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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.

MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 60F10 Large deviations
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References:
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