A stochastic control problem arising from relaxed wealth tracking with a monotone benchmark process

This paper studies a nonstandard stochastic control problem motivated by the optimal consumption in an incomplete market with wealth tracking of a non-decreasing benchmark process. In particular, the monotone benchmark is modelled by the running maximum of a drifted Brownian motion. We consider a relaxed tracking formulation using capital injection such that the wealth compensated by the injected capital dominates the benchmark process at all times. The stochastic control problem is to maximize the expected utility on consumption deducted by the cost of the capital injection under the dynamic floor constraint. By introducing two auxiliary state processes with reflections, an equivalent auxiliary control problem is formulated and studied such that the singular control of capital injection and the floor constraint can be hidden. To tackle the HJB equation with two Neumann boundary conditions, we establish the existence of a unique classical solution to the dual PDE in a separation form using some novel probabilistic representations involving the dual reflected processes and the local time. The proof of the verification theorem on the optimal feedback control can be carried out by some technical stochastic flow analysis of the dual reflected processes and estimations of the optimal control.Comment: Keywords: Non-decreasing benchmark, capital injection, optimal consumption, Neumann boundary conditions, probabilistic representation, reflected diffusion processe

An extended Merton problem with relaxed benchmark tracking

This paper studies a Merton's optimal portfolio and consumption problem in an extended formulation incorporating the tracking of a benchmark process described by a geometric Brownian motion. We consider a relaxed tracking formulation such that that the wealth process compensated by a fictitious capital injection outperforms the external benchmark at all times. The fund manager aims to maximize the expected utility of consumption deducted by the cost of the capital injection, where the latter term can also be regarded as the expected largest shortfall with reference to the benchmark. By introducing an auxiliary state process with reflection, we formulate and tackle an equivalent stochastic control problem by means of the dual transform and probabilistic representation, where the dual PDE can be solved explicitly. On the strength of the closed-form results, we can derive and verify the feedback optimal control in the semi-analytical form for the primal control problem, allowing us to observe and discuss some new and interesting financial implications on portfolio and consumption decision making induced by the additional risk-taking in capital injection and the goal of tracking.Comment: Keywords: Benchmark tracking, capital injection, expected largest shortfall, consumption and portfolio choice, Neumann boundary conditio

Inhomogeneous graph trend filtering via a l2,0 cardinality penalty

We study estimation of piecewise smooth signals over a graph. We propose a $\ell_{2,0}$-norm penalized Graph Trend Filtering (GTF) model to estimate piecewise smooth graph signals that exhibits inhomogeneous levels of smoothness across the nodes. We prove that the proposed GTF model is simultaneously a k-means clustering on the signal over the nodes and a minimum graph cut on the edges of the graph, where the clustering and the cut share the same assignment matrix. We propose two methods to solve the proposed GTF model: a spectral decomposition method and a method based on simulated annealing. In the experiment on synthetic and real-world datasets, we show that the proposed GTF model has a better performances compared with existing approaches on the tasks of denoising, support recovery and semi-supervised classification. We also show that the proposed GTF model can be solved more efficiently than existing models for the dataset with a large edge set.Comment: 21 pages, 3 figures, 4 table

Building explainable graph neural network by sparse learning for the drug-protein binding prediction

Explainable Graph Neural Networks (GNNs) have been developed and applied to drug-protein binding prediction to identify the key chemical structures in a drug that have active interactions with the target proteins. However, the key structures identified by the current explainable GNN models are typically chemically invalid. Furthermore, a threshold needs to be manually selected to pinpoint the key structures from the rest. To overcome the limitations of the current explainable GNN models, we propose our SLGNN, which stands for using Sparse Learning to Graph Neural Networks. Our SLGNN relies on using a chemical-substructure-based graph (where nodes are chemical substructures) to represent a drug molecule. Furthermore, SLGNN incorporates generalized fussed lasso with message-passing algorithms to identify connected subgraphs that are critical for the drug-protein binding prediction. Due to the use of the chemical-substructure-based graph, it is guaranteed that any subgraphs in a drug identified by our SLGNN are chemically valid structures. These structures can be further interpreted as the key chemical structures for the drug to bind to the target protein. We demonstrate the explanatory power of our SLGNN by first showing all the key structures identified by our SLGNN are chemically valid. In addition, we illustrate that the key structures identified by our SLGNN have more predictive power than the key structures identified by the competing methods. At last, we use known drug-protein binding data to show the key structures identified by our SLGNN contain most of the binding sites