A procedure for the numerical approximation of high-dimensional Hamilton-JacobiBellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is kn...

We show the existence of a local solution to a Hamilton–Jacobi–Bellman (HJB) PDE around a critical point where the corresponding Hamiltonian ODE is not hyperbolic, i.e., it has eigenvalues on the imaginary axis. Such problems arise in nonlinear regulation, disturbance rejection, gain scheduling, and linear parameter varying control. The proof is based on an extension of the center manifold theo...

The optimal trade execution problem is formulated in terms of a mean-variance tradeoff, as seen at the initial time. The mean-variance problem can be embedded in a Linear-Quadratic (LQ) optimal stochastic control problem, A semi-Lagrangian scheme is used to solve the resulting non-linear Hamilton Jacobi Bellman (HJB) PDE. This method is essentially independent of the form for the price impact f...

We solve the optimal asset allocation problem using a mean variance approach. The original mean variance optimization problem can be embedded into a class of auxiliary stochastic LinearQuadratic (LQ) problems using the method in (Zhou and Li, 2000; Li and Ng, 2000). We use a finite difference method with fully implicit timestepping to solve the resulting non-linear Hamilton-Jacobi-Bellman (HJB)...

1 We present efficient partial differential equation (PDE) methods for continuous time mean2 variance portfolio allocation problems when the underlying risky asset follows a stochastic 3 volatility process. The standard formulation for mean variance optimal portfolio allocation 4 problems gives rise to a two-dimensional non-linear Hamilton-Jacobi-Bellman (HJB) PDE. We 5 use a wide stencil metho...

To sidestep the curse of dimensionality when computing solutions to Hamilton-Jacobi-Bellman partial differential equations (HJB PDE), we propose an algorithm that leverages a neural network to approximate the value function. We show that our final approximation of the value function generates near optimal controls which are guaranteed to successfully drive the system to a target state. Our fram...

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimizing a cost functional, while satisfying hard constraints on the input. For linear systems the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems the “shape” of the optima...

Abstract. In this article we extend earlier work on the jump-diffusion risk-sensitive asset management problem in a factor model [SIAM J. Fin. Math. 2 (2011) 22-54] by allowing jumps in both the factor process and the asset prices, as well as stochastic volatility and investment constraints. In this case, the HJB equation is a partial integro-differential equation (PIDE). We are able to show th...

This paper is concerned with C0 (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and Hamilton–Jacobi–Bellman (HJB) Cordes coefficients. Motivated by Miranda–Talenti estimate, a discrete analog proved once space on $$(n-1)$$ -dimensional subsimplex (face) $$C^1$$ $$(n-2)$$ subsimplex. The main novelty non-standard methods to introduce an interi...