868 research outputs found
Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions
We study periodic homogenization problems for second-order pde in half-space
type domains with Neumann boundary conditions. In particular, we are interested
in "singular problems" for which it is necessary to determine both the
homogenized equation and boundary conditions. We provide new results for fully
nonlinear equations and boundary conditions. Our results extend previous work
of Tanaka in the linear, periodic setting in half-spaces parallel to the axes
of the periodicity, and of Arisawa in a rather restrictive nonlinear periodic
framework. The key step in our analysis is the study of associated ergodic
problems in domains with similar structure
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
Large Deviations and Importance Sampling for Systems of Slow-Fast Motion
In this paper we develop the large deviations principle and a rigorous
mathematical framework for asymptotically efficient importance sampling schemes
for general, fully dependent systems of stochastic differential equations of
slow and fast motion with small noise in the slow component. We assume
periodicity with respect to the fast component. Depending on the interaction of
the fast scale with the smallness of the noise, we get different behavior. We
examine how one range of interaction differs from the other one both for the
large deviations and for the importance sampling. We use the large deviations
results to identify asymptotically optimal importance sampling schemes in each
case. Standard Monte Carlo schemes perform poorly in the small noise limit. In
the presence of multiscale aspects one faces additional difficulties and
straightforward adaptation of importance sampling schemes for standard small
noise diffusions will not produce efficient schemes. It turns out that one has
to consider the so called cell problem from the homogenization theory for
Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality.
We use stochastic control arguments.Comment: More detailed proofs. Differences from the published version are
editorial and typographica
Large deviations for some fast stochastic volatility models by viscosity methods
We consider the short time behaviour of stochastic systems affected by a
stochastic volatility evolving at a faster time scale. We study the asymptotics
of a logarithmic functional of the process by methods of the theory of
homogenisation and singular perturbations for fully nonlinear PDEs. We point
out three regimes depending on how fast the volatility oscillates relative to
the horizon length. We prove a large deviation principle for each regime and
apply it to the asymptotics of option prices near maturity
On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary
We derive the long time asymptotic of solutions to an evolutive
Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with
ergodic problems recently studied in \cite{bcr}. Our main assumption is an
appropriate degeneracy condition on the operator at the boundary. This
condition is related to the characteristic boundary points for linear operators
as well as to the irrelevant points for the generalized Dirichlet problem, and
implies in particular that no boundary datum has to be imposed. We prove that
there exists a constant such that the solutions of the evolutive problem
converge uniformly, in the reference frame moving with constant velocity ,
to a unique steady state solving a suitable ergodic problem.Comment: 12p
Liouville properties and critical value of fully nonlinear elliptic operators
We prove some Liouville properties for sub- and supersolutions of fully
nonlinear degenerate elliptic equations in the whole space. Our assumptions
allow the coefficients of the first order terms to be large at infinity,
provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We
give two applications. The first is a stabilization property for large times of
solutions to fully nonlinear parabolic equations. The second is the solvability
of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique
critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation
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