论文标题
具有多项式运行时的非线性蒙特卡洛方法，用于高维迭代的嵌套期望
Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations
论文作者
论文摘要
迭代嵌套期望的近似计算是应用程序中反复出现的具有挑战性的问题。 Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential equations (BSDEs), in the numerical approximation of solutions of semilinear parabolic partial differential equations (PDEs), in statistical physics, in optimal stopping problems such as the approximative pricing of American or Bermudan options, in risk measure estimation in mathematical finance, or in decision-making under不确定性。上述应用程序中出现的嵌套期望通常由大量巢组成。但是，标准嵌套蒙特卡洛近似迭代嵌套期望的计算工作在嵌套的数量中成倍增长，并且在多项式时间中是否可以计算出近似计算多重迭代的高维嵌套期望，这仍然是一个悬而未决的问题。在本文中，我们通过提出和研究一类新的全历史递归多层PICARD（MLP）近似方案来解决此问题，以迭代嵌套期望。特别是，我们在适当的假设下证明，这些MLP近似方案可以通过在\ in \ mathbb {n} = \ in \ mathbb {n} = \ {1，2，2，2，3，\ ldots \} $，$ dimension $ d \ n \ n n \ mathbb {n} = \ in \ mathbb {n} = \ in \ mathbb {n}中的嵌套数量上的计算努力来近似计算的嵌套期望。所需近似准确性的$ \ varepsilon \ in（0，\ infty）$的倒数$ \ frac {1} {\ varepsilon} $。
The approximative calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential equations (BSDEs), in the numerical approximation of solutions of semilinear parabolic partial differential equations (PDEs), in statistical physics, in optimal stopping problems such as the approximative pricing of American or Bermudan options, in risk measure estimation in mathematical finance, or in decision-making under uncertainty. Nested expectations which arise in the above named applications often consist of a large number of nestings. However, the computational effort of standard nested Monte Carlo approximations for iterated nested expectations grows exponentially in the number of nestings and it remained an open question whether it is possible to approximately calculate multiply iterated high-dimensional nested expectations in polynomial time. In this article we tackle this problem by proposing and studying a new class of full-history recursive multilevel Picard (MLP) approximation schemes for iterated nested expectations. In particular, we prove under suitable assumptions that these MLP approximation schemes can approximately calculate multiply iterated nested expectations with a computational effort growing at most polynomially in the number of nestings $ K \in \mathbb{N} = \{1, 2, 3, \ldots \} $, in the problem dimension $ d \in \mathbb{N} $, and in the reciprocal $\frac{1}{\varepsilon}$ of the desired approximation accuracy $ \varepsilon \in (0, \infty) $.
