×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI