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Distributional properties of CUSUM stopping times and stopped processes. (English) Zbl 1294.62172
Summary: Let $$\{ Z(t), t\geq 0\}$$ be a stochastic process with stationary independent increments and let $$T_1=\inf\{t\geq 0:Z(t)-\min_{0\leq s\leq t} Z(s)\geq h\}$$, $$h>0$$. Under suitable conditions on $$Z(t)$$, we obtain the joint Laplace transform of $$T_1$$ and $$Z(T_1)$$ by deriving a formula for $$\psi_1(\alpha,\beta )=E\exp (\alpha Z(T_1)-\beta T_1)$$, where $$\beta >0$$ and $$\alpha$$ is a suitable number in some interval containing zero. Furthermore, if $$T_2$$ is $$T_1$$ for $$Z^\ast (t)=-Z(t)$$, then $$\psi_1(\alpha,\beta )$$ leads to a formula for $$\psi_2(\alpha , \beta )=E\exp (\alpha Z(T_2)-\beta T_2)$$. Moreover, if $$T=\min (T_1, T_2)$$, then a formula for $$\psi_1(\alpha,\beta )=E\exp (\alpha Z(T)- \beta T)$$ is also given in terms of $$\psi_1(\alpha,\beta )$$ and $$\psi_2(\alpha ,\beta )$$. Our results provide quite direct and simple derivations of several results due to Taylor and others. Many of the isolated examples with detail theories of their own are unified by the given generalization. A number of examples are discussed.

##### MSC:
 62L10 Sequential statistical analysis 60G40 Stopping times; optimal stopping problems; gambling theory
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