10 research outputs found
General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients
The main purpose of this paper is to discuss detailed the stochastic LQ
control problem with random coefficients where the linear system is a
multidimensional stochastic differential equation driven by a multidimensional
Brownian motion and a Poisson random martingale measure. In the paper, we will
establish the connections of the multidimensional Backward stochastic Riccati
equation with jumps (BSRDEJ in short form) to the stochastic LQ problem and to
the associated Hamilton systems. By the connections, we show the optimal
control have the state feedback representation. Moreover, we will show the
existence and uniqueness result of the multidimensional BSRDEJ for the case
where the generator is bounded linear dependence with respect to the unknowns
martingale term
Dynamic exponential utility indifference valuation
We study the dynamics of the exponential utility indifference value process
C(B;\alpha) for a contingent claim B in a semimartingale model with a general
continuous filtration. We prove that C(B;\alpha) is (the first component of)
the unique solution of a backward stochastic differential equation with a
quadratic generator and obtain BMO estimates for the components of this
solution. This allows us to prove several new results about C_t(B;\alpha). We
obtain continuity in B and local Lipschitz-continuity in the risk aversion
\alpha, uniformly in t, and we extend earlier results on the asymptotic
behavior as \alpha\searrow0 or \alpha\nearrow\infty to our general setting.
Moreover, we also prove convergence of the corresponding hedging strategies.Comment: Published at http://dx.doi.org/10.1214/105051605000000395 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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
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