1,015 research outputs found
Regularity of the Optimal Stopping Problem for Jump Diffusions
The value function of an optimal stopping problem for jump diffusions is
known to be a generalized solution of a variational inequality. Assuming that
the diffusion component of the process is nondegenerate and a mild assumption
on the singularity of the L\'{e}vy measure, this paper shows that the value
function of this optimal stopping problem on an unbounded domain with
finite/infinite variation jumps is in with . As a consequence, the smooth-fit property holds.Comment: To Appear in the SIAM Journal on Control and Optimizatio
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
On the regularity of American options with regime-switching uncertainty
We study the regularity of the stochastic representation of the solution of a
class of initial-boundary value problems related to a regime-switching
diffusion. This representation is related to the value function of a
finite-horizon optimal stopping problem such as the price of an American-style
option in finance. We show continuity and smoothness of the value function
using coupling and time-change techniques. As an application, we find the
minimal payoff scenario for the holder of an American-style option in the
presence of regime-switching uncertainty under the assumption that the
transition rates are known to lie within level-dependent compact sets.Comment: 22 pages, to appear in Stochastic Processes and their Application
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