Deep neural network expressivity for optimal stopping problems
研究了深度神经网络在离散时间马尔可夫过程最优停止问题中的表达速率，证明其能避免维度灾难，为高维美式期权定价等应用提供理论依据。
Abstract This article studies deep neural network expression rates for optimal stopping problems of discrete-time Markov processes on high-dimensional state spaces. A general framework is established in which the value function and continuation value of an optimal stopping problem can be approximated with error at most $\varepsilon $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> by a deep ReLU neural network of size at most $\kappa d^{\mathfrak{q}} \varepsilon ^{-\mathfrak{r}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> <mml:msup> <mml:mi>d</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> </mml:math> . The constants $\kappa ,\mathfrak{q},\mathfrak{r} \geq 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:math> do not depend on the dimension $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> of the state space or the approximation accuracy $\varepsilon $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> . This proves that deep neural networks do not suffer from the curse of dimensionality when employed to approximate solutions of optimal stopping problems. The framework covers for example exponential Lévy models, discrete diffusion processes and their running minima and maxima. These results mathematically justify the use of deep neural networks for numerically solving optimal stopping problems and pricing American options in high dimensions.