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- W2951339070 abstract "For a given L'{e}vy process $X=(X_t)_{tinmathbb{R}_+}$ and for fixed $sin mathbb{R}_{+}cup{infty}$ and $tinmathbb{R}_+$ we analyse the {it future drawdown extremes} that are defined as follows: begin{eqnarray*} overline D^*_{t,s} = sup_{0leq uleq t} inf_{uleq w < t+s}(X_w-X_u), qquadqquad underline D^*_{t,s} = inf_{0leq uleq t} inf_{uleq w < t+s}(X_w-X_u). end{eqnarray*} The path-functionals $overline D^*_{t,s}$ and $underline D^*_{t,s}$ are of interest in various areas of application, including financial mathematics and queueing theory. In the case that $X$ has a strictly positive mean, we find the exact asymptotic decay as $xtoinfty$ of the tail probabilities $mathbb{P}(overline D^*_{t}<x)$ and $mathbb{P}(underline D^*_t<x)$ of $overline D^*_{t}=lim_{stoinfty}overline D^*_{t,s}$ and $underline D^*_{t} = lim_{stoinfty}underline D^*_{t,s}$ both when the jumps satisfy the Cram'er assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the L'{e}vy process $X$ are of single sign and $X$ is not subordinator, we identify the one-dimensional distributions in terms of the scale function of $X$. By way of example, we derive explicit results for the Black-Scholes-Samuelson model." @default.
- W2951339070 created "2019-06-27" @default.
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- W2951339070 date "2014-09-12" @default.
- W2951339070 modified "2023-09-25" @default.
- W2951339070 title "On Future Drawdowns of L'evy processes" @default.
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- W2951339070 doi "https://doi.org/10.48550/arxiv.1409.3780" @default.
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