989 research outputs found
Quasi-Self-Dual Exponential LĂ©vy Processes
The important application of semistatic hedging in financial markets naturally leads to the notion of quasi--self-dual processes. The focus of our study is to give new characterizations of quasi--self-duality. We analyze quasi--self-dual LĂ©vy driven markets which do not admit arbitrage opportunities and derive a set of equivalent conditions for the stochastic logarithm of quasi--self-dual martingale models. Since for nonvanishing order parameter two martingale properties have to be satisfied simultaneously, there is a nontrivial relation between the order and shift parameter representing carrying costs in financial applications. This leads to an equation containing an integral term which has to be inverted in applications. We first discuss several important properties of this equation and, for some well-known LĂ©vy-driven models, we derive a family of closed-form inversion formulae
Implicit renewal theory for exponential functionals of L\'evy processes
We establish a new functional relation for the probability density function
of the exponential functional of a L\'evy process, which allows to
significantly simplify the techniques commonly used in the study of these
random variables and hence provide quick proofs of known results, derive new
results, as well as sharpening known estimates for the distribution. We apply
this formula to provide another look to the Wiener-Hopf type factorisation for
exponential functionals obtained in a series of papers by Pardo, Patie and
Savov, derive new identities in law, and to describe the behaviour of the tail
distribution at infinity and of the distribution at zero in a rather large set
of situations
The hitting time of zero for a stable process
For any two-sided jumping -stable process, where , we
find an explicit identity for the law of the first hitting time of the origin.
This complements existing work in the symmetric case and the spectrally
one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008)
respectively. We appeal to the Lamperti-Kiu representation of
Chaumont-Pant\'i-Rivero (2011) for real-valued self-similar Markov processes.
Our main result follows by considering a vector-valued functional equation for
the Mellin transform of the integrated exponential Markov additive process in
the Lamperti-Kiu representation. We conclude our presentation with some
applications
A Free Boundary Characterisation of the Root Barrier for Markov Processes
We study the existence, optimality, and construction of non-randomised
stopping times that solve the Skorokhod embedding problem (SEP) for Markov
processes which satisfy a duality assumption. These stopping times are hitting
times of space-time subsets, so-called Root barriers. Our main result is,
besides the existence and optimality, a potential-theoretic characterisation of
this Root barrier as a free boundary. If the generator of the Markov process is
sufficiently regular, this reduces to an obstacle PDE that has the Root barrier
as free boundary and thereby generalises previous results from one-dimensional
diffusions to Markov processes. However, our characterisation always applies
and allows, at least in principle, to compute the Root barrier by dynamic
programming, even when the well-posedness of the informally associated obstacle
PDE is not clear. Finally, we demonstrate the flexibility of our method by
replacing time by an additive functional in Root's construction. Already for
multi-dimensional Brownian motion this leads to new class of constructive
solutions of (SEP).Comment: 31 pages, 14 figure
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