40 research outputs found
A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular L\'evy measures
We introduce a Monte Carlo scheme for fully nonlinear parabolic nonlocal
PDE's whose nonlinearity in of Hamilton-Jacobi-Bellman-Isaacs (HJBI for short).
We avoid the difficulties of infinite L\'evy measure by truncation of the
L\'evy integral. The first result provides the convergence of the scheme for
general parabolic nonlinearities. The second result provides bounds on the rate
of convergence for concave (or equivalently convex) nonlinearities. For both
results, it is crucial to choose truncation of the infinite L\'evy measure
appropriately dependent on the time discretization. We also introduce a Monte
Carlo Quadrature method to approximate the nonlocal term in the HJBI
nonlinearity.Comment: Keywords: Viscosity solution, nonlocal PDE, Monte Carlo approximatio
Approximation schemes for mixed optimal stopping and control problems with nonlinear expectations and jumps
We propose a class of numerical schemes for mixed optimal stopping and
control of processes with infinite activity jumps and where the objective is
evaluated by a nonlinear expectation. Exploiting an approximation by switching
systems, piecewise constant policy timestepping reduces the problem to nonlocal
semi-linear equations with different control parameters, uncoupled over
individual time steps, which we solve by fully implicit monotone approximations
to the controlled diffusion and the nonlocal term, and specifically the
Lax-Friedrichs scheme for the nonlinearity in the gradient. We establish a
comparison principle for the switching system and demonstrate the convergence
of the schemes, which subsequently gives a constructive proof for the existence
of a solution to the switching system. Numerical experiments are presented for
a recursive utility maximization problem to demonstrate the effectiveness of
the new schemes
A penalty scheme for monotone systems with interconnected obstacles: convergence and error estimates
We present a novel penalty approach for a class of quasi-variational
inequalities (QVIs) involving monotone systems and interconnected obstacles. We
show that for any given positive switching cost, the solutions of the penalized
equations converge monotonically to those of the QVIs. We estimate the
penalization errors and are able to deduce that the optimal switching regions
are constructed exactly. We further demonstrate that as the switching cost
tends to zero, the QVI degenerates into an equation of HJB type, which is
approximated by the penalized equation at the same order (up to a log factor)
as that for positive switching cost. Numerical experiments on optimal switching
problems are presented to illustrate the theoretical results and to demonstrate
the effectiveness of the method.Comment: Accepted for publication (in this revised form) in SIAM Journal on
Numerical Analysi
Precise Error Bounds for Numerical Approximations of Fractional HJB Equations
We prove precise rates of convergence for monotone approximation schemes of
fractional and nonlocal Hamilton-Jacobi-Bellman (HJB) equations. We consider
diffusion corrected difference-quadrature schemes from the literature and new
approximations based on powers of discrete Laplacians, approximations which are
(formally) fractional order and 2nd order methods. It is well-known in
numerical analysis that convergence rates depend on the regularity of
solutions, and here we consider cases with varying solution regularity: (i)
Strongly degenerate problems with Lipschitz solutions, and (ii) weakly
non-degenerate problems where we show that solutions have bounded fractional
derivatives of order between 1 and 2. Our main results are optimal error
estimates with convergence rates that capture precisely both the fractional
order of the schemes and the fractional regularity of the solutions. For
strongly degenerate equations, these rates improve earlier results. For weakly
non-degenerate problems of order greater than one, the results are new. Here we
show improved rates compared to the strongly degenerate case, rates that are
always better than 1/2
Value iteration convergence of ε-monotone schemes for stationary Hamilton-Jacobi equations
International audienceWe present an abstract convergence result for the xed point approximation of stationary Hamilton{Jacobi equations. The basic assumptions on the discrete operator are invariance with respect to the addition of constants, "-monotonicity and consistency. The result can be applied to various high-order approximation schemes which are illustrated in the paper. Several applications to Hamilton{Jacobi equations and numerical tests are presented
Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem
This paper is devoted to establish continuous dependence estimates for the
ergodic problem for Bellman operators (namely, estimates of (v_1-v_2) where v_1
and v_2 solve two equations with different coefficients). We shall obtain an
estimate of ||v_1-v_2||_\infty with an explicit dependence on the
L^\infty-distance between the coefficients and an explicit characterization of
the constants and also, under some regularity conditions, an estimate of
||v_1-v_2||_{C^2(\R^n)}.
Afterwards, the former result will be crucial in the estimate of the rate of
convergence for the homogenization of Bellman equations. In some regular cases,
we shall obtain the same rate of convergence established in the monographs
[11,26] for regular linear problems