arXiv subjects

Zimu Zhu

Publications and source records attributed to Zimu Zhu.

11 recordsLinked to original sources

arXiv2026

Optimal Comfortable Consumption under Epstein-Zin utility

We solve the optimal portfolio choice problem under Epstein--Zin utility with a time-varying consumption constraint, where closed-form expressions for neither the primal nor the dual value function are available. We establish the dynamic programming principle for the value function and prove that it is a viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation. We further establish the $C^2$ regularity of the value function and derive a verification theorem using stochastic perturbation techniques. Finally, we provide an explicit characterization of the constrained region. The proposed methodology extends naturally to other constrained portfolio choice problems under the Epstein--Zin utility.

math.OC↗

arXiv2026

Stackelberg Games with a Robust Leader

In this paper we study a Stackelberg game with one leader and multiple followers. Given the leader's control, the followers solve a Nash game with possibly multiple equilibria. We consider a robust leader who considers the worst scenario, namely the followers would select the equilibrium worst for the leader. By using the weak formulation, the problem induces a zero sum game problem with open loop controls, which is time inconsistent. We shall characterize the last problem through an HJB equation on the Wasserstein space of probability measures. The principal-agent problem with one principal and multiple agents can be viewed as a special case of our problem, but with certain constraints.

math.OC↗

arXiv2026

DeepPAAC: A New Deep Galerkin Method for Principal-Agent Problems

We consider numerical resolution of principal-agent (PA) problems in continuous time. We formulate a generic PA model with continuous and lump payments and a multi-dimensional strategy of the agent. To tackle the resulting Hamilton-Jacobi-Bellman equation with an implicit Hamiltonian we develop a novel deep learning method: the Deep Principal-Agent Actor Critic (DeepPAAC) Actor-Critic algorithm. DeepPAAC is able to handle multi-dimensional states and controls, as well as constraints. We investigate the role of the neural network architecture, training designs, loss functions, etc. on the convergence of the solver, presenting five different case studies.

math.NA↗

arXiv2026

A General Model for Continuous Time Principal-Agent Problem Under Hidden Action

In this paper, we study a general continuous-time Principal-Agent (PA) problem, where the agent privately makes effort and consumption decisions over time under a contract with payment schemes both in continuous time and in lump sums. In particular, we allow the continuous payment process to be a controlled diffusion, which is directly related to the pay-to-performance sensitivity (PPS) in the empirical literature. In solving the agent's problem, we propose a new sufficient condition that directly yields a solution to the agent's problem, without requiring a separate verification step for the solution obtained from the first-order approach. We also present an example which can be solved explicitly.

q-fin.MF↗

arXiv2025

Optimal Consumption-Investment with Epstein-Zin Utility under Leverage Constraint

We study optimal portfolio choice under Epstein-Zin recursive utility in the presence of general leverage constraints. We first establish that the optimal value function is the unique viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, by developing a new dynamic programming principle under constraints. We further demonstrate that the value function admits smoothness and characterize the optimal consumption and investment strategies. In addition, we derive explicit solutions for the optimal strategy and explicitly delineate the constrained and unconstrained regions in several special cases of the leverage constraint. Finally, we conduct a comparative analysis, highlighting the differences relative to the classical time-separable preferences and to the setting without leverage constraints.

q-fin.PM↗

arXiv2024

A Dynamic Principal Agent Problem with One-sided Commitment

In this paper we consider a principal agent problem where the agent is allowed to quit, by incurring a cost. When the current agent quits the job, the principal will hire a new one, possibly with a different type. We characterize the principal's dynamic value function, which could be discontinuous at the boundary, as the (unique) minimal solution of an infinite dimensional system of HJB equations, parametrized by the agent's type. This dynamic problem is time consistent in certain sense. Some interesting findings are worth mentioning. First, self-enforcing contracts are typically suboptimal. The principal would rather let the agent quit and hire a new one. Next, the standard contract for a committed agent may also be suboptimal, due to the presence of different types of agents in our model. The principal may prefer no commitment from the agent, then she can hire a cheaper one from the market at a later time by designing the contract to induce the current agent to quit. Moreover, due to the cost incurring to the agent, the principal will see only finitely many quittings.

math.OC↗

arXiv2023

Smoothness of the Value Function for Optimal Consumption Model with Consumption-Wealth Utility and Borrowing Constraint

This paper studies an optimal consumption-investment problem for an investor whose instantaneous utility depends on both consumption and wealth, and the investor faces a general borrowing constraint that the investment amount in the risky asset does not exceed an exogenous function of the wealth. We show that the value function is second-order smooth and present the optimal consumption-investment policy in a feedback form. Moreover, when the risky investment amount is bounded above by a fixed constant, we show that under certain conditions, the constraint is binding if and only if an endogenous threshold bounds the portfolio wealth, and we determine the endogenous wealth threshold with the smooth fit condition. Our results encompass several well-developed portfolio choice models and imply new applications.

q-fin.PM↗

arXiv2023

Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems

The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great power in solving high-dimensional forward-backward stochastic differential equations (FBSDEs), and inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which can not be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [Wang, Chen, Sudjianto, Liu and Shen, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide the rigorous theory for the backward deep BSDE method. Specifically, 1. We derive the a posteriori error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and; 2. We give an upper bound of the loss function, which can be sufficiently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory.

math.PR↗

arXiv2021

A Portfolio Choice Problem Under Risk Capacity Constraint

This paper studies an optimal investing problem for a retiree facing longevity risk and living standard risk. We formulate the investing problem as a portfolio choice problem under a time-varying risk capacity constraint. We derive the optimal investment strategy under the specific condition on model parameters in terms of second-order ordinary differential equations. We demonstrate an endogenous number that measures the expected value to sustain the spending post-retirement. The optimal portfolio is nearly neutral to the stock market movement if the portfolio's value is higher than this number; but, if the portfolio is not worth enough to sustain the retirement spending, the retiree actively invests in the stock market for the higher expected return. Besides, we solve an optimal portfolio choice problem under a leverage constraint and show that the optimal portfolio would lose significantly in stressed markets. This paper shows that the time-varying risk capacity constraint has important implications for asset allocation in retirement.

q-fin.PM↗

arXiv2017

Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial and Outlook

Building on the scientific understanding and technological infrastructure of single-mode fibers, multimode fibers are being explored as a means of adding new degrees of freedom to optical technologies such as telecommunications, fiber lasers, imaging, and measurement. Here, starting from a baseline of single-mode nonlinear fiber optics, we introduce the growing topic of multimode nonlinear fiber optics. We demonstrate a new numerical solution method for the system of equations that describes nonlinear multimode propagation, the generalized multimode nonlinear Schrodinger equation. This numerical solver is freely available, and includes a number of multimode fiber analysis tools. It features a significant parallel computing speed-up on modern graphical processing units, translating to orders-of-magnitude speed-up over the split-step Fourier method. We demonstrate its use with several examples in graded- and step-index multimode fibers. Finally, we discuss several key open directions and questions, whose answers could have significant scientific and technological impact.

physics.optics↗

arXiv2016

Observation of Multimode Solitons in Few-Mode Fiber

We experimentally isolate and directly observe multimode solitons in few-mode graded-index fiber. By varying the input energy and modal composition of the launched pulse, we observe a continuous variation of multimode solitons with different spatiotemporal properties. They exhibit an energy-volume relation that is distinct from those of single-mode and fully spatiotemporal solitons.

physics.optics↗