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Large deviations for the time of ruin. (English) Zbl 0947.60048
Summary: Let $$\{Y_n\mid n=1,2,\dots\}$$ be a stochastic process and $$M$$ a positive real number. Define the time of ruin by $$T= \inf \{n\mid Y_n> M\}$$ $$(T= +\infty$$ if $$Y_n\leq M$$ for $$n= 1,2,\dots)$$. We are interested in the ruin probabilities for large $$M$$. Define the family of measures $$\{P_M\mid M>0\}$$ by $$P_M (B)=\text{P}(T/M\in B)$$ for $$B\in {\mathcal B}$$ ($${\mathcal B} =$$ Borel sets of $$\mathbb{R}$$). We prove that for a wide class of processes $$\{Y_n\}$$, the family $$\{P_M\}$$ satisfies a large deviations principle. The rate function will correspond to the approximation P$$(T/M\approx x)\approx\text{P}(Y_{\lceil xM\rceil}/M\approx 1)$$ for $$x> 0$$. We apply the result to a simulation problem.

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