Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University 508 YiFu Lou talk 06/

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1 Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University 508 YiFu Lou talk 06/04/ Page 1

2 Outline 508 YiFu Lou talk 06/04/ Page 2

3 Outline Examples 508 YiFu Lou talk 06/04/ Page 2

4 Outline Examples 508 YiFu Lou talk 06/04/ Page 2

5 Theorem Let {N i (t), t 0} be a HPP(λ i ) (λ i > 0, i = 1, 2,, n) and they are independent each other, then N(t) = n i=1 N i(t) is HPP( n i=1 λ i). 508 YiFu Lou talk 06/04/ Page 3

6 Theorem Let {N(t), t 0} be a HPP(λ), under the condition τ 1 = x 1,, τ n = x n (0 < x 1 < < x n ), then conditional density of τ n+k (k 1) is = f(x n+k x 1,, x n ) = f(x n+k x n ) { λ k (k 1)! (x n+k x n ) k 1 e λ(x n+k x n ), x n+k > x n, 0, x n+k x n, that is, τ n+k τ n τ 1 = x 1,, τ n = x n Γ(k, λ). 508 YiFu Lou talk 06/04/ Page 4

7 Theorem Let {N(t), t 0} be a HPP(λ), 0 < s < t, under the condition N(t) = n (n 1), N(s) B ( n, s t ). 508 YiFu Lou talk 06/04/ Page 5

8 Lemma Let {N(t), t 0} be a HPP(λ), then (1) The joint probability density of first n arrival times τ 1,, τ n and event {N(t) = n} (t > 0, n 1) is f(x 1,, x n ; N(t) = n) = { λ n e λt, 0 < x 1 < < x n < t, 0, otherwise. 508 YiFu Lou talk 06/04/ Page 6

9 Lemma Let {N(t), t 0} be a HPP(λ), then (1) The joint probability density of first n arrival times τ 1,, τ n and event {N(t) = n} (t > 0, n 1) is f(x 1,, x n ; N(t) = n) = { λ n e λt, 0 < x 1 < < x n < t, 0, otherwise. (2) The joint probability density of τ 1,, τ n, τ n+1,, x n+k (k 1) and event {N(t) = n} is = f(x 1,, x n, x n+1,, x n+k, N(t) = n) { λ n+k e λx n+k, 0 < x 1 < < x n < t < x n+1 < < x n+k, 0, otherwise. 508 YiFu Lou talk 06/04/ Page 6

10 Theorem {N(t), t 0} is a HPP(λ) if and only if (1) t > 0, N(t) is a poisson distribution with parameter λt; and (2) t > 0, n 1, under the condition N(t) = n, the first n arrival times τ 1,, τ n and U (1),, U (n) have the same distribution, where U 1,, U n are independent each other and have uniformly distributed on (0, t), and U (1),, U (n) are the order quantities of U 1,, U n. 508 YiFu Lou talk 06/04/ Page 7

11 Theorem {N(t), t 0} is a HPP(λ),under the condition τ n = t(t > 0, n 1), the first n 1 arrival times τ 1,, τ n 1 and U (1),, U (n 1) have the same distribution, where U 1,, U n 1 are independent each other and have uniformly distributed on (0, t), and U (1),, U (n 1) are the order quantities of U 1,, U n YiFu Lou talk 06/04/ Page 8

12 Theorem {N(t), t 0} is a HPP(λ),under the condition N(t) = n(t > 0, n 1), (1) Conditional density of τ k (1 k n) is = f(x k N(t) = n) { n! (k 1)!(n k)!t ( xk ) k 1 ( t 1 x k ) n k t, 0 < xk < t, 0, otherwise, 508 YiFu Lou talk 06/04/ Page 9

13 Theorem {N(t), t 0} is a HPP(λ),under the condition N(t) = n(t > 0, n 1), (1) Conditional density of τ k (1 k n) is = f(x k N(t) = n) { n! (k 1)!(n k)!t ( xk ) k 1 ( t 1 x k ) n k t, 0 < xk < t, 0, otherwise, (2) Conditional density of τ n+k (k 1) is = f(x n+k N(t) = n) { λ k (k 1)! (x n+k t) k 1 e λ(xn+k t), x n+k > t, 0, x n+k t, 508 YiFu Lou talk 06/04/ Page 9

14 Examples Example YiFu Lou talk 06/04/ Page 10

15 Examples Example Example YiFu Lou talk 06/04/ Page 10

16 non homogeneous PP Definition Suppose that a stochastic process {N(t), t 0} satisfies the following conditions: (1) {N(t), t 0} is a SP with independent increments and non negative integer values, and N(0) = 0; (2) For any 0 s < t, increments N(t) N(s) of SP has a poisson distribution with parameter Λ(t) Λ(s), i.e. P {N(t) N(s) = k} = [Λ(t) Λ(s)]k k! e { [Λ(t) Λ(s)]}, k = 0, 1, 2,, where Λ(t) = t 0 v(u)du. Then {N(t), t 0} is said to be a non homogeneous Poisson process with intensity function v(t), NHPP(v(t)), for short. 508 YiFu Lou talk 06/04/ Page 11

17 Definition If {N(t), t 0} is NHPP(v(t)), one dimensional distribution of the process N(t) is a poisson distribution with parameter Λ(t), t > YiFu Lou talk 06/04/ Page 12

18 Definition If {N(t), t 0} is NHPP(v(t)), one dimensional distribution of the process N(t) is a poisson distribution with parameter Λ(t), t > 0. n-dimensional distribution = P{N(t 1 ) = i 1, N(t 2 ) = i 2,, N(t n ) = i n } { Π n [Λ(t j ) Λ(t j 1 )] i j i j 1 j=1 (i j i j 1 )! e { [Λ(t j) Λ(t j 1 )]}, 0 = i 0 i n 0, otherwise. 508 YiFu Lou talk 06/04/ Page 12

19 Definition Mean function µ(t) = EN(t) = Λ(t), t 0, 508 YiFu Lou talk 06/04/ Page 13

20 Definition Mean function µ(t) = EN(t) = Λ(t), t 0, Variance Function σ 2 (t) = DN(t) = Λ(t), t 0, 508 YiFu Lou talk 06/04/ Page 13

21 Definition Mean function µ(t) = EN(t) = Λ(t), t 0, Variance Function σ 2 (t) = DN(t) = Λ(t), t 0, Relative function R(s, t) = E[N(t)N(s)] = Λ(s)Λ(t) + Λ(min(s, t)), s, t 0, 508 YiFu Lou talk 06/04/ Page 13

22 Definition Covariance function Γ(s, t) = Cov(N(s), N(t)) = Λ(min(s, t)), s, t 0, 508 YiFu Lou talk 06/04/ Page 14

23 Definition Covariance function Γ(s, t) = Cov(N(s), N(t)) = Λ(min(s, t)), s, t 0, Characteristic function of N(t) N(s)(0 s < t) is ϕ st (u) = exp{[λ(t) Λ(s)](e iu 1)} (i = 1). 508 YiFu Lou talk 06/04/ Page 14

24 Definition NHPP(λ) is a continuous time homogeneous Markov process with discrete state space, the transition probability functions are p ij (s, s + t) = P{N(s + t) = j N(s) = i} = (s 0, t > 0). { [Λ(s+t) Λ(s)] j i (j i)! exp{ [Λ(t) Λ(s)]}, j i, 0, j < i 508 YiFu Lou talk 06/04/ Page 15

25 Theorem If {N(t), t 0} is NHPP(v(t)), then (1) There are only finitely many particles in any finitely interval (a, a + t], that is k=0 P{N(a + t) N(a) = k} = 1; 508 YiFu Lou talk 06/04/ Page 16

26 Theorem If {N(t), t 0} is NHPP(v(t)), then (1) There are only finitely many particles in any finitely interval (a, a + t], that is k=0 P{N(a + t) N(a) = k} = 1; (2) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) = 1} = v(t) t + o( t); 508 YiFu Lou talk 06/04/ Page 16

27 Theorem If {N(t), t 0} is NHPP(v(t)), then (1) There are only finitely many particles in any finitely interval (a, a + t], that is k=0 P{N(a + t) N(a) = k} = 1; (2) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) = 1} = v(t) t + o( t); (3) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) 2} = o( t); 508 YiFu Lou talk 06/04/ Page 16

28 Theorem Suppose that v(t) is continuous on t = 0. If counting process {N(t), t 0} satisfies the following conditions: 508 YiFu Lou talk 06/04/ Page 17

29 Theorem Suppose that v(t) is continuous on t = 0. If counting process {N(t), t 0} satisfies the following conditions: (1) {N(t), t 0} is process with independent increments, and N(0) = 0; 508 YiFu Lou talk 06/04/ Page 17

30 Theorem Suppose that v(t) is continuous on t = 0. If counting process {N(t), t 0} satisfies the following conditions: (1) {N(t), t 0} is process with independent increments, and N(0) = 0; (2) There are only finitely many particles in any finitely interval (a, a + t], that is k=0 P{N(a + t) N(a) = k} = 1; 508 YiFu Lou talk 06/04/ Page 17

31 (3) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) = 1} = v(t) t + o( t); 508 YiFu Lou talk 06/04/ Page 18

32 (3) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) = 1} = v(t) t + o( t); (4) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) 2} = o( t), 508 YiFu Lou talk 06/04/ Page 18

33 (3) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) = 1} = v(t) t + o( t); (4) t > 0 and sufficiently enough small t, we have P{N(t + t) N(t) 2} = o( t), then {N(t), t 0} is NHPP(v(t)). 508 YiFu Lou talk 06/04/ Page 18

34 Theorem {N(t), t 0} is NHPP(v(t)) if and only if n 1, the joint density of the first n arrival times τ 1,, τ n is = f(x 1,, x n ) { [Π n i=1 v(x i)] exp[ Λ(x n )], 0 < x 1 < < x n, 0, otherwise. 508 YiFu Lou talk 06/04/ Page 19

35 Acknowledgement Thanks you all for your attention. 508 YiFu Lou talk 06/04/ Page 20

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