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 ${\bar{D}}_{t,s}^{\ast }$ 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 $x\to \mathrm{\infty }$ of the tail probabilities $\mathbb{P}({\bar{D}}_t^{\ast }<x)$ 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.