Deep learning for ranking response surfaces with applications to optimal stopping problems

In this paper, we propose deep learning algorithms for ranking response surfaces with applications to optimal stopping problems in financial mathematics. The problem of ranking response surfaces is motivated by estimating optimal feedback policy maps in stochastic control problems, aiming to efficiently find the index associated with the minimal response across the entire continuous input space $\mathcal {X} \subseteq \mathbb {R}^d $X⊆Rd. By considering points in $\mathcal {X} $X as pixels and indices of the minimal surfaces as labels, we recast the problem as an image segmentation problem which assigns a label to every pixel in an image with pixels with the same label sharing certain characteristics. This provides an alternative method for efficiently solving the problem instead of using sequential design as in our previous work [R. Hu and M. Ludkovski, Sequential design for ranking response surfaces. SIAM/ASA J. Uncertain. Quantif., 2017, 5, 212–239]. Deep learning algorithms are scalable, parallel and model-free, i.e. no parametric assumptions are needed for response surfaces. Considering ranking response surfaces as image segmentation allows one to use a broad class of deep neural networks (NNs), e.g. feed-forward NNs, UNet, SegNet, DeconvNet, which have been widely applied and numerically proven to possess good performance in the field. We also systematically study the dependence of deep learning algorithms on the input data generated on uniform grids or by sequential design sampling and observe that the performance of deep learning is not sensitive to the noise and location (close to/away from boundaries) of training data. We present a few examples, including synthetic ones and the Bermudan option pricing problem, to show the efficiency and accuracy of this method. We also simulate a 10-dimensional example to demonstrate robustness, while non-learning algorithms in general have difficulties in such high dimensions.