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 0 0 0 q 1 r 1 p 1 0 0. B............. C @ 0 0 q L 1 r L 1 p L 1 A 0 0 q L r L
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
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  j2x  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 )
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)
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)
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
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
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
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 }
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 0 0 2 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
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
Linear vacuous mixtures Consider state space X := {0,...,4}, e i = e = 0.4 and Q q P = 0 1 0.55 0.45 0 0 0 0.3 0.5 0.2 0 0 B 0 0.3 0.5 0.2 0 C @ 0 0 0.3 0.5 0.2A 0 0 0 0.6 0.4 then, for all i 2 X \{0,L} p r Q we calculate lower and upper expected return times i i!i i!i 0 1.584 91.41 1 1.526 24.956 2 1.678 17.845 3 1.656 79.71 4 2.037 503.724
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!1 2.5 4!3 1.666 1!2 3.889 3!2 2.051 2!3 4.814 2!1 2.169 3!4 5.432 1!0 2.206 0!1 6.666 4!3 5 1!2 43.333 3!2 12 2!3 226.666 2!1 23.2 3!4 1143.333 1!0 41.12
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