4,857 research outputs found
A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps
We consider a stochastic functional delay differential equation, namely an
equation whose evolution depends on its past history as well as on its present
state, driven by a pure diffusive component plus a pure jump Poisson
compensated measure. We lift the problem in the infinite dimensional space of
square integrable Lebesgue functions in order to show that its solution is an
valued Markov process whose uniqueness can be shown under standard
assumptions of locally Lipschitzianity and linear growth for the coefficients.
Coupling the aforementioned equation with a standard backward differential
equation, and deriving some ad hoc results concerning the Malliavin derivative
for systems with memory, we are able to derive a non--linear Feynman--Kac
representation theorem under mild assumptions of differentiability
Stochastic Optimal Control with Delay in the Control: solution through partial smoothing
Stochastic optimal control problems governed by delay equations with delay in
the control are usually more difficult to study than the the ones when the
delay appears only in the state. This is particularly true when we look at the
associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even in the
simplified setting (introduced first by Vinter and Kwong for the deterministic
case) the HJB equation is an infinite dimensional second order semilinear
Partial Differential Equation (PDE) that does not satisfy the so-called
"structure condition" which substantially means that "the noise enters the
system with the control." The absence of such condition, together with the lack
of smoothing properties which is a common feature of problems with delay,
prevents the use of the known techniques (based on Backward Stochastic
Differential Equations (BSDEs) or on the smoothing properties of the linear
part) to prove the existence of regular solutions of this HJB equation and so
no results on this direction have been proved till now.
In this paper we provide a result on existence of regular solutions of such
kind of HJB equations and we use it to solve completely the corresponding
control problem finding optimal feedback controls also in the more difficult
case of pointwise delay. The main tool used is a partial smoothing property
that we prove for the transition semigroup associated to the uncontrolled
problem. Such results holds for a specific class of equations and data which
arises naturally in many applied problems
An infinite-dimensional approach to path-dependent Kolmogorov equations
In this paper, a Banach space framework is introduced in order to deal with
finite-dimensional path-dependent stochastic differential equations. A version
of Kolmogorov backward equation is formulated and solved both in the space of
paths and in the space of continuous paths using the associated
stochastic differential equation, thus establishing a relation between
path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is
shown how to establish a connection between such Kolmogorov equation and the
analogue finite-dimensional equation that can be formulated in terms of the
path-dependent derivatives recently introduced by Dupire, Cont and Fourni\'{e}.Comment: Published at http://dx.doi.org/10.1214/15-AOP1031 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Recommendations and illustrations for the evaluation of photonic random number generators
The never-ending quest to improve the security of digital information
combined with recent improvements in hardware technology has caused the field
of random number generation to undergo a fundamental shift from relying solely
on pseudo-random algorithms to employing optical entropy sources. Despite these
significant advances on the hardware side, commonly used statistical measures
and evaluation practices remain ill-suited to understand or quantify the
optical entropy that underlies physical random number generation. We review the
state of the art in the evaluation of optical random number generation and
recommend a new paradigm: quantifying entropy generation and understanding the
physical limits of the optical sources of randomness. In order to do this, we
advocate for the separation of the physical entropy source from deterministic
post-processing in the evaluation of random number generators and for the
explicit consideration of the impact of the measurement and digitization
process on the rate of entropy production. We present the Cohen-Procaccia
estimate of the entropy rate as one way to do this. In order
to provide an illustration of our recommendations, we apply the Cohen-Procaccia
estimate as well as the entropy estimates from the new NIST draft standards for
physical random number generators to evaluate and compare three common optical
entropy sources: single photon time-of-arrival detection, chaotic lasers, and
amplified spontaneous emission
Geometric and projection effects in Kramers-Moyal analysis
Kramers-Moyal coefficients provide a simple and easily visualized method with
which to analyze stochastic time series, particularly nonlinear ones. One
mechanism that can affect the estimation of the coefficients is geometric
projection effects. For some biologically-inspired examples, these effects are
predicted and explored with a non-stochastic projection operator method, and
compared with direct numerical simulation of the systems' Langevin equations.
General features and characteristics are identified, and the utility of the
Kramers-Moyal method discussed. Projections of a system are in general
non-Markovian, but here the Kramers-Moyal method remains useful, and in any
case the primary examples considered are found to be close to Markovian.Comment: Submitted to Phys. Rev.
Mild solutions of semilinear elliptic equations in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable
sense) for a class of semilinear elliptic equations in Hilbert spaces. The
notion of regularity is based on the concept of -derivative, which is
introduced and discussed. A result of existence and uniqueness of solutions is
stated and proved under the assumption that the transition semigroup associated
to the linear part of the equation has a smoothing property, that is, it maps
continuous functions into -differentiable ones. The validity of this
smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck
transition semigroup and for the case of invertible diffusion coefficient
covering cases not previously addressed by the literature. It is shown that the
results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite
horizon optimal stochastic control problems in infinite dimension and that, in
particular, they cover examples of optimal boundary control of the heat
equation that were not treatable with the approaches developed in the
literature up to now
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