7,374 research outputs found
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
Discrete time McKean-Vlasov control problem: a dynamic programming approach
We consider the stochastic optimal control problem of nonlinear mean-field
systems in discrete time. We reformulate the problem into a deterministic
control problem with marginal distribution as controlled state variable, and
prove that dynamic programming principle holds in its general form. We apply
our method for solving explicitly the mean-variance portfolio selection and the
multivariate linear-quadratic McKean-Vlasov control problem
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