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Stochastic linear quadratic optimal control problems
 Appl. Math. Optim
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A general theory of Markovian time inconsistent stochastic control problems. Available at SSRN: http://ssrn.com/abstract=1694759
, 2010
"... We develop a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for Nash subgame perfect equilibrium points. F ..."
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We develop a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for Nash subgame perfect equilibrium points. For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard HamiltonJacobiBellman equation, in the form of a system of nonlinear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. All known examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We also prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem coincides with the equilibrium control and value function respectively for the time inconsistent problem. We also study some concrete examples. Key words: Time consistency, time inconsistent control, dynamic programming, time inconsistency, stochastic control, hyperbolic discounting, meanvariance,
A HamiltonJacobiBellman approach to optimal trade execution
, 2009
"... The optimal trade execution problem is formulated in terms of a meanvariance tradeoff, as seen at the initial time. The meanvariance problem can be embedded in a LinearQuadratic (LQ) optimal stochastic control problem, A semiLagrangian scheme is used to solve the resulting nonlinear Hamilton Ja ..."
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Cited by 17 (2 self)
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The optimal trade execution problem is formulated in terms of a meanvariance tradeoff, as seen at the initial time. The meanvariance problem can be embedded in a LinearQuadratic (LQ) optimal stochastic control problem, A semiLagrangian scheme is used to solve the resulting nonlinear Hamilton Jacobi Bellman (HJB) PDE. This method is essentially independent of the form for the price impact functions. Provided a strong comparision property holds, we prove that the numerical scheme converges to the viscosity solution of the HJB PDE. Numerical examples are presented in terms of the efficient trading frontier and the trading strategy. The numerical results indicate that in some cases there are many different trading strategies which generate almost identical efficient frontiers.
Mean–variance optimal adaptive execution
 Applied Mathematical Finance
, 2011
"... Electronic trading of equities and other securities makes heavy use of “arrival price ” algorithms, that balance the market impact cost of rapid execution against the volatility risk of slow execution. In the standard formulation, meanvariance optimal trading strategies are static: they do not modi ..."
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Cited by 12 (1 self)
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Electronic trading of equities and other securities makes heavy use of “arrival price ” algorithms, that balance the market impact cost of rapid execution against the volatility risk of slow execution. In the standard formulation, meanvariance optimal trading strategies are static: they do not modify the execution speed in response to price motions observed during trading. We show that substantial improvement is possible by using dynamic trading strategies, and that the improvement is larger for large initial positions. We develop a technique for computing optimal dynamic strategies to any desired degree of precision. The asset price process is observed on a discrete tree with a arbitrary number of levels. We introduce a novel dynamic programming technique in which the control variables are not only the shares traded at each time step, but also the maximum expected cost for the remainder of the program; the value function is the variance of the remaining program. The resulting adaptive strategies are “aggressiveinthemoney”: they accelerate the execution when the price moves in the trader’s favor, spending parts of the trading gains to reduce risk.
Timeinconsistent stochastic linear–quadratic control
 SIAM Journal on Control and Optimization
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On Efficiency of MeanVariance based Portfolio Selection in DC Pension Schemes. Collegio Carlo Alberto
, 2010
"... www.carloalberto.org/working_papers © 2010 by Elena Vigna. Any opinions expressed here are those of the authors and not those of theCollegio Carlo Alberto. On efficiency of meanvariance based portfolio selection in DC pension schemes ..."
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www.carloalberto.org/working_papers © 2010 by Elena Vigna. Any opinions expressed here are those of the authors and not those of theCollegio Carlo Alberto. On efficiency of meanvariance based portfolio selection in DC pension schemes
Numerical methods for nonlinear PDEs in finance
 in Handbook of Computational Finance
, 2012
"... Many problems in finance can be posed in terms of an optimal stochastic control. Some wellknown examples include transaction cost/uncertain volatility models [17, 2, 25], passport options [1, 26], unequal borrowing/lending costs in option pricing [9], risk control in reinsurance [23], optimal with ..."
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Many problems in finance can be posed in terms of an optimal stochastic control. Some wellknown examples include transaction cost/uncertain volatility models [17, 2, 25], passport options [1, 26], unequal borrowing/lending costs in option pricing [9], risk control in reinsurance [23], optimal withdrawals in
A deterministic linear quadratic timeinconsistent optimal control problem
 Mathematical Control and Related Fields
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Timeconsistent meanvariance portfolio selection in discrete and continuous time, Finance and Stochastics
, 2013
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Optimal MeanVariance Selling Strategies
"... Assuming that the stock price X follows a geometric Brownian motion with drift µ ∈ IR and volatility σ> 0, and letting Px denote a probability measure under which X starts at x> 0, we study the dynamic version of the nonlinear meanvariance optimal stopping problem sup EXt(Xτ) − c VarXt(Xτ) ..."
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Assuming that the stock price X follows a geometric Brownian motion with drift µ ∈ IR and volatility σ> 0, and letting Px denote a probability measure under which X starts at x> 0, we study the dynamic version of the nonlinear meanvariance optimal stopping problem sup EXt(Xτ) − c VarXt(Xτ)