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46
Markowitz’s Mean–Variance Portfolio Selection with Regime Switching: A Continuous Time Model,
 SIAM J. Control Optim.
, 2003
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The general structure of optimal investment and consumption with small transaction costs, preprint
, 2013
"... We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and timevarying but small transaction costs, we derive simple formal asymptotics for the opt ..."
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Cited by 11 (7 self)
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We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and timevarying but small transaction costs, we derive simple formal asymptotics for the optimal policy and welfare. These reveal the roles of the investors’ preferences as well as the market and cost dynamics, and also lead to a fully dynamic model for the implied trading volume. In frictionless models that can be solved in closed form, explicit formulas for the leadingorder corrections due to small transaction costs obtain.
Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation
"... We consider a stochastic linear–quadratic (LQ) problem with possible indefinite cost weighting matrices for the state and the control. An outstanding open problem is to identify an appropriate Riccatitype equation whose solvability is equivalent to the solvability of this possibly indefinite LQ pro ..."
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Cited by 10 (2 self)
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We consider a stochastic linear–quadratic (LQ) problem with possible indefinite cost weighting matrices for the state and the control. An outstanding open problem is to identify an appropriate Riccatitype equation whose solvability is equivalent to the solvability of this possibly indefinite LQ problem. In this paper we introduce a new type of differential Riccati equation, called the generalized (differential) Riccati equation, which in turn provides a complete solution to the indefinite LQ problem. Moreover, all the optimal feedback/openloop controls can be identified via the solution to this Riccati equation.
Timeinconsistent stochastic linear–quadratic control
 SIAM Journal on Control and Optimization
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Tracking a financial benchmark using a few assets. Operat
, 2006
"... We study the problem of tracking a financial benchmark — a continuously compounded growth rate or a stock market index — by dynamically managing a portfolio consisting of a small number of traded stocks in the market. In either case, we formulate the tracking problem as an instance of the stochastic ..."
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Cited by 6 (1 self)
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We study the problem of tracking a financial benchmark — a continuously compounded growth rate or a stock market index — by dynamically managing a portfolio consisting of a small number of traded stocks in the market. In either case, we formulate the tracking problem as an instance of the stochastic linear quadratic control (SLQ), involving indefinite cost matrices. As the SLQ formulation involves a discounted objective over an infinite horizon, we first address the issue of stabilizability. We then use semidefinite programming (SDP) as a computational tool to generate the optimal feedback control. We present numerical examples involving stocks traded at Hong Kong and New York Stock Exchanges, to illustrate the various features of the model and its performance.
Randomized portfolio selection with constraints
 Pacific Journal of Optimization
"... In this paper we propose to deal with the combinatorial difficulties in meanvariance portfolio selection, caused by various side constraints, by means of randomization. As examples of such side constraints, we consider in this paper the following two models. In the first model, an investor is inter ..."
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Cited by 4 (0 self)
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In this paper we propose to deal with the combinatorial difficulties in meanvariance portfolio selection, caused by various side constraints, by means of randomization. As examples of such side constraints, we consider in this paper the following two models. In the first model, an investor is interested in a small, compact portfolio, in the sense that it involves only a small number of securities. The second model explicitly requires that each security involved in the portfolio need to have a substantial presence if it is present at all, implicitly restricting the number of them given the budget constraints. These constraints are motivated by practical considerations in the face of management and informational costs in investment. By incorporating such side constraints, however, the meanvariance model becomes very hard to solve. We resort to the method of randomization for finding good approximation solutions. Extensive numerical experiments show that randomization is indeed a viable alternative for solving such hard investment models, for which the combinatorial complexity in the constraints makes it quite hopeless to find an exact solution, while good approximate solutions in fact already serve the purpose quite well given the approximative nature of the models.
Risksensitive control with HARA utility
 IEEE Transactions on Automatic Control
, 2001
"... Abstract—In this paper, a control methodology based on the hyperbolic absolute risk averse (HARA) utility function is presented as an alternative to the exponentialofanintegral approach to finding robust controllers. This work is inspired by the intuition that HARA controllers, while being robust ..."
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Cited by 2 (0 self)
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Abstract—In this paper, a control methodology based on the hyperbolic absolute risk averse (HARA) utility function is presented as an alternative to the exponentialofanintegral approach to finding robust controllers. This work is inspired by the intuition that HARA controllers, while being robust, may give better performance than exponential controllers in normal situations. The HARA problem is shown to be equivalent to a certain differential game, and the asymptotic properties of the HARA problem and this differential game are studied. As an example, a linearquadratic HARA problem is studied, where the problem of finding a robust HARA controller is proved to be equivalent to solving a standard linearquadratic problem for a system with a higher noise intensity. This reveals an interesting relationship between robustness and uncertainty. Index Terms—Differential games, HARA utility function, risksensitive control, upper/lower Isaacs equations, viscosity solutions. I.
STOCHASTIC LINEARQUADRATIC CONTROL WITH CONIC CONTROL CONSTRAINTS ON AN INFINITE TIME HORIZON
, 2004
"... This paper is concerned with a stochastic linearquadratic (LQ) control problem in the infinite time horizon where the control is constrained in a given, arbitrary closed cone, the cost weighting matrices are allowed to be indefinite, and the state is scalarvalued. First, the (meansquare, conic) ..."
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Cited by 2 (0 self)
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This paper is concerned with a stochastic linearquadratic (LQ) control problem in the infinite time horizon where the control is constrained in a given, arbitrary closed cone, the cost weighting matrices are allowed to be indefinite, and the state is scalarvalued. First, the (meansquare, conic) stabilizability of the system is defined, which is then characterized by a set of simple conditions involving linear matrix inequalities (LMIs). Next, the issue of wellposedness of the underlying optimal LQ control, which is necessitated by the indefiniteness of the problem, is addressed in great detail, and necessary and sufficient conditions of the wellposedness are presented. On the other hand, to address the LQ optimality two new algebraic equations à la Riccati, called extended algebraic Riccati equations (EAREs), along with the notion of their stabilizing solutions, are introduced for the first time. Optimal feedback control as well as the optimal value are explicitly derived in terms of the stabilizing solutions to the EAREs. Moreover, several cases when the stabilizing solutions do exist are discussed and algorithms of computing the solutions are presented. Finally, numerical examples are provided to illustrate the theoretical results established.