Results 1  10
of
20
3 3Dto3D drawing systems
 IEEE Transactions on Automatic Control
, 2003
"... Abstract — We study a discretetime version of Markowitz’s meanvariance portfolio selection problem where the market parameters depend on the market mode (regime) that jumps among a finite number of states. The random regime switching is delineated by a finitestate Markov chain, based on which a d ..."
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Cited by 17 (5 self)
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Abstract — We study a discretetime version of Markowitz’s meanvariance portfolio selection problem where the market parameters depend on the market mode (regime) that jumps among a finite number of states. The random regime switching is delineated by a finitestate Markov chain, based on which a discretetime Markov modulated portfolio selection model is presented. Such models either arise from multiperiod portfolio selections or result from numerical solution of continuoustime problems. The natural connections between discretetime models and their continuoustime counterpart are revealed. Since the Markov chain frequently has a large state space, to reduce the complexity, an aggregated process with smaller state space is introduced and the underlying portfolio selection is formulated as a twotimescale problem. We prove that the process of interest yields a switching diffusion limit using weak convergence methods. Next, based on the optimal control of the limit process obtained from our recent work, we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality. Index Terms — Markowitz’s meanvariance portfolio selection, discretetime model, Markov chain, switching diffusion, linearquadratic problem, singular perturbation. I.
Quadratic hedging and meanvariance portfolio selection with random parameters in an incomplete market
 Math. Opers. Res., Vol 29, No
, 2004
"... in an incomplete market ..."
Dynamic meanvariance portfolio selection with noshorting constraints
 SIAM J. Control Optim
"... Abstract. This paper is concerned with meanvariance portfolio selection problems in continuoustime under the constraint that shortselling of stocks is prohibited. The problem is formulated as a stochastic optimal linearquadratic (LQ) control problem. However, this LQ problem is not a conventional ..."
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Cited by 6 (1 self)
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Abstract. This paper is concerned with meanvariance portfolio selection problems in continuoustime under the constraint that shortselling of stocks is prohibited. The problem is formulated as a stochastic optimal linearquadratic (LQ) control problem. However, this LQ problem is not a conventional one in that the control (portfolio) is constrained to take nonnegative values due to the noshorting restriction, and thereby the usual Riccati equation approach (involving a “completion of squares”) does not apply directly. In addition, the corresponding Hamilton–Jacobi–Bellman (HJB) equation inherently has no smooth solution. To tackle these difficulties, a continuous function is constructed via two Riccati equations, and then it is shown that this function is a viscosity solution to the HJB equation. Solving these Riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original meanvariance problem. An example illustrating these results is also presented.
Continuous time mean variance asset allocation: a time consistent strategy. Working
, 2009
"... We develop a numerical scheme for determining the optimal asset allocation strategy for timeconsistent, continuous time, mean variance optimization. Any type of constraint can be applied to the investment policy. The optimal policies for timeconsistent and precommitment strategies are compared. W ..."
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Cited by 4 (3 self)
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We develop a numerical scheme for determining the optimal asset allocation strategy for timeconsistent, continuous time, mean variance optimization. Any type of constraint can be applied to the investment policy. The optimal policies for timeconsistent and precommitment strategies are compared. When realistic constraints are applied, the efficient frontiers for the precommitment and timeconsistent strategies are similar, but the optimal investment strategies are quite different.
Dynamic meanvariance asset allocation
 Working Paper, London Business School
, 2007
"... Toronto and University of Warwick for helpful comments. All errors are our responsibility. ..."
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Cited by 4 (0 self)
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Toronto and University of Warwick for helpful comments. All errors are our responsibility.
Numerical solution of the HamiltonJacobiBellman formulation for continuous time mean variance asset allocation. Forthcoming
 in the Journal of Economic Dynamics and Control
, 2009
"... We solve the optimal asset allocation problem using a mean variance approach. The original mean variance optimization problem can be embedded into a class of auxiliary stochastic LinearQuadratic (LQ) problems using the method in (Zhou and Li, 2000; Li and Ng, 2000). We use a finite difference metho ..."
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Cited by 3 (3 self)
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We solve the optimal asset allocation problem using a mean variance approach. The original mean variance optimization problem can be embedded into a class of auxiliary stochastic LinearQuadratic (LQ) problems using the method in (Zhou and Li, 2000; Li and Ng, 2000). We use a finite difference method with fully implicit timestepping to solve the resulting nonlinear HamiltonJacobiBellman (HJB) PDE, and present the solutions in terms of an efficient frontier and an optimal asset allocation strategy. The numerical scheme satisfies sufficient conditions to ensure convergence to the viscosity solution of the HJB PDE. We handle various constraints on the optimal policy. Numerical tests indicate that realistic constraints can have a dramatic
Optimal Trade Execution: A Mean–QuadraticVariation Approach
, 2009
"... We propose the use of a mean–quadraticvariation criteria to determine an optimal trading strategy in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian Motio ..."
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Cited by 2 (0 self)
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We propose the use of a mean–quadraticvariation criteria to determine an optimal trading strategy in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian Motion (GBM). We also derive the HJB PDE assuming that the trading horizon is small and that the underlying process can be approximated by Arithmetic Brownian Motion (ABM). The exact solution of the ABM formulation is in fact identical to the priceindependent approximate optimal control for the meanvariance objective function in [2]. The GBM mean–quadraticvariation optimal trading strategy is in general a function of the asset price. However, for short term trading horizons, the control determined under the ABM assumption is an excellent approximation.
Digital Portfolio Theory
"... Abstract. The Modern Portfolio Theory of Markowitz maximized portfolio expected return subject to holding total portfolio variance below a selected level. Digital Portfolio Theory is an extension of Modern Portfolio Theory, with the added dimension of memory. Digital Portfolio Theory decomposes the ..."
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Cited by 1 (1 self)
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Abstract. The Modern Portfolio Theory of Markowitz maximized portfolio expected return subject to holding total portfolio variance below a selected level. Digital Portfolio Theory is an extension of Modern Portfolio Theory, with the added dimension of memory. Digital Portfolio Theory decomposes the portfolio variance into independent components using the signal processing decomposition of variance. The risk or variance of each security’s return process is represented by multiple periodic components. These periodic variance components are further decomposed into systematic and unsystematic parts relative to a reference index. The Digital Portfolio Theory model maximizes portfolio expected return subject to a set of linear constraints that control systematic, unsystematic, calendar and noncalendar variance. The paper formulates a single period, digital signal processing, portfolio selection model using crosscovariance constraints to describe covariance and autocorrelation characteristics. Expected calendar effects can be optimally arbitraged by controlling the memory or autocorrelation characteristics of the efficient portfolios. The Digital Portfolio Theory optimization model is compared to the Modern Portfolio Theory model and is used to find efficient portfolios with zero calendar risk for selected periods. Key words: portfolio optimization, portfolio theory, digital signal processing, calendar anomalies 1.
Comparison of Mean Variance Like Strategies for Optimal Asset Allocation Problems ∗
, 2010
"... We determine the optimal dynamic investment policy for a mean quadratic variation objective function by numerical solution of a nonlinear HamiltonJacobiBellman (HJB) partial differential equation (PDE). We compare the efficient frontiers and optimal investment policies for three mean variance like ..."
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We determine the optimal dynamic investment policy for a mean quadratic variation objective function by numerical solution of a nonlinear HamiltonJacobiBellman (HJB) partial differential equation (PDE). We compare the efficient frontiers and optimal investment policies for three mean variance like strategies: precommitment mean variance, timeconsistent mean variance, and mean quadratic variation, assuming realistic investment constraints (e.g. no bankruptcy, finite shorting, borrowing). When the investment policy is constrained, the efficient frontiers for all three objective functions are similar, but the optimal policies are quite different.