138 research outputs found
Sufficient stochastic maximum principle in a regime-switching diffusion model
We prove a sufficient stochastic maximum principle for the optimal control of
a regime-switching diffusion model. We show the connection to dynamic
programming and we apply the result to a quadratic loss minimization problem,
which can be used to solve a mean-variance portfolio selection problem
The joint law of the extrema, final value and signature of a stopped random walk
A complete characterization of the possible joint distributions of the
maximum and terminal value of uniformly integrable martingale has been known
for some time, and the aim of this paper is to establish a similar
characterization for continuous martingales of the joint law of the minimum,
final value, and maximum, along with the direction of the final excursion. We
solve this problem completely for the discrete analogue, that of a simple
symmetric random walk stopped at some almost-surely finite stopping time. This
characterization leads to robust hedging strategies for derivatives whose value
depends on the maximum, minimum and final values of the underlying asset
On the flow-level stability of data networks without congestion control: the case of linear networks and upstream trees
In this paper, flow models of networks without congestion control are
considered. Users generate data transfers according to some Poisson processes
and transmit corresponding packet at a fixed rate equal to their access rate
until the entire document is received at the destination; some erasure codes
are used to make the transmission robust to packet losses. We study the
stability of the stochastic process representing the number of active flows in
two particular cases: linear networks and upstream trees. For the case of
linear networks, we notably use fluid limits and an interesting phenomenon of
"time scale separation" occurs. Bounds on the stability region of linear
networks are given. For the case of upstream trees, underlying monotonic
properties are used. Finally, the asymptotic stability of those processes is
analyzed when the access rate of the users decreases to 0. An appropriate
scaling is introduced and used to prove that the stability region of those
networks is asymptotically maximized
Travelling Randomly on the Poincar\'e Half-Plane with a Pythagorean Compass
A random motion on the Poincar\'e half-plane is studied. A particle runs on
the geodesic lines changing direction at Poisson-paced times. The hyperbolic
distance is analyzed, also in the case where returns to the starting point are
admitted. The main results concern the mean hyperbolic distance (and also the
conditional mean distance) in all versions of the motion envisaged. Also an
analogous motion on orthogonal circles of the sphere is examined and the
evolution of the mean distance from the starting point is investigated
Local time and the pricing of time-dependent barrier options
A time-dependent double-barrier option is a derivative security that delivers
the terminal value at expiry if neither of the continuous
time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time
interval . Using a probabilistic approach we obtain a decomposition of
the barrier option price into the corresponding European option price minus the
barrier premium for a wide class of payoff functions , barrier functions
and linear diffusions . We show that the barrier
premium can be expressed as a sum of integrals along the barriers of
the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair
of functions solves a system of Volterra integral
equations of the first kind. We find a semi-analytic solution for this system
in the case of constant double barriers and briefly discus a numerical
algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic
On inversions and Doob -transforms of linear diffusions
Let be a regular linear diffusion whose state space is an open interval
. We consider a diffusion which probability law is
obtained as a Doob -transform of the law of , where is a positive
harmonic function for the infinitesimal generator of on . This is the
dual of with respect to where is the speed measure of
. Examples include the case where is conditioned to stay above
some fixed level. We provide a construction of as a deterministic
inversion of , time changed with some random clock. The study involves the
construction of some inversions which generalize the Euclidean inversions.
Brownian motion with drift and Bessel processes are considered in details.Comment: 19 page
Short-time Gibbsianness for Infinite-dimensional Diffusions with Space-Time Interaction
We consider a class of infinite-dimensional diffusions where the interaction
between the components is both spatial and temporal. We start the system from a
Gibbs measure with finite-range uniformly bounded interaction. Under suitable
conditions on the drift, we prove that there exists such that the
distribution at time is a Gibbs measure with absolutely summable
interaction. The main tool is a cluster expansion of both the initial
interaction and certain time-reversed Girsanov factors coming from the
dynamics
Stochastic B\"acklund transformations
How does one introduce randomness into a classical dynamical system in order
to produce something which is related to the `corresponding' quantum system? We
consider this question from a probabilistic point of view, in the context of
some integrable Hamiltonian systems
Stochastic resonance-free multiple time-step algorithm for molecular dynamics with very large time steps
Molecular dynamics is one of the most commonly used approaches for studying
the dynamics and statistical distributions of many physical, chemical, and
biological systems using atomistic or coarse-grained models. It is often the
case, however, that the interparticle forces drive motion on many time scales,
and the efficiency of a calculation is limited by the choice of time step,
which must be sufficiently small that the fastest force components are
accurately integrated. Multiple time-stepping algorithms partially alleviate
this inefficiency by assigning to each time scale an appropriately chosen
step-size. However, such approaches are limited by resonance phenomena, wherein
motion on the fastest time scales limits the step sizes associated with slower
time scales. In atomistic models of biomolecular systems, for example,
resonances limit the largest time step to around 5-6 fs. In this paper, we
introduce a set of stochastic isokinetic equations of motion that are shown to
be rigorously ergodic and that can be integrated using a multiple time-stepping
algorithm that can be easily implemented in existing molecular dynamics codes.
The technique is applied to a simple, illustrative problem and then to a more
realistic system, namely, a flexible water model. Using this approach outer
time steps as large as 100 fs are shown to be possible
Numerical Schemes for Rough Parabolic Equations
This paper is devoted to the study of numerical approximation schemes for a
class of parabolic equations on (0, 1) perturbed by a non-linear rough signal.
It is the continuation of [8, 7], where the existence and uniqueness of a
solution has been established. The approach combines rough paths methods with
standard considerations on discretizing stochastic PDEs. The results apply to a
geometric 2-rough path, which covers the case of the multidimensional
fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201
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