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Finite and infinite time ruin probabilities in a stochastic economic environment. (English) Zbl 1047.60040
Summary: Let \((A_1,B_1,L_1), (A_2,B_2,L_2),\dots \) be a sequence of independent and identically distributed random vectors. For \(n\in \mathbb N\), denote \[ Y_n = B_1 + A_1B_1 +A_1A_2B_3+\cdots +A_1\cdots A_{n-1}B_n+A_1\cdots A_nL_n. \] For \(M>0\), define the time of ruin by \(T_M = \inf \,\{n\mid Y_n>M\}\) \((T_M=+\infty \), if \(Y_n\leq M\) for \(n=1,2,\dots )\). We are interested in the ruin probabilities for large \(M\). Our objective is to give reasons for the crude estimates \({\mathbf P}(T_M\leq x \log M)\approx M^{-R(x)}\) and \({\mathbf P}(T_M< \infty )\approx M^{-w}\) where \(x>0\) is fixed and \(R(x)\) and \(w\) are positive parameters. We also prove an asymptotic equivalence \({\mathbf P}(T_M<\infty )\sim CM^{-w}\) with a strictly positive constant \(C\). Similar results are obtained in an analogous continuous time model.

60G40 Stopping times; optimal stopping problems; gambling theory
60F10 Large deviations
Full Text: DOI
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