207 research outputs found
Solvability conditions for indefinite linear quadratic optimal stochastic control problems and associated stochastic Riccati equations
A linear quadratic optimal stochastic control problem with random
coefficients and indefinite state/control weight costs is usually linked to an
indefinite stochastic Riccati equation (SRE) which is a matrix-valued quadratic
backward stochastic differential equation along with an algebraic constraint
involving the unknown. Either the optimal control problem or the SRE is
solvable only if the given data satisfy a certain structure condition that has
yet to be precisely defined. In this paper, by introducing a notion of
subsolution for the SRE, we derive several novel sufficient conditions for the
existence and uniqueness of the solution to the SRE and for the solvability of
the associated optimal stochastic control problem.Comment: 17 page
On solvability of an indefinite Riccati equation
This note concerns a class of matrix Riccati equations associated with
stochastic linear-quadratic optimal control problems with indefinite state and
control weighting costs. A novel sufficient condition of solvability of such
equations is derived, based on a monotonicity property of a newly defined set.
Such a set is used to describe a family of solvable equations.Comment: 11 page
Open-Loop and Closed-Loop Solvabilities for Stochastic Linear Quadratic Optimal Control Problems
This paper is concerned with a stochastic linear quadratic (LQ, for short)
optimal control problem. The notions of open-loop and closed-loop solvabilities
are introduced. A simple example shows that these two solvabilities are
different. Closed-loop solvability is established by means of solvability of
the corresponding Riccati equation, which is implied by the uniform convexity
of the quadratic cost functional. Conditions ensuring the convexity of the cost
functional are discussed, including the issue that how negative the control
weighting matrix-valued function R(s) can be. Finiteness of the LQ problem is
characterized by the convergence of the solutions to a family of Riccati
equations. Then, a minimizing sequence, whose convergence is equivalent to the
open-loop solvability of the problem, is constructed. Finally, an illustrative
example is presented.Comment: 40 page
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