57 research outputs found
Semi-Lagrangian schemes for linear and fully non-linear Hamilton-Jacobi-Bellman equations
We consider the numerical solution of Hamilton-Jacobi-Bellman equations
arising in stochastic control theory. We introduce a class of monotone
approximation schemes relying on monotone interpolation. These schemes converge
under very weak assumptions, including the case of arbitrary degenerate
diffusions. Besides providing a unifying framework that includes several known
first order accurate schemes, stability and convergence results are given,
along with two different robust error estimates. Finally, the method is applied
to a super-replication problem from finance.Comment: to appear in the proceedings of HYP201
Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations
We obtain non-symmetric upper and lower bounds on the rate of convergence of
general monotone approximation/numerical schemes for parabolic Hamilton Jacobi
Bellman Equations by introducing a new notion of consistency. We apply our
general results to various schemes including finite difference schemes,
splitting methods and the classical approximation by piecewise constant
controls
Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type
We study the uniqueness, existence, and properties of bounded distributional
solutions of the initial value problem problem for the anomalous diffusion
equation . Here
can be any nonlocal symmetric degenerate elliptic operator including the
fractional Laplacian and numerical discretizations of this operator. The
function is only assumed to be continuous
and nondecreasing. The class of equations include nonlocal (generalized) porous
medium equations, fast diffusion equations, and Stefan problems. In addition to
very general uniqueness and existence results, we obtain -contraction and
a priori estimates. We also study local limits, continuous dependence, and
properties and convergence of a numerical approximation of our equations.Comment: To appear in "Advances in Mathematics
On distributional solutions of local and nonlocal problems of porous medium type
We present a theory of well-posedness and a priori estimates for bounded
distributional (or very weak) solutions of
where is merely continuous and
nondecreasing and is the generator of a general
symmetric L\'evy process. This means that can have
both local and nonlocal parts like e.g.
. New uniqueness results
for bounded distributional solutions of this problem and the corresponding
elliptic equation are presented and proven. A key role is played by a new
Liouville type result for . Existence and a priori
estimates are deduced from a numerical approximation, and energy type estimates
are also obtained.Comment: 6 pages. Minor revision. Added details to Step 2 of the proof of
Theorem 3.
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
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