Optimal control of multiscale systems using reduced-order models

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"--rather than "first optimize, then reduce"--is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls

Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems

We study linear-quadratic stochastic optimal control problems with bilinear state dependence for which the underlying stochastic differential equation (SDE) consists of slow and fast degrees of freedom. We show that, in the same way in which the underlying dynamics can be well approximated by a reduced order effective dynamics in the time scale limit (using classical homogenziation results), the associated optimal expected cost converges in the time scale limit to an effective optimal cost. This entails that we can well approximate the stochastic optimal control for the whole system by the reduced order stochastic optimal control, which is clearly easier to solve because of lower dimensionality. The approach uses an equivalent formulation of the Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs (FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares Monte Carlo algorithm and show its applicability by a suitable numerical example

Analytical expansions for parabolic equations

We consider the Cauchy problem associated with a general parabolic partial differential equation in $d$ dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provide convergence results for arbitrarily large time intervals.Comment: 23 page

An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians

We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac and the Hopf-Lax formulae. We establish the convergence of the scheme and determine the convergence rate, combining Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation.Comment: 24 page