Birth-death chain. X n. X k:n k,n 2 N k apple n. X k: L 2 N. x 1:n := x 1,...,x n. E n+1 ( x 1:n )=E n+1 ( x n ), 8x 1:n 2 X n.

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Georgina Mosley
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2 Birth-death chains Birth-death chain special type of Markov chain Finite state space X := {0,...,L}, with L 2 N X n X k:n k,n 2 N k apple n Random variable and a sequence of variables, with and A sequence can be infinite as well X k: Sequence of state values x 1:n := x 1,...,x n in X n Markov condition E n+1 ( x 1:n )=E n+1 ( x n ), 8x 1:n 2 X n where for E n+1 ( x n ) is the expectation operator with p.m.f p time-homogeneous P = p(x n+1 x n ) 0 1 r 0 p q 1 r 1 p B q L 1 r L 1 p L 1 A 0 0 q L r L

3 Imprecise birth-death chains Consider a matrix P with p.m.f. not precisely known For every i 2 X, the p.m.f. of the i row belong to a credal set M i and consists of elements f i of the form f i ( j)= 8 q i if j = i 1 >< if j = i r i p i if j = i + 1 >: 0 otherwise i 2 X \{0,L} f 0 ( j)= 8 >< r 0 if j = 0 p 0 if j = 1 >: 0 otherwise f L ( j)= 8 >< q L if j = L 1 r L if j = L >: 0 otherwise Positivity assumption: r 0, p 0,r L,q L and q i,r i, p i for all i 2 X \{0,L} strictly positive

4 Imprecise Markov condition Lower and upper expectations of real-valued function f on X E( f i) := min E fi ( f )= min f i ( j) f ( j) f i 2M i f i 2M i E( f i) := max f i 2M i E fi ( f )= max f i 2M i Â j2x Â j2x f i ( j) f ( j) and for all x 1:n 2 X n, the imprecise Markov condition is E n+1 ( x 1:n )=E n+1 ( x n ) := E( x n )

5 Global uncertainty models Based on the notion of submartingales, we derive global uncertainty models These models satisfy a version of the Law of Iterated expectation X For every n 2 N and every real-valued function g on X N E n+1:1 (g(x n+1:1 ) i) =E n+2:1 (g(x n+2:1 ) i). (time-homogeneity) By defining f 0 on X by f 0 (i 0 ) := E n+2: (g(i 0,X n+2: ) i 0 ) for all i 0 2 X, then E n+1: (g(x n+1: ) i)=e n+1 ( f 0 i)=e( f 0 i)

6 First passage time The first passage time from i to j with i, j 2 X is ( 1 X n+1 = j t i! j (i,x n+1: ) := 1 + t Xn+1! j(x n+1,x n+2: ) X n+1 6= j = 1 + I j c(x n+1 )t Xn+1! j(x n+1,x n+2: ) where I j c is the indicator function of j c := X \{j} For i = j, we have the return time Due to time-homogeneity t i! j,n := E n+1: (t i! j (i,x n+1: ) i) and t i! j,n := E n+1: (t i! j (i,x n+1: ) i) will be denoted by t i! j and t i! j t i! j t i! j Due to positivity assumption and are real-valued and strictly positive and have the form t i! j = 1 + E(I j ct! j i) and t i! j = 1 + E(I j ct! j i)

7 Lower expected upward first passage time The first passage time from i to j with i, j 2 X and i < j t 0!1 = 1 p 0 For all i 2 X \{0,L}, we have that min {q i t i 1!i p i t i!i+1 } = 1 f i 2M i M i For all satisfying the positivity assumption, with i 2 X \{0,L}, and c a real constant, then min {qc pµ} is strictly decreasing in µ f i 2M i

8 Lower expected upward first passage time min {q i t i 1!i p i t i!i+1 } = 1 f i 2M i We can calculate t i!i+1 recursively Using a bisection method, as long as we have calculated t i 1!i Moreover, For all i 2 X \{0,L}, s.t i + 1 < j, we have that t i! j = t i!i+1 + t i+1! j For all i 2 X, such that i < j, we have that t i! j = j 1 Â k=i t k!k+1

9 Lower expected downward first passage time The first passage time from i to j with i, j 2 X and i > j Similarly to the upward case t L!L 1 = 1 q L For all i 2 X \{0,L}, we have that min { q i t i!i 1 + p i t i+1!i } = 1 f i 2M i i 1 For all i 2 X, such that i > j, we have that t i! j = Â t k+1!k k= j

10 Lower expected return time The first passage time from i to j with i, j 2 X and i = j Combining the results from expected upward with these of downward first passage times t 0!0 = 1 + min f 0 2M 0 {p 0 t 1!0 } = 1 + p 0 t 1!0 t L!L = 1 + min f L 2M L {q L t L 1!L } = 1 + q L t L 1!L and for all i 2 X \{0,L} t i!i = 1 + min f i 2M i {q i t i 1!i + p i t i+1!i }

11 Linear vacuous mixtures The set M i is a subset of the simplex S X For any i 2 X, S Xi is the subset of S X containing p.m.f. f i Given precise f0,f L,f and e i 2 [0,1) for any i i 2 X M 0 = (1 e 0 )f 0 + e 0f 0 0 : f S X 0 M L = (1 e L )f L + e Lf 0 L : f 0 L 2 S X L and for all i 2 X \{0,L} M i = (1 e i )f i + e i f 0 i : f 0 i 2 S X i

12 Linear vacuous mixtures We can also define q i :=(1 e i )q i and q i :=(1 e i )q i + e i for all i 2 X \{0} p i :=(1 e i )p i and p i :=(1 e i )p i + e i for all i 2 X \{L} Expected lower upward, downward first passage and return times t i!i+1 = Â i k=0 ì =k+1 q` i m=k p m t i!i 1 = Â L k=i k 1 `=i p` k m=i q m t i!i = 1 + q i t i 1!i + p i t i+1!i

13 Linear vacuous mixtures Consider state space X := {0,...,4}, e i = e = 0.4 and Q q P = B A then, for all i 2 X \{0,L} p r Q we calculate lower and upper expected return times i i!i i!i

14 General example Consider state space X := {0,...,4} M 0 is determined by p 0 2 [0.15,0.4] and M L by q L 2 [0.2,0.6] For all i 2 X \{0,L}, M i is characterised by triplets of the form Q q (q i,r i, p i ) (0.65, 0.15, 0.2), (0.6, 0.25, 0.15), (0.5, 0.4, 0.1), (0.43, 0.45, 0.12), (0.33, 0.5, 0.17), (0.27, 0.43, 0.3), (0.25, 0.35, 0.4), (0.3, 0.25, 0.45), (0.4, 0.17, 0.43), (0.55, 0.1, 0.35) ) p r lower and upper expected upward and downward first passage times 0! ! ! ! ! ! ! ! ! !3 5 1! !2 12 2! ! ! !

15 Conclusions and future work Simple methods for computing lower and upper expected first passage and return times Applying similar methods to other type of chains, e.g. Bonus-Malus systems Applying similar methods to continuous time systems

Calculating Bounds on Expected Return and First Passage Times in Finite-State Imprecise Birth-Death Chains

9th International Symposium on Imprecise Probability: Theories Applications, Pescara, Italy, 2015 Calculating Bounds on Expected Return First Passage Times in Finite-State Imprecise Birth-Death Chains