( ) (Appendix) *

Transcription

1 Frisch Slutzky 1930 = ( ) 4 1

2 ( ) (Appendix) * A 42 *1 ( )

3 B 45 3

4 (2 binomial distribution) 2 n n S n S n S n 0, 1, 2,..., n S n p ( ) n P (S n = k) = p k (1 p) n k, k = 1, 2,..., n. k ( ) n f(x) = p x (1 p) n x, x = 1, 2,..., n (1) x n p 2.2 ( Poisson distribution ) λ λ f(x) = { λ x e λ x!, x = 0, 1, 2,..., 0, e λ = x=0 λ x x!, f(x) = 1 x=0 n p 2 λ = np p n λ 2.3 (2 ) X n p 2 f(x) = p) n x, x = 0, 1, 2,..., n ( ) n p x (1 x n = 1 1 (X = 1 ) p (X = 0 ) 1 p EX = 0 f(x = 0) + 1 f(x = 1) = p, 4

5 1 n > 1 X {0, 1, 2,..., n} n ( ) n EX = j p j (1 p) n j. j j=0 EX = np, (2) *2. 2 X 1, X 2,, X n ( 1 ) p S n = X 1 + X X n S n 2 EX i = p, (i = 1, 2,..., n) ES n = E(X 1 + X X n ) = 2.4 ( ) n EX i = np. i=1 λ f(x) = λ x e λ /x! EX = j=1 j λj j! e λ = 2.5 (2 ) j=1 λ j (j 1)! e λ = λe λ j=0 λ j j! = λe λ e λ = λ, ES n = np VarS n = Var(X 1 + X X n ) = VarX 1 + VarX VarX n = nvarx 1. X 1 = 0, 1 X 2 1 = X 1 EX 2 1 = EX 1 = p VarX 1 = EX 2 1 (EX 1 ) 2 = p p 2 = p(1 p) 2 VarS n = np(1 p) X λ EX = λ, VarX = λ 2.6 ( normal distribution ) µ σ 2 f(x) = 1 (x µ)2 exp{ 2πσ 2σ 2 }, *2 Hoel, Port and Stone,An Introduction to Probability Theory 5

6 V V (lognormal distribution) X µ σ 2 V = e X V f(v) = 1 (ln v µ)2 exp{ 2πσv 2σ 2 }, v 0 EV = exp{µ + σ2 σ2 }, VarV = exp{2(µ )}[exp{σ2 } 1] 2.7 ( exponential distribution) f(x) = λe λx, x 0; f(x) = 0, x < 0 x λ (exponential distribution) F (x) = 1 e λx, x 0 EX = 1 λ, VarX = 1 λ ( gamma distribution) α > 0 λ > 0 Γ(x; α, λ) Γ(x; α, λ) = λα Γ(α) xα 1 e λx, x > 0, Γ(α) = 0 x α 1 e x dx α {Y 1, Y 2,.., Y α } λ X = Y 1 + Y Y α Γ(x; α, λ) 2.2 X t X t X(t) t T T t X t {X t, t T } X t *3 *3 A S t T {X t (ω) A} Ω (σ ) F T P F T X t (Ω, F T, P ). X t : Ω S (X t S, t T ) 6

7 {X t, t T } T (, ) T T = {0, 1, 2...} P Pr X S S {0, 1, 2,...} {0, ±1, ±2,,...} S = (, ) S k k t T X t t {X t, t T } (sample function) n X n X n, n = 1, 2,..., {1, 2, 3, 4, 5, 6} 2.9 ( Wiener process) (i). 0 < t 0 < t 1 < < t n {t k } X t1 X t0, X t2 X t1,..., X tn X tn 1 (ii). Pr(X t X s < x) = (iii). X x 2πB(t s) u 2 exp{ }du, (t > s, B > 0). 2B(t s) R. N. B ( ) EX t = 0, VarX t = Bt, 2.10 ( Poisson process) (i). 0 < t 0 < t 1 < < t n {t k } X t1 X t0, X t2 X t1,..., X tn X tn 1 λ(t s) {λ(t s)}k (ii). Pr(X t X s = k) = e, k = 0, 1, 2,..., (t > s, λ > 0) k! 7

8 (iii). X 0 0. EX t = λt, VarX t = λt, (3) (0, t] N(t) α N(t) (Markov processes) t 1 < t 2 < < t n < t x 1, x 2,, x n, x Pr{X t < x X t1 = x 1, X t2 = x 2,..., X tn = x n } = Pr{X t < x X tn = x n } {X t, t T } P (x, t n ; y, t n+1 ) = Pr{X(t n+1 ) = y X(t n ) = x} P (x, t n ; y, t n+1 ) t n+1 t n 2. (Processes with independent increments) t 1 < t 2 < < t n X t2 X t1, X t3 X t2,, X tn X tn 1 {X t } ( ) 3. (Martingales) {X t } E{ X t } < t 1 < t 2 < < t n < t a 1, a 2,, a n E(X t X t1 = a 1,, X tn = a n ) = a n 8

9 X t T = {0, 1, 2, } (discrete-time Markov chains) S = {0, 1, 2, } n x n n + 1 x n+1 P (n, x n ; n + 1, x n+1 ) Pr{X n+1 = x n+1 X n = x n } (transition probability) P (n, x n ; n + 1, x n+1 ) n P x,y P (n, x; n + 1, y) = P (0, x; 1, y) P x,y x y P x,y 0, x, y S, P x,y = 1, x S y=0 (4) (5) (4)(5) P x,y ( ) π 0 (x) = Pr(X 0 = x), x S, π 0 π 0 (x) 0, x S π 0 (x) = 1 x {X 0, X 1, X n } X 0, X 1 Pr(X 0 = x 0, X 1 = x 1 ) = Pr(X 0 = x 0 ) Pr(X 1 = x 1 X 0 = x 0 ) = π 0 (x 0 )P x0,x 1 Pr(X 0 = x 0, X 1 = x 1, X 2 = x 2 ) = Pr(X 0 = x 0, X 1 = x 1 ) Pr(X 2 = x 2 X 1 = x 1, X 0 = x 0 ) = π 0 (x 0 )P x0,x 1 Pr(X 2 = x 2 X 1 = x 1 ) = π 0 (x 0 )P x0,x 1 P x1,x 2 9

10 Pr(X 0 = x 0,, X n = x n ) = π 0 (x 0 )P x0,x 1 P xn 1,x n (6) 3.1 ( random walk) {ξ 1, ξ 2,...} f(ξ i ) X 0 ξ i X n = X 0 + ξ 1 + ξ ξ n {X n n 0} P x,y = Pr(X n+1 = y X n = x) = Pr(ξ n+1 = y x) = f(y x) ξ i { 1, 0, 1} f(1) = p, f( 1) = q, f(0) = r p + q + r = 1, p > 0, q > 0, r > 0 p, y = x + 1, q, y = x 1, P x,y = r, y = x, 0, 3.2 ( birth and death chains) p x, y = x + 1, q P x,y = x, y = x 1, r x, y = x, 0, p x + q x + r x = 1, p x > 0, q x > 0, r x > 0 x x + 1 x x 1 p x, q x, r x x p 0 = 1(q 0 = 0, r 0 = 0) r 0 = 1(p 0 = 0, q 0 = 0) ( branching chains) 0 n n + 1 X n n 10

11 ξ ξ f X n 0 P x,y = Pr(ξ 1 + ξ ξ x = y) P 1,y = f(y) 3.2 {X n, n 0} P x,y S n (n ) (n-step transition function) Pr(X n+1 = x n+1,, X n+m = x n+m X 0 = x 0, X 1 = x 1,, X n = x n ) = Pr(X 0 = x 0,, X n+m = x n+m ) Pr(X 0 = x 0,, X n = x n ) = π 0(x 0 )P x0,x 1 P xn+m 1,x n+m π 0 (x 0 )P x0,x 1 P xn 1,x n = P xn,x n+1 P xn+m 1,x n+m S A 0,, A n 1 Pr(X n+1 = x n+1,, X n+m = x n+m X 0 A 0,, X n 1 A n 1, X n = x n ) = P xn,x n+1 P xn+m 1,x n+m B 1,, B m S Pr(X n+1 B 1,, X n+m B m X 0 A 0,, X n 1 A n 1, X n = x) = P x,y1 P y1,y 2 P ym 1,y m. y 1 B 1 y m B m B 1 = = B m 1 = S B m = {y} x m y Pr(X n+m = y X 0 A 0,, X n 1 A n 1, X n = x) = P x,y1 P y1,y 2 P ym 1,y y 1 S y m 1 S x m y P x,y (m) P x,y (m) = Pr(X n+m = y X n = x) = Pr(X m = y X 0 = x) = P x,y1 P y1,y 2 P ym 1,y, m 2. (7) y 1 S y m 1 S P x,y (1) = P x,y P x,y (0) = { 1, x = y, 0, x y, 11

12 n P x,y (n + m) = Pr(X n+m = y X 0 = x) = Pr(X n = z X 0 = x) Pr(X n+m = y X n = z) z S = P x,z (n) Pr(X n+m = y X n = z) z S P x,y (n + m) = z S P x,z (n)p z,y (m). (8) - (Chapman-Kolmogorov equation) n Pr(X n = y) = Pr(X 0 = x, X n = y) = Pr(X 0 = x) Pr(X n = y X 0 = x) x S x S Pr(X n = y) = x S π 0 (x)p x,y (n) (9) ( inventory model) ξ n n n = 0, 1,..., ξ n Pr(ξ n = k) = p k, k = 0, 1, 2,..., p k >0 k=0 = 1 s S S n () X n X n X n = S, S 1,..., +1, 0, 1, 2,..., X n { Xn ξ X n+1 = n+1, s < X n < S, S ξ n+1, X n < s, ξ n X n { Pr(ξn+1 = x y), s < x < S, P x,y = Pr(X n+1 = y X n = x) = Pr(ξ n+1 = S y), x < s, 12

13 0, 1, 2 Pr(ξ = 0) = 0.5, Pr(ξ n = 1) = 0.4, Pr(ξ n = 2) = 0.1 s = 0, S = 2 2, 1 X n 2, 1, 0, 1 P 1,0 P 1,0 = Pr(X n+1 = 0 X n = 1) = Pr(ξ n+1 = 1) = 0.4 P 1,1 = Pr(X n+1 = 1 X n = 1) = Pr(ξ n+1 = 0) = 0.5 P = ( Markov chains in genetics) S. Wright 2 a, A 1 a A 2N a A 2N 2N a x a p x = x 2N A q x = 1 x 2N n a X n X n S = {0, 1, 2,..., 2N} 2 ( 2N Pr(X n+1 = y X n = x) = P x,y = y ) p y xq 2N y x, q; x, y = 0, 1,..., 2N (10) X n = 0 X n = 2N A X n = 0 13

14 a X n = 2N a A α A a β a p x = x (1 α) + (1 x 2N 2N )β A q x = x 2N α + (1 x )(1 β) 2N (10) αβ > 0 n X n 3.4 S = {0, 1,..., N} P = P i,j k P k (regular) π(x) 0, x S x π(x) = 1 lim P x,y(n) = π(y) > 0, y = 0, 1,..., N n y π(y) 3.1 P S = {0, 1,..., N} π = (π 0, π 1,..., π N ) π(x)p x,y = π(y), x S y S X n n π π *4 3.6 (11) *4 Taylor and Karlin(1998),pp

15 S = {0, 1, 2} P = π 0.4π(0) π(1) π(2) = π(0) 0.50π(0) π(1) π(2) = π(1) 0.10π(0) π(1) π(2) = π(2) π(0) + π(1) + π(2) = 1 π(0) = , π(1) = , π(2) = P 4 = (13) P 8 = (14) π π(x) {X n } x ( ) m x m 1 1 m k=0 δ Xk,x δ x,y { 1, y = x, δ x,y = 0, y x x E[ 1 m m 1 k=0 δ Xk,x X 0 = i] = 1 m m 1 k=0 lim n P i,x (n) = π(x) lim n m 1 1 m k=0 P i,x (k) = π(x) E[δ Xk,x X 0 = i] = 1 m 15 m 1 k=0 Pr(X k = x X 0 = i) = 1 m m 1 k=0 P i,x (k) (12)

16 P x,y (n) > 0 n y x (accessible) x y (communicate) (equivalence relation) (irreducible) x P x,x (n) > 0 n x (period) d(x) n f x,x (n) f x,x (n) = Pr(X n = x, X k x, k = 1, 2,..., n 1 X 0 = x) x n x f x,x (1) = P x,x n x k x n k x P x,x (n) = n f x,x (k)p x,x (n k), n 1 k=0 fx, x(0) = 0 x x f x,x f x,x (n) = lim n=0 N n=0 N f x,x (n) f x,x = 1 x (recurrent) (transient) 3.2 P x,x (n) = n=1 x P x,x (n) < n=1 x 3.3 lim P x,x(n) = n 1 n=0 nf x,x(n) 16

17 y lim P y,x(n) = lim P x,x(n) n n Karlin and Taylor(1975) Taylor and Karlin(1998) 3.7 (1 ) {0, ±1, ±2, } P x,x+1 = p, P x,x 1 = q, 0 < p < 1, p + q = P 0,0 (2n + 1) = 0, n = 0, 1,..., ( ) 2n P 0,0 (2n) = p n q n = (2n)! n n!n! pn q n Stirling n! n n+1/2 e n 2π P 0,0 (2n) (pq)n 2 2n πn = (4pq)n πn pq = p(1 p) < 1/4 p = q = 1/2 p = 1/2 P 0,0 (2n) = n=0 p = q = 1/ S t T = [0, ) X 0 x 0 τ 1 > 0 17

18 X(τ 1 ) = x 1 ( x 0 ) τ 2 (> τ 1 ) X(τ 2 ) = x 2 ( x 1 ) τ 1, τ 2, (jump times) S 1 = τ 1 0, S 2 = τ 2 τ 1, S 3 = τ 3 τ 2, (holding times) X(t) X(t) x 0, 0 < t < τ 1, x 1, τ 1 < t < τ 2, X(t) = x 2, τ 2 < t < τ 3,. X(t) (right-continuous) (jump process) x n ( n ) τ n+1 = x n ( (absorbing state) (non-absorbing state) lim τ n < n X(t) (explosion) X(t) lim τ n = n S n x t Pr(τ < t) F x (t) t < 0 F x (t) = 0 x y (transition probability) p xy p xy = 1; p xx = 0. y S x τ 1 x X(τ 1 ) = y τ 1 F x y p xy τ 1 X(τ 1 ) P x (τ 1 < t, X(τ 1 ) = y) Pr(τ 1 < t, X(τ 1 ) = y X(0) = x) = F x (t)p xy, P x ( ) x Pr( X(0) = x) y y y x y P x (τ 1 < s, X(τ 1 ) = y, τ 2 τ 1 <t, X(τ 2 ) = z) = F x (s)p xy F y (t)p yz 18

19 x p xy = δ xy δ xy δ xy = { 1, y = x, 0, y x x t y P x,y (t) P x,y (t) = P x (X(t) = y) = Pr(X(t) = y X(0) = x) P x,y (t) = 1 y S P x,y (0) = δ xy π 0 (x) 0, x S P (X(t) = y) = x S π 0 (x)p x,y (t), π 0 (x) = 1. x S {Y n = X(τ n )} (jump chains) Y n Y n Y n (holding times) P x (τ 2 > t + s τ 1 > s) = P x (τ 2 τ 1 > t), s, t 0, (15) 1 F x (t + s) 1 F x (s) = 1 F x (t) ( ) *5 τ : Ω [0, ] w x (0 < w x < ) Eτ = 1/w x f x (t) = { wx e wxt, t 0, 0, t < 0 *5 Norris(1996)

20 P x (τ t) = 1 F x (t) = e w xt. t w x e w xs ds = e w xt. x w x = 0 ξ 1, ξ 2,..., ξ n α 1, α 2,..., α n min(ξ 1,, ξ n ) α α n Pr(ξ k = min(ξ 1,..., ξ n )) = α k α α n, k = 1, 2,..., n Pr(min(ξ 1,..., ξ n ) > t) η k = min(ξ j : j k) Pr(ξ k = min(ξ 1,..., ξ n )) = Pr(ξ k < η k ) ( ) 0 < t 1 < t 2 < t n x 1, x 2,, x n S Pr(X(t n ) = x n X(t 1 ) = x 1,..., X(t n 1 ) = x n 1 ) = P xn 1,x n (t n t n 1 ) P x,y (t) Pr(X(t) = y X(0) = x) P x (X(t) = y) Pr(X(t 2 ) = x 2,..., X(t n ) = x n X(t 1 ) = x 1 ) = P x1,x 2 (t 2 t 1 ) P xn 1,x n (t n t n 1 ) P x (X(t) = z, X(t + s) = y) = P x,z (t)p z,y (s), t 0, s 0, P x,y (t + s) = P x (X(t) = z, X(t + s) = y), z S P x,y (t + s) = z S P x,z (t)p z,y (s), s 0, t 0, (16) (Chapman-Kolmogorov) 4.2 ( )P x,y (t) P x,y (t) = δ xy e w xt + t 0 w x e w xs { z x p xz P z,y (t s)}ds, t 0. (17) 20

21 x P x,y (t) = δ xy, t 0. x x t y P x,y (t) = P x (X(t) = y) = P x (τ > t, X(t) = y) + P x (τ<t, X(t) = y) 1 t P x (τ > t, X(t) = y) = (1 F x (t))δ xy = δ xy e w xt 2 t τ x t τ y s x z t s y P x (X(s) = z, X(t) = y) = w x e wxs p xz P z,y (t s) t z t y P x (X(τ) = z, τ<t, X(t) = y) = P x (τ<t, X(t) = y) = t 0 t 0 w x e w xs p xz P z,y (t s)ds. w x e w xs { z x p xz P z,y (t s)}ds (17) (17) s t r P x,y (t) = δ xy e w xt + w x e w xt (18) t t 0 w x e w xr { z x p xz P z,y (r)}dr, t 0 (18) dp x,y (t) = w x P x,y (t) + w x p xz P z,y (t), t 0 (19) z x t 0 dp x,y (0) = w x P x,y (0) + w x p xz P z,y (0) z x = w x δ xy + w x p xz δ zy z x = w x δ xy + w x p xy w xy w xy dp x,y(0), x, y S 21

22 { wx p w xy = xy, y x, w x, x = y. w xy = w x = w xx y x (20) w xy, x, y S (infinitesimal parameters) ( ) (19) dp x,y (t) = z w xz P z,y (t), t 0, (21) (backward equation) (16) s dp x,y (t + s) = z P x,z (t) dp z,y(s) ds dp x,y (t) dp x,y (t) = z = z P x,z (t) dp z,y(0) ds P x,z (t)w zy, t 0 (22) (forward equation) (master equation) t X(t) = y Pr(X(t) = y) = π 0 (z)p z,y (t) z S X(0) = x(0) Pr(X(t) = y) = z S δ x(0),z P z,y (t) Pr(X(t) = y) = P x(0),y (t) P 0,y x (t) = P x,y (t) P x (t) P 0,x (t) t w x,y (t) Pr(X(t + h) = y X(t) = x) Pr(X(t) = y X(t) = x) = w x,y (t) h + o(h) (23) w x,y (t) dp x,y(t) 22

23 y (23) P x,y (h) P x,y (0) = w x,y (t) h + o(h) y S y S y S w x,y (t) = 0 y S w x,x (t) = y x w x,y (t). (24) Pr(X(s) = y) = x S Pr(X(t) = x) Pr(X(s) = y X(t) = x) Pr(X(t + h) = y) = x S Pr(X(t) = x) Pr(X(t + h) = y X(t) = x) = x S Pr(X(t) = x)[pr(x(t) = y X(t) = x) + w x,y (t) h + o(h)] = Pr(X(t) = y) + x S Pr(X(t) = x)[w x,y (t) h + o(h)]. h 0 dp y (t) = x S P x (t)w x,y (t). x S P x (t)w x,y (t) = x y 2 (24) w y,y (t) = x y w y,x (t) P x (t)w x,y (t) + P y (t)w y,y (t) dp y (t) = x y P x (t)w x,y (t) x y P y (t)w y,x (t) (25) (25) 1 y 2 y w x,y (t) = w xy, t 0 23

24 4.3 S = {0, 1, 2,...} (X(t)) t 0 λ Y n = X(τ n ) = n λ τ 0 = 0, τ n = S 1 + S S n τ n n () (X(t)) t < λ < 3 (i). (X(t)) t 0 S 1, S 2, λ Y n = n, n = 1, 2, ; (ii). (X(t)) t 0 h 0 Pr(X(t + h) X(t) = 0) = 1 λh + o(h), Pr(X(t + h) X(t) = 1) = λh + o(h); (iii). (X(t)) t 0 t X(t) λt (X(t)) t 0 λ (i) (ii) (iii) *6 () P j (t) = Pr(X(t) = j) P j (t + h) = (1 λh)p j (t) + λp j 1 (t)h + o(h), j 1, P 0 (t + h) = (1 λh)p 0 (t) + o(h). h 0 dp j (t) = λp j (t) + λp j 1 (t), j 1, (26) dp 0 (t) = λp 0 (t), (27) P j (0) = δ 0,j λt (λt)j P j (t) = e j! (28) *6 Norris(1997) 24

25 *7 (X t ) t 0 (Y t ) t 0 λ µ (X t + Y t ) t 0 (λ + µ) 4.1 h Pr(B(t + h) B(t) = 1) βh + o(h) Pr(R(t + h) R(t) = 1) ρh + o(h) t n k k Pr(B(t) = k) = e βt (βt) k /k!, n k Pr(R(t) = n k) = e ρt (ρt) n k /(n k)! t n (β + ρ)t Pr(B(t) + R(t) = n) = e (β+ρ)t (βt + ρt) n /n! n k Pr(B(t) = k B(t) + R(t) = n) = Pr(B(t) = k, R(t) = n k)/ Pr(B(t) + R(t) = n) ( ) n β ρ Pr(B(t) = k B(t) + R(t) = n) = ( k β + ρ )k ( β + ρ )n k. n β/(β + ρ) w x = λ { 1, y = x + 1, p x,y = 0, y x + 1 λ, y = x + 1 w x,y = λ, y = x 0, *7 Karlin and Taylor(1975),pp

26 (21) (22) dp x,y (t) = λp x,y (t) + λp x+1,y (t), dp x,y (t) = λp x,y 1 (t) λp x,y (t), (29) P x,y (0) = δ x,y *8 (29) P x,y (t) = e λt P x,y (0) + λ t P x,y (0) = δ xy y > x y P x,y (t) = λ t 0 0 e λ(t s) P x,y 1 (s)ds, t 0. P x,x (t) = Pr(τ 1 > t) = e λt, t 0, y = x + 1 P x,x+1 (t) = λ y = x + 2 P x,x+2 (t) = λ t 0 t 0 e λ(t s) P x,y 1 (s)ds, t 0. e λ(t s) e λs ds = λte λt, t 0. e λ(t s) λse λs ds = (λt)2 2 e λt, t 0. P x,y (t) = (λt)y x e λt, y x, t 0, (y x)! W = (w ij : i, j S) S W (i). i S 0 < w ii < ; (ii). i j w ij 0 ; (iii). i, w ij = 1. j S W λ λ 0 W = 0 λ λ λ *8 Appendix 26

27 W S W P (t) = e tw (P (t) : t 0) *9 (i). s, t P (s + t) = P (s)p (t); (ii). (P (t) : t 0) d P (t) = P (t)w, P (0) = I; (iii). (P (t) : t 0) d P (t) = W P (t), P (0) = I; (iv). k = 0, 1, 2,..., ( d kp (t) = W ) t=0 k λ (pure birth processes) x λ x, x 0 h Pr(X(t + h) = x X(t) = x) = 1 λ x h + o(h) Pr(X(t + h) = x + 1 X(t) = x) = λ x h + o(h). X(0) = x S 1, S 2,... λ x, λ x+1,... Y n = x + n (26) (27) dp y (t) = λ y P y (t) + λ y 1 P y 1 (t), y 1, dp 0 (t) = λ 0 P 0 (t), P y (0) = δ xy. (29) dp x,y (t) = λ y 1 P x,y 1 (t) λ y P x,y (t), (30) λ x (e λxt e λx+1t ), λ P x,x+1 (t) = x+1 λ x, λ x+1 λ x λ x e λxt, λ x+1 = λ x. *9 Norris(1997) 27

28 4.2 ( ) λ x = xλ, x 0, (26,27) dp y (t) = λyp y (t) + λ(y 1)P y 1 (t), y 1. x = 1 P y (t) = e λt (1 e λt ) y 1, y 1, (4.6) dp x,y (t) = λ(y 1)P x,y 1 (t) λyp x,y (t), ( ) y 1 P x,y (t) = e xλt (1 e λt ) y k, y x, t 0. y x 4.4 S = {0, 1, 2,..., d} S = {0, 1, 2,..., } ( ) (birth and death processes) y x > 1 w xy = 0. x x 1 x + 1 w x,x+1 w x,x 1 λ x = w x,x+1, µ x = w x,x 1 w x p xy (20) w x,x+1 + w x,x 1 = w x = w xx w xx = (λ x + µ x ), w x = λ x + µ x. 28

29 p xy p xy = w xy /w x, x y µ x λ x +µ x, y = x 1, λ p xy = x λ x +µ x, y = x + 1,. 0, µ 0 = 0 λ d = 0 A, B λ x <A + Bx, (i). h Pr(X(t + h) = x + 1 X(t) = x) = λ x h + o(h); (ii). h Pr(X(t + h) = x 1 X(t) = x) = µ x h + o(h); x 0. (iii). h Pr(X(t + h) = x X(t) = x) = 1 (λ x + µ x )h + o(h); (iv). µ 0 = 0, λ 0 > 0, µ x, λ x > 0, x 1. W (infinitesimal generator ) λ 0 λ µ 1 (λ 1 + µ 1 ) λ 1 0 W = 0 µ 2 (λ 2 + µ 2 ) λ µ 3 (λ 3 + µ 3 )..... * 10 (21)(22) dp x,y (t) = µ x P x 1,y (t) (λ x + µ x )P x,y (t) + λ x P x+1,y (t), (31) dp x,y (t) = λ y 1 P x,y 1 (t) (λ y + µ y )P x,y (t) + µ y+1 P x,y+1 (t). (32) λ 1 = 0 P x,y (t) y = x + 1, x + 2, * 11 (25) dp y (t) = P y 1 (t)w y 1,y (t) + P y+1 (t)w y+1,y P y (t)w y,y+1 (t) P y (t)w y,y 1 (t) λ µ dp y (t) = λ y 1 P y 1 (t) + µ y+1 P y+1 (t) (λ y + µ y )P y (t), y = 1, 2,, (33) dp 0 (t) = µ 1 P 1 (t) λ y P 0 (t), (34) *10 Taylor and Karlin(1998) *11 (2003) 29

30 4.3 ( : ) x λ x = λx, µ x = µx (λ, µ > 0) E Ef(n) = f(n + 1), E 1 f(n) = f(n 1). P n (t) = (E 1)µ n P n (t) + (E 1 1)λ n P n (t), (35) 4.5 π(x) = 1, π(x) 0, (36) x π(x)p x,y (t) = π(y), y S, t 0, (37) x S π (stationary probability distribution) lim P x,y(t) = π(y), t y S * 12 X(t) t π (37) x π(x) dp x,y(t) t = 0 π(x)w xy = 0, x = 0. y S π lim t P x,y (t) = π(y) dp x,y (t)/ = 0 (21) (22) w xz P z,y = 0, P x,z w z,y = 0 z z (38) *12 30

31 4.4 ( ) X(t), t 0 {0, 1, 2,...} (irreducible) x λ x µ x (38) X π(x) π(1)µ 1 π(0)λ 0 = 0, π(y + 1)µ y+1 π(y)(λ y + µ y ) + π(y 1)λ y 1 = 0, y 1. π(y + 1)µ y+1 π(y)λ y = π(y)µ y π(y 1)λ y 1, y 1, π(y + 1)µ y+1 π(y)λ y = 0 π(y + 1) = λ y µ y+1 π(y), y 0. x S π(x) = λ 0 λ x 1 µ 1 µ x π(0), x 1. π x = { 1, x = 0, λ 0 λ x 1 /µ 1 µ x, x 1, π(x) = π x π(0), x 0, π x = x x=1 λ 0 λ x 1 µ 1 µ x <, π(x) = π x y=0 π, x 0. y (39) S = {0, 1, d} π(x) = π x d y=0 π, 0 < x < d, y 31

32 (25) P x w x,y = P y w y,x x y x y y y P x w x,y = P y w y,x, x, y S * (linear birth-death process with immigration) λ n = λn + α; µ n = µn, α > 0 λ 0 = α P k (t) P 0 (t) = (α + λ(k 1))P k 1 (t) (α + (λ + µ)k)p k (t) + µ(k + 1)P k+1 (t), k 1, = µp 1 (t) αp 0 (t). ( ) (α + λ(k 1))π k 1 (α + (λ + µ)k)π k + µ(k + 1)π k+1 = 0. (α + λ(k 1))π k 1 µkπ k = (α + λk)π k µ(k + 1)π k+1 k 1 k k k + 1 π k α + λ(k 1) = π k 1 µk ( ) θ + k 1 π k = γ k (1 γ) θ, k θ = α/λ, γ = λ/µ * 14. *13 van Kampen(1992),chapter 5 *14 (2003) 6 32

33 (H ) * 15 H (i). H(t) 0 P n (t) = π n H = 0. (ii). P n (t) π n dh(t)/ < 0. (iii). P n (t) = π n dh(t)/ = 0. 3 H(t) = n S P n (t) ln P n(t) π n 4.6 = ( ) = dp j (t) = λjp j (t) + λ(j 1)P j 1 (t), j 1. (40) P j (0) = δ 1,j P j (t) G(z, t) = P j (t)z j j=0 (40) G t = λz(z 1) G z 1 = dz λz(z 1) z 1 e λt = c z c ϕ G(z, t) = ϕ(c) G(z, t) = ϕ( z 1 e λt ) z *15 van Kampne(1992), chapter 5 Weidlich(2000), chapter 10 33

34 * 16 ϕ G(z, 0) = z z = ϕ( z 1 ). z ϕ(x) = 1 1 x. G(z, t) = ze λt 1 (1 e λt )z z G(z, t) = e λt z z j (1 e λt ) j z j, (1 e λt )z < 1. j=0 P j (t) = e λt (1 e λt ) j 1, j 1, * 17 ( P j (t) G t = α(z 1)G + (λz µ)(z 1) G z P j (t) * S P (x, t n ; y, t n+1 ) P (x, t n ; y, t n+1 ) = Pr(X(t n+1 ) = y X(t n ) = x), n = 0, 1, 2,.. *16 Appendix *17 (1968) 4 *18 (2003) 6 34

35 t n+1 t n Pr(y, t n+2 x, t n ; z, t n+1 ) = P (x, t n ; z, t n+1 )P (z, t n+1 ; y, t n+2 ) P (x, t n ; y, t n+2 ) = P (x, t n ; y, t n+2 ) = Pr(y, t n+2 x, t n ; z, t n+1 )dz P (x, t n ; z, t n+1 )P (z, t n+1 ; y, t n+2 )dz (41) P (z, t; y, t + τ ) = (1 a 0 τ )δ y,z + τ W (y z) + o(τ ) (42) W (y z) z y ( w z,y ) P (x, t; y, t + τ + τ ) = P (z, t + τ; y, t + τ + τ )P (x, t; z, t + τ)dz P (x, t; y, t + τ + τ ) = (1 a 0 τ )P (x, t + τ; y, t + τ + τ ) + τ W (y z)p (x, t; z, t + τ)dz τ 0 P (x, t; y, t + τ) = {W (y z)p (x, t; z, t + τ) W (z y)p (x, t; y, t + τ)}dz τ a 0 (x) = W (z x)dz X(t 0 ) = x 0 Pr(X(t) = y) = P (y, t) = Pr(X(t) = y) δ x0,zp (z, t 0 ; y, t)dz P (y, t) = {W (y z)p (z, t) W (z y)p (y, t)}dz (43) t 35

36 X(t) ( 1 t (diffusion process) ϵ > 0 1 lim Pr( X(t + h) X(t) > ϵ X(t) = x) = 0, x S. (44) h 0 h ϵ (44) Dynkin lim h 0 lim h 0 1 E[X(t + h) X(t) X(t) = x] = µ(x, t), h (45) 1 h E[(X(t + h) X(t))2 X(t) = x] = σ 2 (x, t) (46) µ σ (45) µ(x, t) (drift coefficients) (infinitesimal mean), (46) σ 2 (x, t) (diffusion coefficients) (infinitesimal variance) X(t) (44) (45) (46) µ(x, t) σ 2 (x, t) X(t) * ( Brownian process) ()X(t) E[X(t + h) X(t)] 0 V ar[x(t + h) X(t)] Bh Bh = σ 2 h E[X(t + h) X(t) X(t) = x] = 0, E[(X(t + h) X(t)) 2 X(t) = x] = σ 2 h Dynkin (, + ) µ(x) = 0 σ 2 (x) = σ 2 µ(x) = µ 0 µ = ( Ornstein-Uhlenbeck process) µ(x) = αx σ 2 (x) = σ 2 * 20 *19 Karlin and Taylor(1981) 15 *20 Karlin and Taylor(1981) pp

37 5.1 X(t) µ(x) σ 2 (x) g(x) x 2 g (x) Y (t) = g[x(t)] Y (t) µ Y (y) = 1 2 σ2 (x)g (x) + µ(x)g (x), σy 2 (y) = σ 2 (x)[g (x)] 2 µ Y σy ( geometric Brawnian motion) X(t) µ σ 2 Y (t) = exp{x(t)} (0, ) t 0 < t 1 < < t n Y (t 1 )/Y (t 0 ), Y (t 2 )/Y (t 1 ),..., Y (t n )/Y (t n 1 ) y = g(x) = exp{x} g (x) = g (x) = y Y (t) µ Y (y) = ( 1 2 σ2 + µ)y, σy 2 (y) = σ 2 y (41) F (x, τ; y, t) F (x, τ; y, t) Pr(X(t)<y X(τ) = x) F (x, τ; y, t) = y P (x, τ; z, t)dz 5.2 F (x, τ; y, τ + t) = Pr(X(t)<y X(0) = x) = F (x, y, t) F (x, y, t) (44)(45)(46) F (x, y, t), 2 F (x, y, t) x x 2 37

38 (x, t) F (x, y, t) t F (x, y, t) = 1 2 σ2 (x) 2 F (x, y, t) + µ(x) F (x, y, t) (47) x2 x * 21 F (x, y, t) P (y, t) van Kampen(1992) W (y x) r = y x W (x; r) W (x + r x) P (y, t) = P (y r, t)w (y r; r)dr P (y, t)w (y; r)dr t W (x; r) r r > δ, W (x; r) 0, x < δ, W (x + x; r) W (x; r), δ > 0 1 P (y, t) = t P (y, t)w (y; r)dr r y {W (y; r)p (y, t)}dr P (y, t)w (y; r)dr + o(r 2 ), r 2 2 {W (y; r)p (y, t)}dr y2 r a ν (y) = r ν W (y; r)dr P (y, t) t = y {a 1(y)P (y, t)} y 2 {a 2(y)P (y, t)}. (48) *21 6 Karlin and Taylor(1981) 15 38

39 W (x y) a 1, a 2 a 1 (x), a 2 (x) a 1 (x) = µ(x), a 2 (x) = σ 2 (x). 5.4 ( ) µ(x) = 0 σ 2 (x) = σ 2 P (y, t) t = σ2 2 2 P (y, t) y2 D a 2 /2 D = σ 2 /2 P (y, t) = 1 4πDt exp[ y2 4Dt ] ( ) X 2 (t) = 2Dt t t + t t X(t + t) X(t) µ(x, t) t + σ(x, t) B(t) (49) X(t) = x t B(t) B(t) = B(t + t) B(t) t E[ B(t)] = 0, Var[ B(t)] = t µ(x, t) σ(x, t) 2 µ(x, t) σ(x, t) B(t) 1 X(t). (49) X(t) µ(x, t) σ 2 (x, t). E[ X] 1 lim = lim E[µ(x, t) t + σ(x, t) B(t)] = µ(x, t) t 0 t t 0 t 39

40 . E[ X 2 ] 1 lim = lim t 0 t t 0 t Var[σ(x, t) B(t)] = σ2 (x, t). lim t 0 X/ t. (49) dx(t) = µ(x(t), t) + σ(x(t), t)db(t) (50) ( ) ds = S[µ + σdb(t)] S(t) t B(t) 1 ds = µ + σdb(t) S ln S(t) = µt + σb(t) S(0) S(t) = S(0) exp[µt + σb(t)] Y (t) (50) X(t) Y (t) = f(x(t), t) f(x(t), t) db(t) 2 f dy (t) = df(x(t), t) = f(x(t), t) dx(t) + x f(x(t), t) f(x(t), t) t f(x(t), t) dx(t) + 1 x t 2 dx(t) 2 = µ(x(t), t) µ(X(t), t)σ(x(t), t)db + σ 2 (X(t), t)db(t) 2 x 2 dx(t) 2 2 f(x(t), t) t (51) µ(x(t), t) µ(X(t), t)σ(x(t), t)db + σ 2 (X(t), t) (52) (50) (51) dy (t) = [ f(x(t), t) f(x(t), t) µ(x(t), t) + x t + 2 f(x(t), t) x 2 σ 2 (X(t), t)] f(x(t), t) σ(x(t), t)db(t) x + (53) 40

41 [+ ] 2 0 (53) Y (τ) Y (0) = τ 0 f(x(t), t) f(x(t), t) [ µ(x(t), t) f(x(t), t) x t 2 x 2 σ 2 (X(t), t)] τ f(x(t), t) + σ(x(t), t)db(t) 0 x 2 ( ) (53) Appendix A x t t x, dx, d 2 x 2, t x, dx ϕ(t, x, dx ) = 0 1 dx = f(t, x) (54) 1 (54) t dx = f(t) t x = f(t) + C dx = f(x) t = dx f(x) + C 41

42 dx = φ(t) ψ(x) ψ(x)dx = φ(t) + C A.1 dx = 2x 1 dx = 2 x ln x = 2t + ln C x = Ce 2t C f(t) g(t) dx + f(t)x = g(t) (55) 1 0 dx + f(t)x = 0 (55) ( ) ( ) dx = f(t)x x = C exp{ t t 0 f(r)dr} C (55) C t x = C(t) exp{ t t 0 f(r)dr} (55) C(t) x = exp{ t t 0 f(r)dr}[ t t 0 exp{ r t 0 f(u)du}g(r)dr + x 0 ] (56) 42

43 x 0 t = t 0 x (56) t t t r x = x 0 exp{ f(r)dr} + exp{ f(r)dr} exp{ f(u)du}g(r)dr t 0 t 0 t 0 t 0 A.2 dx = λx + g(t) t = 0 x(0) = x 0 (56) t x(t) = x 0 e λt + e λt e λr g(r)dr A.3 dx = λx, dy = x + λy. 0 t = 0 x = x 0, y = y 0 1 x x(t) = x 0 e λt 2 t y(t) = y 0 e λt + e λt e λr x(r)dr 0 t t e λt e λr x(r)dr = e λt x 0 dr = x 0 te λt 0 y(t) = y 0 e λt + x 0 te λt 0 43

44 B z x y z x = p, z y = q x y p, q f(x, y, z, p, q) = 0 z = F (x, y) z = F (x, y) p + q = 0 z = F (x y) F y = 0 xz = a z = a x y OG D OG az cx = 0, bz cy = 0 az cx = C 1, bz cy = C 2 C 1, C 2 D C 1 C 2 F (C 1, C 2 ) = 0 F (az cx, bz cy) = 0 44

45 F x + p F z = 0, F y + q F z = 0 (ap c)f 1 + bpf 2 = 0, aqf 1 + (bq c)f 2 = 0 F 1 1 F F 2 2 F 1, F 2 ap + bq = c F f(x, y, z) z z + g(x, y, z) x y = h(x, y, z) (57) P (x, y, z) P PQ PQ (f(x, y, z), g(x, y, z), h(x.y, z)) P S z = F (x, y) P S PN (p, q, 1) (57) fp + gq h = 0 PQ PN PQ S dx f(x, y, z) = dy g(x, y, z) = y = ϕ(x, C 1, C 2 ), z = ψ(x, C 1, C 2 ) dz (58) h(x, y, z) u(x, y, z) = C 1, v(x, y, z) = C 2 F (u, v) = 0 (59) F (57) (58) F (u, v) = 0 (58) (59) u = ϕ(v) ϕ 45

46 B.1 w k,k+1 = α + λk G(z, t) G t λz(z 1) G z = α(z 1)G (t, z) G (t, z) (57) f = 1, g = λz(z 1), h = α(z 1)G (58) 1 = dz λz(z 1) = 2 dg α(z 1) ln G(z, t) = α λ ln z + ln C 1 ln[g(z, t)z α/λ ] = ln C 1 1 z z 1 = C 2e λt C 1, C 2 ϕ C 1 = ϕ(c 2 ) G(z, t)z α/λ z = ϕ( z 1 e λt ) ( ) G(z, t) = z α/λ z ϕ( z 1 e λt ) (1),(5),(6) (6) (1). Paul G. Hoel, Sidney C. Port, and Charles J. Stone(1972), Introduction to Stochastic Processes, Houghton Mifflin. (2). Samuel Karlin and Howard M. Taylor(1975), A First Course in Stochastic Processes, Academic Press. 46

47 (3). Samuel Karlin and Howard M. Taylor(1981), A Seccond Course in Stochastic Processes, Academic Press. (4). J.R.Norris(1997), Markov Chains, Cambridge University Press. (5). Howard M. Taylor and Samuel Karlin(1998), An Introduction to Stochastic Modeling, Academic Press. (6). (1968), (7). N. G. Van Kampen(1992), Stochastic Processes in Physics and Chemistry, North-Holland. (3) (1). Masanao Aoki(1996), New Approaches to Macroeconomic Modeling,Cambridge University Press. (2). Masanao Aoki(202), Modeling Aggregate Behavior and Fluctuations in Economics, Cambridge University Press. (3). (2003) (4). Wolfgang Weidlich and Gunter Haag(1983), Concepts and Models of a Quantitative Sociology,Springer- Verlag. (5). Wolfgang Weidlich(2000), Sociodynamics:A Systematic Approach to Mathematical Modelling in the Social Sciences, Dover Publication. 47