Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations

We propose a novel non-compact, positivity-preserving scheme for linear non-divergence form parabolic equations. Based on the Feynman-Kac formula, the solution is expressed as a conditional expectation of an associated diffusion process. Instead of using compact Markov chain approximations, we employ a wide stencil scheme to approximate the conditional expectation, ensuring consistency and positivity preservation. This method is effective for anisotropic diffusion with mixed derivatives, where classical schemes often fail unless the covariance matrix is diagonally dominated. A key feature of our framework is its robust treatment of boundary conditions, which avoids the accuracy loss commonly encountered in BZ and semi-Lagrangian schemes.

This paper establishes the existence, uniqueness, and global $C^{1,β}$ regularity of positive classical solutions to a class of quasilinear Hamilton--Jacobi--Bellman (HJB) equations with Dirichlet boundary conditions on bounded convex domains. The core technical contribution is a constructive existence proof based on a weighted linear monotone iteration scheme. This scheme's stability and convergence are rigorously established through the construction of adaptive sub- and super-solutions leveraging the torsion function of the domain. Additionally, we provide a complete probabilistic derivation of the quasilinear PDE from the framework of controlled Itô diffusions, formally bridging the gap between stochastic optimal control theory and elliptic regularity analysis. Our results extend beyond the classical quadratic cost regime to the wider class of sub-quadratic growth source terms. Finally, we demonstrate the utility of this theoretical framework through high-precision numerical impleme …

We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in t …

In a previous work (Akian, Fodjo, 2016), we introduced a lower complexity probabilistic max-plus numerical method for solving fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. This method was based on the idempotent expansion properties obtained by McEneaney, Kaise and Han (2011) and on the numerical probabilistic method proposed by Fahim, Touzi and Warin (2011) for solving some fully nonlinear parabolic partial differential equations. A difficulty of the algorithm of Fahim, Touzi and Warin is in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence of the algorithm. Here, we propose a new "probabilistic scheme" which is monotone under rather weak assumptions, including the case of strongly elliptic PDE with bounded coefficients. This allows us to apply our probabilistic max-plus method in mo …

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.