The least squares Monte Carlo algorithm has become a popular tool for solving stochastic control problems, mainly due to its ability to handle multiple stochastic state variables. However, a configuration that remains challenging is when dealing with high dimensional controls, due to the exponential complexity of grid search methods or the mathematical difficulty of deriving and solving high-dimensional first-order optimality conditions. This paper proposes an efficient least squares Monte Carlo algorithm that is suitable for handling high-dimensional controls. In particular, we first approximate backward recursive dynamic programs on a coarse grid of controls , then use a combined technique of local control regression and adaptive refinement grids to improve the optimal control estimates. We numerically show that the local control regression is more accurate and more efficient than global control regression , and that the overall computational runtime scales polynomially with respect to the dimension of the control. Finally, we further validate our method by solving a portfolio optimization problem with twelve risky assets with transaction cost, liquidity cost and market impact, which involves twelve exogenous risk factors (asset returns) and thirteen endogenous risk factors (portfolio value and asset prices subject to market impact) .
The least squares Monte Carlo algorithm has become popular for solving portfolio optimization problems. A simple approach is to approximate the value functions on a discrete grid of portfolio weights , then use control regression to generalize the discrete estimates. However, the classical global control regression can be expensive and inaccurate. To overcome this difficulty, we introduce a local control regression technique, combined with adaptive grids. We show that choosing a coarse grid for local regression can produce sufficiently accurate results .
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a popular tool for solving stochastic control problems, mainly due to its ability to handle multiple stochastic state variables. However, a configuration that remains challenging is when dealing with high dimensional controls, due to the exponential complexity of grid search methods or the mathematical difficulty of deriving and solving high-dimensional first-order optimality conditions. This paper proposes an efficient least squares Monte Carlo algorithm that is suitable for handling high-dimensional controls. In particular, we first approximate backward recursive dynamic programs on a coarse grid of controls
popular for solving portfolio optimization problems. A simple approach is to approximate the value functions on a discrete grid of portfolio weights
R
a combined technique of local control regression and adaptive refinement grids to improve the optimal control estimates. We numerically show that the local control regression is more accurate and more efficient than
control regression to generalize the discrete estimates. However, the classical
R
, and that the overall computational runtime scales polynomially with respect to the dimension of the control. Finally, we further validate our method by solving a portfolio optimization problem with twelve risky assets with transaction cost, liquidity cost and market impact, which involves twelve exogenous risk factors (asset returns) and thirteen endogenous risk factors (portfolio value and asset prices subject to market impact)
can be expensive and inaccurate. To overcome this difficulty, we introduce a local control regression technique, combined with adaptive grids. We show that choosing a coarse grid for local regression can produce sufficiently accurate results