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首页> 外文期刊>Numerical Methods for Partial Differential Equations: An International Journal >Continuous Time Mean-Variance Optimal Portfolio Allocation Under Jump Diffusion: An Numerical Impulse Control Approach
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Continuous Time Mean-Variance Optimal Portfolio Allocation Under Jump Diffusion: An Numerical Impulse Control Approach

机译：跳跃扩散下的连续时间均值-方差最优投资组合分配：一种数值脉冲控制方法

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摘要

We present efficient partial differential equation methods for continuous time mean-variance portfolio allocation problems when the underlying risky asset follows a jump-diffusion. The standard formulation of mean-variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one-dimensional (1D) nonlinear Hamilton-Jacobi-Bellman (HJB) partial integrodifferential equation (PIDE) with the control present in the integrand of the jump term, and thus is difficult to solve efficiently. To preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 2D impulse control problem, 1D for each asset in the portfolio, namely the bond and the stock. We then develop a numerical scheme based on a semi-Lagrangian timestepping method, which we show to be monotone, consistent, and stable. Hence, assuming a strong comparison property holds, the numerical solution is guaranteed to converge to the unique viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of realistic financial modeling, such as different borrowing and lending interest rates, transaction costs, and constraints on the portfolio, such as maximum limits on borrowing and solvency.
机译：当基础风险资产遵循跳跃扩散时，我们为连续时间均方差投资组合分配问题提出了有效的偏微分方程方法。标准差的均方差最优投资组合分配问题的公式，其中总财富是潜在的随机过程，产生了带有控制的一维（1D）非线性Hamilton-Jacobi-Bellman（HJB）偏积分微分方程（PIDE）存在于跳跃项的整数中，因此难以有效解决。为了保持对跳变项的有效处理，我们将资产分配问题公式化为2D脉冲控制问题，即投资组合中每个资产（即债券和股票）的1D冲动控制问题。然后，我们基于半拉格朗日时间步长法开发了一种数值方案，该方案证明是单调的，一致的和稳定的。因此，假设具有很强的比较性质，就可以保证数值解收敛到相应HJB PIDE的唯一粘度解。数值算例验证了所提出数值框架的正确性。我们还将讨论对现实财务模型的有效边界的影响，例如不同的借贷利率，交易成本以及对投资组合的限制，例如对借贷和偿付能力的最大限制。

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