We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear price impact. In our model, cross‐impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second‐order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.

中文翻译：

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较小的非线性价格影响的渐近性：多维案例的PDE方法

我们从非线性价格影响较小的消费中提供了多维效用最大化问题的价值函数的渐近展开。在我们的模型中，允许资产之间的交叉影响。在价格影响较小的极限中，我们确定值函数在其无摩擦版本附近的渐近展开。前导校正的特征是与遍历控制问题有关的非线性二阶PDE和线性抛物线PDE。我们在多元几何布朗运动价格模型上说明了我们的结果。