On Future Drawdowns of Lévy processes

Abstract

For a given Lévy 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 ${\overline{D}}_{t,s}^{\ast }$ and ${\underline{D}}_{t,s}^{\ast }$ 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 $\mathbb{P}({\overline{D}}_t^{\ast }<x)$ and $\mathbb{P}({\underline{D}}_t^{\ast }<x)$ of ${\overline{D}}_t^{\ast }={\lim }_{s\to \infty }{\overline{D}}_{t,s}^{\ast }$ and ${\underline{D}}_t^{\ast }={\lim }_{s\to \infty }{\underline{D}}_{t,s}^{\ast }$ both when the jumps satisfy the Cramér assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the Lévy 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.

References in corpus (1)

• On magnitude, asymptotics and duration of drawdowns for Lévy models

Cited by in corpus (1)

• The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends