On Future Drawdowns of L\'evy processes

For a given L\'{e}vy process $X=(X_t)_{t\in\mathbb{R}_+}$ and for fixed $s\in \mathbb{R}_{+}\cup\{\infty\}$ and $t\in\mathbb{R}_+$ we analyse the {\it future drawdown extremes} that are defined as follows: \begin{eqnarray*} \overline D^*_{t,s} = \sup_{0\leq u\leq t} \inf_{u\leq w < t+s}(X_w-X_u), \qquad\qquad \underline D^*_{t,s} = \inf_{0\leq u\leq t} \inf_{u\leq 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 $x\to\infty$ of the tail probabilities $\mathbb{P}(\overline D^*_{t}<x)$ and $\mathbb{P}(\underline D^*_t<x)$ of $\overline D^*_{t}=\lim_{s\to\infty}\overline D^*_{t,s}$ and $\underline D^*_{t} = \lim_{s\to\infty}\underline D^*_{t,s}$ 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.