Annales Academić Scientiarum Fennicć
Mathematica
Volumen 32, 2007, 523-543

ON A FUNCTION-THEORETIC RUIN PROBLEM

Gerd Jensen and Christian Pommerenke

Sensburger Allee 22 a, D-14055 Berlin, Germany; cg.jensen 'at' t-online.de

Technische Universität, Institut für Mathematik
D-10623 Berlin, Germany; pommeren 'at' math.tu-berlin.de

Abstract. Let \varphi : D \to D be analytic, m \in N and g₀ \in H². We define the g_n \in H² by

(*) g_n+1(z) = (g_n(z) \varphi(z)z^-m)⁺ for n \in N₀

where (...)⁺ denotes the analytic part of the Laurent series. We derive explicit formulas for the coefficients b_n,k of the g_n.

The recursion (*) comes from the study of the random variables

S_n+1 = S₀ + X₁ + ... + X_n - mn,

where the X_\nu are i.i.d. with generating function \varphi. Ruin occurs when S_n becomes negative. We have b_n,k = P(S_n = k, not yet ruined).

2000 Mathematics Subject Classification: Primary 30B10, 60E05, 91B30.

Key words: Generating functions, Laurent separation, ruin, barrier.

Reference to this article: G. Jensen and Ch. Pommerenke: On a function-theoretic ruin problem. Ann. Acad. Sci. Fenn. Math. 32 (2007), 523-543.