Optimal Capital Requirements in Financial Networks with Fire Sales

I explore and analyze a problem of finding the socially optimal capital requirements for financial institutions considering two distinct channels of contagion: direct exposures among the institutions, as represented by a network and fire sales externalities, which reflect the negative price impact of massive liquidation of assets.These two channels amplify shocks from individual financial institutions to the financial system as a whole and thus increase the risk of joint defaults amongst the interconnected financial institutions; this is often referred to as systemic risk. In the model, there is a trade-off between reducing systemic risk and raising the capital requirements of the financial institutions. The policymaker considers this trade-off and determines the optimal capital requirements for individual financial institutions. I provide a method for finding and analyzing the optimal capital requirements that can be applied to arbitrary network structures and arbitrary distributions of investment returns.

In particular, I first consider a network model consisting only of direct exposures and show that the optimal capital requirements can be found by solving a stochastic linear programming problem. I then extend the analysis to financial networks with default costs and show the optimal capital requirements can be found by solving a stochastic mixed integer programming problem. The computational complexity of this problem poses a challenge, and I develop an iterative algorithm that can be efficiently executed. I show that the iterative algorithm leads to solutions that are nearly optimal by comparing it with lower bounds based on a dual approach. I also show that the iterative algorithm converges to the optimal solution.

Finally, I incorporate fire sales externalities into the model. In particular, I am able to extend the analysis of systemic risk and the optimal capital requirements with a single illiquid asset to a model with multiple illiquid assets. The model with multiple illiquid assets incorporates liquidation rules used by the banks. I provide an optimization formulation whose solution provides the equilibrium payments for a given liquidation rule.

I further show that the socially optimal capital problem using the ``socially optimal liquidation" and prioritized liquidation rules can be formulated as a convex and convex mixed integer problem, respectively. Finally, I illustrate the results of the methodology on numerical examples and

discuss some implications for capital regulation policy and stress testing.

A Molecular-scale Programmable Stochastic Process Based On Resonance Energy Transfer Networks: Modeling And Applications

While molecular and cellular processes are often modeled as stochastic processes, such as Brownian motion, chemical reaction networks and gene regulatory networks, there are few attempts to program a molecular-scale process to physically implement stochastic processes. DNA has been used as a substrate for programming molecular interactions, but its applications are restricted to deterministic functions and unfavorable properties such as slow processing, thermal annealing, aqueous solvents and difficult readout limit them to proof-of-concept purposes. To date, whether there exists a molecular process that can be programmed to implement stochastic processes for practical applications remains unknown.

In this dissertation, a fully specified Resonance Energy Transfer (RET) network between chromophores is accurately fabricated via DNA self-assembly, and the exciton dynamics in the RET network physically implement a stochastic process, specifically a continuous-time Markov chain (CTMC), which has a direct mapping to the physical geometry of the chromophore network. Excited by a light source, a RET network generates random samples in the temporal domain in the form of fluorescence photons which can be detected by a photon detector. The intrinsic sampling distribution of a RET network is derived as a phase-type distribution configured by its CTMC model. The conclusion is that the exciton dynamics in a RET network implement a general and important class of stochastic processes that can be directly and accurately programmed and used for practical applications of photonics and optoelectronics. Different approaches to using RET networks exist with vast potential applications. As an entropy source that can directly generate samples from virtually arbitrary distributions, RET networks can benefit applications that rely on generating random samples such as 1) fluorescent taggants and 2) stochastic computing.

By using RET networks between chromophores to implement fluorescent taggants with temporally coded signatures, the taggant design is not constrained by resolvable dyes and has a significantly larger coding capacity than spectrally or lifetime coded fluorescent taggants. Meanwhile, the taggant detection process becomes highly efficient, and the Maximum Likelihood Estimation (MLE) based taggant identification guarantees high accuracy even with only a few hundred detected photons.

Meanwhile, RET-based sampling units (RSU) can be constructed to accelerate probabilistic algorithms for wide applications in machine learning and data analytics. Because probabilistic algorithms often rely on iteratively sampling from parameterized distributions, they can be inefficient in practice on the deterministic hardware traditional computers use, especially for high-dimensional and complex problems. As an efficient universal sampling unit, the proposed RSU can be integrated into a processor / GPU as specialized functional units or organized as a discrete accelerator to bring substantial speedups and power savings.