Zhipeng Huang

Zhipeng Huang, Cornelis W. Oosterlee

In this paper, we investigate the Markovian iteration method for solving coupled forward-backward stochastic differential equations (FBSDEs) with a fully coupled drift term of the form $b(t,X_t,Y_t,Z_t)$. An FBSDE system typically involves three stochastic processes: the forward process $X$, the backward process $Y$ representing the solution, and the $Z$ process corresponding to the scaled derivative of $Y$. Previous work by Bender and Zhang (2008) established convergence results for iterative schemes for $Y$-coupled FBSDEs. However, extending these results to equations with $Z$ coupling presents significant challenges, particularly in obtaining a uniform control of the Lipschitz constants of the decoupling fields across iterations and time steps within a fixed-point framework. To overcome this issue, we propose a novel differentiation-based method for handling the $Z$ process. This approach enables better control of the Lipschitz constants of decoupling fields, facilitating the well-posedness of the discretized FBSDE system with fully coupled drift. We rigorously prove the convergence of our Markovian iteration method in this more complex setting. Finally, we develop an efficient algorithm for computing the resulting numerical scheme, and numerical experiments confirm the theoretical findings and demonstrate the effectiveness and accuracy of the proposed methodology.

Mohammadreza Nemati, Zhipeng Huang, Kevin S. Xu

Some of the simplest, yet most frequently used predictors in statistics and machine learning use weighted linear combinations of features. Such linear predictors can model non-linear relationships between features by adding interaction terms corresponding to the products of all pairs of features. We consider the problem of accurately estimating coefficients for interaction terms in linear predictors. We hypothesize that the coefficients for different interaction terms have an approximate low-dimensional structure and represent each feature by a latent vector in a low-dimensional space. This low-dimensional representation can be viewed as a structured regularization approach that further mitigates overfitting in high-dimensional settings beyond standard regularizers such as the lasso and elastic net. We demonstrate that our approach, called LIT-LVM, achieves superior prediction accuracy compared to elastic net and factorization machines on a wide variety of simulated and real data, particularly when the number of interaction terms is high compared to the number of samples. LIT-LVM also provides low-dimensional latent representations for features that are useful for visualizing and analyzing their relationships.