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Department of Applied Mathematics and Statistics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark, www.math.ku.dk/ams Niels Richard Hansen February 23, 2006 1 Exercises This exercise is about the random walk ascending and descending ladder height distribution when the increment distribution has ﬁnite support concentrated on a lattice. In this case the two distributions can be computed by ﬁnding roots in a polynomial equation. The solution is based on the Wiener-Hopf factorisation identity. We need to introduce some notation. Let n, m ≥ 1 be given and assume that P(X1 = k) = pk , k ∈ {−m, −m + 1, . . . , n − 1, n} n where k=−m pk = 1 and p−m , pn > 0. Let p+,k = P(Sτ+ = k, τ+ < ∞), k = 1, 2, . . . , n and p−,k = P(Sτ− = k, τ− < ∞), k = 0, −1, . . . , −m. and deﬁne for z ∈ C n p+ (z) = p+,k z k k=1 and m p− (z) = p−,−k z −k , z = 0, k=0 together with n p(z) = pk z k , z = 0. k=−m Question 1.1. Show that the Wiener-Hopf identity (Theorem 8.41) implies that (1 − p+ (z))(1 − p− (z)) = 1 − p(z) (1) for z ∈ C\{0}. Hint: First argue for |z| = 1 then use an extension argument. Question 1.2. Argue that if the random walk has negative drift then |p− (z)| < 1 for |z| > 1. Then argue that q(z) = z m (1 − p(z)) is a polynomial, and from the identity above that q has precisely n roots with absolute value > 1, and that these roots are also the roots of 1 − p+ (z). 2 1. Exercises The roots of q can be computed from q numerically, and we can single out those with absolute value > 1. Let α1 , . . . , αn with |αi | > 1 denote these roots. The ascending ladder height distribution given in terms of the point probabilities p+,1 , . . . , p+,n is on the other hand not known. Question 1.3. Show that p+,k , k = 1, . . . , n is a solution to the following system of linear equations: n αk p+,k = 1, i i = 1, . . . , n. k=1 Under what conditions does there exist a unique solution to these equations? Question 1.4. Argue that q has a total of n+m roots = 0 with m of them having absolute value ≤ 1, that q− (z) = z m (1 − p− (z)) is a polynomial, and that the m roots of q with absolute value ≤ 1 are precisely the roots of q− . Let β1 , . . . , βm with |βi | ≤ 1 denote the roots of q with absolute value ≤ 1. Question 1.5. Show that p−,k , k = 0, −1, . . . , −m is a solution to the following system of linear equations: m α−k p−,−k = 1, i i = 1, . . . , n. k=0 Under what conditions does there exist a unique solution to these equations? Hint: By the negative drift you know the extra equation m p−,−k = 1. k=0

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