We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of the associated backward stochastic differential equation as the loss function. Our proposed framework yields prices and deltas for the entire spacetime, not only at a given point (e.g. t = 0). The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer. Our numerical simulations demonstrate these contributions, and show that the proposed neural network framework outperforms state-of-the-art approaches in high dimensions.

Invasive plants threaten biodiversity and cause huge economic losses. It is thought that global change factors (GCFs) associated with climate change (including shifts in temperature, precipitation, nitrogen, and atmospheric CO₂) will amplify their impacts. However, only few studies assessed mixed factors on plant invasion. We collated the literature on plant responses to GCFs to explore independent, combined, and interactive effects on performance and competitiveness of native and invasive plants. From 176 plant species, our results showed that: (1) when native and invasive plants are affected by both independent and multiple GCFs, there is an overall positive effect on plant performance, but a negative effect on plant competitiveness; (2) under increased precipitation or in combination with temperature, most invasive plants gain advantages over natives; and (3) interactions between GCFs on plant performance and competitiveness were mostly synergistic or antagonistic. Our results indicate that native and invasive plants may be affected by independent or combined GCFs, and invasive plants likely gain advantages over native plants. The interactive effects of factors on plants were non-additive, but the advantages of invasive plants may not increase indefinitely. Our findings show that inferring the impacts of climate change on plant invasion from factors individually could be misleading. More mixed factor studies are needed to predict plant invasions under global change.

Editorial: Global Changes and Plant Invasions