78 research outputs found
A class of recursive optimal stopping problems with applications to stock trading
In this paper we introduce and solve a class of optimal stopping problems of
recursive type. In particular, the stopping payoff depends directly on the
value function of the problem itself. In a multi-dimensional Markovian setting
we show that the problem is well posed, in the sense that the value is indeed
the unique solution to a fixed point problem in a suitable space of continuous
functions, and an optimal stopping time exists. We then apply our class of
problems to a model for stock trading in two different market venues and we
determine the optimal stopping rule in that case.Comment: 35 pages, 2 figures. In this version, we provide a general analysis
of a class of recursive optimal stopping problems with both finite-time and
infinite-time horizon. We also discuss other application
Dynkin games with incomplete and asymmetric information
We study the value and the optimal strategies for a two-player zero-sum
optimal stopping game with incomplete and asymmetric information. In our
Bayesian set-up, the drift of the underlying diffusion process is unknown to
one player (incomplete information feature), but known to the other one
(asymmetric information feature). We formulate the problem and reduce it to a
fully Markovian setup where the uninformed player optimises over stopping times
and the informed one uses randomised stopping times in order to hide their
informational advantage. Then we provide a general verification result which
allows us to find the value of the game and players' optimal strategies by
solving suitable quasi-variational inequalities with some non-standard
constraints. Finally, we study an example with linear payoffs, in which an
explicit solution of the corresponding quasi-variational inequalities can be
obtained.Comment: 31 pages, 5 figures, small changes in the terminology from game
theor
Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon
We consider optimal stopping problems with finite-time horizon and
state-dependent discounting. The underlying process is a one-dimensional linear
diffusion and the gain function is time-homogeneous and difference of two
convex functions. Under mild technical assumptions with local nature we prove
fine regularity properties of the optimal stopping boundary including its
continuity and strict monotonicity. The latter was never proven with
probabilistic arguments. We also show that atoms in the signed measure
associated with the second order spatial derivative of the gain function induce
geometric properties of the continuation/stopping set that cannot be observed
with smoother gain functions (we call them \emph{continuation bays} and
\emph{stopping spikes}). The value function is continuously differentiable in
time without any requirement on the smoothness of the gain function.Comment: 41 pages, 2 figures; added more details and fixed some technical
problems; main results remain unchange
A change of variable formula with applications to multi-dimensional optimal stopping problems
We derive a change of variable formula for functions
whose second order spatial
derivatives may explode and not be integrable in the neighbourhood of a surface
that splits the state
space into two sets and . The formula is tailored
for applications in problems of optimal stopping where it is generally very
hard to control the second derivatives of the value function near the optimal
stopping boundary. Differently to other existing papers on similar topics we
only require that the surface be monotonic in each variable and we formally
obtain the same expression as the classical It\^o's formula.Comment: 19 pages; added examples in Section
Dynamic programming principle for classical and singular stochastic control with discretionary stopping
We prove the dynamic programming principle (DPP) in a class of problems where
an agent controls a -dimensional diffusive dynamics via both classical and
singular controls and, moreover, is able to terminate the optimisation at a
time of her choosing, prior to a given maturity. The time-horizon of the
problem is random and it is the smallest between a fixed terminal time and the
first exit time of the state dynamics from a Borel set. We consider both the
cases in which the total available fuel for the singular control is either
bounded or unbounded. We build upon existing proofs of DPP and extend results
available in the traditional literature on singular control (e.g., Haussmann
and Suo, SIAM J. Control Optim., 33, 1995) by relaxing some key assumptions and
including the discretionary stopping feature. We also connect with more general
versions of the DPP (e.g., Bouchard and Touzi, SIAM J. Control Optim., 49,
2011) by showing in detail how our class of problems meets the abstract
requirements therein
A numerical scheme for stochastic differential equations with distributional drift
In this paper we present a scheme for the numerical solution of stochastic
differential equations (SDEs) with distributional drift. The approximating
process, obtained by the scheme, converges in law to the (virtual) solution of
the SDE in a general multi-dimensional setting. When we restrict our attention
to the case of a one-dimensional SDE we also obtain a rate of convergence in a
suitable -norm. Moreover, we implement our method in the one-dimensional
case, when the drift is obtained as the distributional derivative of a sample
path of a fractional Brownian motion. To the best of our knowledge this is the
first paper to study (and implement) numerical solutions of SDEs whose drift
cannot be expressed as a function of the state.Comment: 35 pages, 8 figure
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