755 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
Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations
The paper reminds the basic ideas of stochastic calculus via regularizations
in Banach spaces and its applications to the study of strict solutions of
Kolmogorov path dependent equations associated with "windows" of diffusion
processes. One makes the link between the Banach space approach and the so
called functional stochastic calculus. When no strict solutions are available
one describes the notion of strong-viscosity solution which alternative (in
infinite dimension) to the classical notion of viscosity solution.Comment: arXiv admin note: text overlap with arXiv:1401.503
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
Structured populations with distributed recruitment: from PDE to delay formulation
In this work first we consider a physiologically structured population model
with a distributed recruitment process. That is, our model allows newly
recruited individuals to enter the population at all possible individual
states, in principle. The model can be naturally formulated as a first order
partial integro-differential equation, and it has been studied extensively. In
particular, it is well-posed on the biologically relevant state space of
Lebesgue integrable functions. We also formulate a delayed integral equation
(renewal equation) for the distributed birth rate of the population. We aim to
illustrate the connection between the partial integro-differential and the
delayed integral equation formulation of the model utilising a recent spectral
theoretic result. In particular, we consider the equivalence of the steady
state problems in the two different formulations, which then leads us to
characterise irreducibility of the semigroup governing the linear partial
integro-differential equation. Furthermore, using the method of
characteristics, we investigate the connection between the time dependent
problems. In particular, we prove that any (non-negative) solution of the
delayed integral equation determines a (non-negative) solution of the partial
differential equation and vice versa. The results obtained for the particular
distributed states at birth model then lead us to present some very general
results, which establish the equivalence between a general class of partial
differential and delay equation, modelling physiologically structured
populations.Comment: 28 pages, to appear in Mathematical Methods in the Applied Science
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