +2β)) dx2 dx α

82 73 / 74 G =(1+x 2 ) d2 d +(2(α +1)x +2β)) dx2 dx α

83 74 / 74

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

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