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On Future Drawdowns of L'evy processes | | E. J. Baurdoux
; Z. Palmowski
; M.R. Pistorius
; | | Date: |
12 Sep 2014 | | Abstract: | For a given stochastic process $X=(X_t)_{tin R_+}$ the {em future drawdown
process} $D^*=(D^*_{t,s})_{t,sin R_+}$ is defined by egin{equation*}
D^*_{t,s} = inf_{tleq u < t+s}(X_u-X_t). end{equation*} For fixed
non-negative $s$, two path-functionals that describe the fluctuations of
$D^*_{t,s}$ are the running supremum $(overline D^*_{t,s})_{tin R_+}$ and
running infimum $(underline D^*_{t,s})_{tin R_+}$ of $D^*_{cdot,s}$,
egin{eqnarray*} overline D^*_{t,s} = sup_{0leq uleq t} D^*_{u,s},
qquadqquad underline D^*_{t,s} = inf_{0leq uleq t} D^*_{u,s}.
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 $X$ is a L’{e}vy process
with strictly positive mean, we find the exact asymptotic decay as $x oinfty$
of the tail probabilities $P(overline D^*_{t}<x)$ and $P(underline D^*_t<x)$
of $overline D^*_{t}=lim_{s oinfty}overline D^*_{t,s}$ and $underline
D^*_{t} = lim_{s oinfty}underline D^*_{t,s}$ both when the jumps satisfy
the Cram’er assumption and in the heavy-tailed case. Furthermore, when the
jumps of the L’{e}vy process $X$ are of single sign, 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. | | Source: | arXiv, 1409.3780 | | Services: | Forum | Review | PDF | Favorites | |
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