161 research outputs found
Time-Inconsistent Optimal Control Problems and the Equilibrium HJB Equation
A general time-inconsistent optimal control problem is considered for
stochastic differential equations with deterministic coefficients. Under
suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the
equilibrium value function of the problem. Well-posedness and some properties
of such an equation is studied, and time-consistent equilibrium strategies are
constructed. As special cases, the linear-quadratic problem and a generalized
Merton's portfolio problem are investigated.Comment: 51 page
Forward-Backward Evolution Equations and Applications
Well-posedness is studied for a special system of two-point boundary value
problem for evolution equations which is called a forward-backward evolution
equation (FBEE, for short). Two approaches are introduced: A decoupling method
with some brief discussions, and a method of continuation with some substantial
discussions. For the latter, we have introduced Lyapunov operators for FBEEs,
whose existence leads to some uniform a priori estimates for the mild solutions
of FBEEs, which will be sufficient for the well-posedness. For some special
cases, Lyapunov operators are constructed. Also, from some given Lyapunov
operators, the corresponding solvable FBEEs are identified.Comment: 52 page
A Deterministic Affine-Quadratic Optimal Control Problem
A Deterministic affine quadratic optimal control problem is considered. Due
to the nature of the problem, optimal controls exist under some very mild
conditions. Further, it is shown that under some assumptions, the value
function is differentiable and therefore satisfies the corresponding
Hamilton-Jacobi-Bellman equation in the classical sense. Moreover, the
so-called quasi-Riccati equation is derived and any optimal control admits a
state feedback representation.Comment: 30 page
Linear Quadratic Stochastic Differential Games: Open-Loop and Closed-Loop Saddle Points
In this paper, we consider a linear quadratic stochastic two-person zero-sum
differential game. The controls for both players are allowed to appear in both
drift and diffusion of the state equation. The weighting matrices in the
performance functional are not assumed to be definite/non-singular. A necessary
and sufficient condition for the existence of a closed-loop saddle point is
established in terms of the solvability of a Riccati differential equation with
certain regularity. It is possible that the closed-loop saddle point fails to
exist, and at the same time, the corresponding Riccati equation admits a
solution (which does not have needed regularity). Also, we will indicate that
the solution of the Riccati equation may be non-unique.Comment: 28 page
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