Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the state-of-the-art pricing and hedging of financial derivatives. In this article we prove that semilinear heat equations with gradient-dependent nonlinearities can be approximated under suitable assumptions with computational complexity that grows polynomially both in the dimension and the reciprocal of the accuracy.

中文翻译：

---

具有梯度相关非线性的高维半线性抛物线微分方程的多级 Picard 近似

抛物线偏微分方程 (PDE) 和后向随机微分方程 (BSDE) 具有广泛的应用。特别是，具有梯度相关非线性的高维偏微分方程经常出现在最先进的金融衍生品定价和对冲中。在本文中，我们证明了具有梯度相关非线性的半线性热方程可以在适当的假设下近似，计算复杂度在维度和精度的倒数上呈多项式增长。