arXiv subjects

Xun Li

Publications and source records attributed to Xun Li.

At least 73 records · Page 4Linked to original sources

arXiv2024

Weak Closed-loop Solvability of Linear Quadratic Stochastic Optimal Control Problems with Partial Information

This paper investigates a linear quadratic stochastic optimal control (LQSOC) problem with partial information. Firstly, by introducing two Riccati equations and a backward stochastic differential equation (BSDE), we solve this LQSOC problem under standard positive semidefinite assumptions. Secondly, by means of a perturbation approach, we study open-loop solvability of this problem when the weighting matrices in the cost functional are indefinite. Thirdly, we investigate weak closed-loop solvability of this problem and prove the equivalence between open-loop and weak closed-loop solvabilities. Finally, we give an example to illustrate the way for obtaining a weak closed-loop optimal strategy.

math.OC↗

arXiv2024

Zero-sum stochastic linear-quadratic Stackelberg differential games of Markovian regime-switching system

This paper investigates a zero-sum stochastic linear-quadratic (SLQ, for short) Stackelberg differential game problem, where the coefficients of the state equation and the weighting matrices in the performance functional are regulated by a Markov chain. By utilizing the findings in \citet{Zhang.X.2021_ILQM}, we directly present the feedback representation to the rational reaction of the follower. For the leader's problem, we derive the optimality system through the variational method and study its unique solvability from the Hilbert space point of view. We construct the explicit optimal control for the leader based on the solution to coupled differential Riccati equations (CDREs, for short) and obtain the solvability of CDREs under the one-dimensional framework. Finally, we provide two concrete examples to illustrate the results developed in this paper.

math.OC↗

arXiv2024

Stochastic linear-quadratic differential game with Markovian jumps in an infinite horizon

This paper investigates a two-person non-homogeneous linear-quadratic stochastic differential game (LQ-SDG, for short) in an infinite horizon for a system regulated by a time-invariant Markov chain. Both non-zero-sum and zero-sum LQ-SDG problems are studied. It is shown that the zero-sum LQ-SDG problem can be considered a special non-zero-sum LQ-SDG problem. The open-loop Nash equilibrium point of non-zero-sum (zero-sum, respectively) LQ-SDG problem is characterized by the solvability of a system of constrained forward-backward stochastic differential equations (FBSDEs, for short) in an infinite horizon and the convexity (convexity-concavity, respectively) of the performance functional and their corresponding closed-loop Nash equilibrium strategy are characterized by the solvability of a system of constrained coupled algebra Riccati equations (CAREs, for short) with certain stabilizing conditions. In addition, the closed-loop representation of open-loop Nash equilibrium point for non-zero-sum (zero-sum, respectively) LQ-SDG is provided by the non-symmetric (symmetric, respectively) solution to a system CAREs. At the end of this paper, we provide three concrete examples and solve their open-loop/closed-loop Nash equilibrium strategy based on the obtained results.

math.OC↗

arXiv2024

Infinite time horizon stochastic recursive control problems with jumps: dynamic programming and stochastic verification theorems

This paper is devoted to studying an infinite time horizon stochastic recursive control problem with jumps, where infinite time horizon stochastic differential equation and backward stochastic differential equation with jumps describe the state process and cost functional, respectively. For this, the first is to explore the wellposedness and regularity of these two equations in $L^p$-sense ($p\geq2$). By establishing the dynamic programming principle, we relate the value function of the control problem with integral-partial differential equation of HJB type in the sense of viscosity solutions. On the other hand, stochastic verification theorems are also studied to provide sufficient conditions to verify the optimality of the given admissible controls. Such a study is carried out in the framework of classical solutions but also in that of viscosity solutions. Our work emphasizes important differences from the approach for finite time horizon problems. In particular, we have to work in an $L^p$-setting for $p>4$ in order to study the verification theorem in viscosity sense.

math.OC↗

arXiv2024

Two person non-zero-sum linear-quadratic differential game with Markovian jumps in infinite horizon

This paper investigates an inhomogeneous non-zero-sum linear-quadratic (LQ, for short) differential game problem whose state process and cost functional are regulated by a Markov chain. Under the $L^2$ stabilizability framework, we first provide a sufficient condition to ensure the $L^2$-integrability of the state process and study a class of linear backward stochastic differential equation (BSDE, for short) in infinite horizon. Then, we seriously discuss the LQ problem and show that the closed-loop optimal control is characterized by the solutions to coupled algebra Riccati equations (CAREs, for short) with some stabilizing conditions and a linear BSDE. Based on those results, we further analyze the non-zero-sum stochastic differential game problem and give the closed-loop Nash equilibrium through the solution to a system of two cross-coupled CAREs and two cross-coupled BSDEs. Finally, some related numerical

math.OC↗

arXiv2024

Two system transformation data-driven algorithms for linear quadratic mean-field games

This paper studies a class of continuous-time linear quadratic (LQ) mean-field game problems. We develop two system transformation data-driven algorithms to approximate the decentralized strategies of the LQ mean-field games. The main feature of the obtained data-driven algorithms is that they eliminate the requirement on all system matrices. First, we transform the original stochastic system into an ordinary differential equation (ODE). Subsequently, we construct some Kronecker product-based matrices by the input/state data of the ODE. By virtue of these matrices, we implement a model-based policy iteration (PI) algorithm and a model-based value iteration (VI) algorithm in a data-driven fashion. In addition, we also demonstrate the convergence of these two data-driven algorithms under some mild conditions. Finally, we illustrate the practicality of our algorithms via two numerical examples.

math.OC↗

arXiv2024

GMXPolymer: a generated polymerization algorithm based on GROMACS

This work introduces a method for generating generalized structures of amorphous polymers using simulated polymerization and molecular dynamics equilibration, with a particular focus on amorphous polymers. The techniques and algorithms used in this method are described in the main text, and example input scripts are provided for the GMXPolymer code, which is based on the GROMACS molecular dynamics package. To demonstrate the efficacy of our method, we apply it to different glassy polymers exhibiting varying degrees of functionality, polarity, and rigidity. The reliability of the method is validated by comparing simulation results to experimental data for various structural and thermal properties, all of which show excellent agreement.

cond-mat.soft↗

arXiv2024

Optimal consumption with loss aversion and reference to past spending maximum

This paper studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the non-negative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the utility with respect to consumption, allowing us to focus on an auxiliary HJB variational inequality on the strength of concavification principle and dynamic programming arguments. By applying the dual transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in piecewise closed-form and some thresholds of the wealth variable are obtained. The optimal consumption and investment control can be derived in the piecewise feedback form. The rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analysis and financial implications are also presented.

math.OC↗

arXiv2023

Discrete-Time Mean-Variance Strategy Based on Reinforcement Learning

This paper studies a discrete-time mean-variance model based on reinforcement learning. Compared with its continuous-time counterpart in \cite{zhou2020mv}, the discrete-time model makes more general assumptions about the asset's return distribution. Using entropy to measure the cost of exploration, we derive the optimal investment strategy, whose density function is also Gaussian type. Additionally, we design the corresponding reinforcement learning algorithm. Both simulation experiments and empirical analysis indicate that our discrete-time model exhibits better applicability when analyzing real-world data than the continuous-time model.

q-fin.MF↗

arXiv2023

Solving Coupled Nonlinear Forward-backward Stochastic Differential Equations: An Optimization Perspective with Backward Measurability Loss

This paper aims to extend the BML method proposed in Wang et al. [22] to make it applicable to more general coupled nonlinear FBSDEs. We interpret BML from the fixed-point iteration perspective and show that optimizing BML is equivalent to minimizing the distance between two consecutive trial solutions in a fixed-point iteration. Thus, this paper provides a theoretical foundation for an optimization-based approach to solving FBSDEs. We also empirically evaluate the method through four numerical experiments.

math.OC↗

arXiv2023

Reinforcement Learning for Stochastic LQ Control of Discrete-Time Systems with Multiplicative Noises

This paper considers a stochastic linear quadratic problem for discrete-time systems with multiplicative noises over an infinite horizon. To obtain the optimal solution, we propose an online iterative algorithm of reinforcement learning based on Bellman dynamic programming principle. The algorithm avoids the direct calculation of algebra Riccati equations. It merely takes advantage of state trajectories over a short interval instead of all iterations, significantly simplifying the calculation process. Under the stabilizable initial values, numerical examples shed light on our theoretical results.

math.OC↗

arXiv2023

Dynamic Programming for Indefinite Stochastic McKean-Vlasov LQ Control Problem under Input Constraints

In this note, we study a class of indefinite stochastic McKean-Vlasov linear-quadratic (LQ in short) control problem under the control taking nonnegative values. In contrast to the conventional issue, both the classical dynamic programming principle (DPP in short) and the usual Riccati equation approach fail. We tackle these difficulties by extending the state space from $\mathbb{R}$ to probability measure space, afterward derive the the corresponding the infinite dimensional Hamilton--Jacobi--Bellman (HJB in short) equation. The optimal control and value function can be obtained basing on two functions constructed via two groups of novelty ordinary differential equations satisfying the HJB equation mentioned before. As an application, we revisit the mean-variance portfolio selection problems in continuous time under the constraint that short-selling of stocks is prohibited. The investment risk and the capital market line can be captured simultaneously.

math.OC↗

arXiv2023

Differential Multi-probe Thermal Transport Measurements of Multi-walled Carbon Nanotubes grown by Chemical Vapor Deposition

Carbon nanotubes (CNTs) are quasi-1D nanostructures that display both high thermal conductivity and intriguing low-dimensional phonon transport phenomena. In comparison to the advances made in the theoretical calculation of the lattice thermal conductivity of CNTs, thermal transport measurements of CNTs have been limited by either the poor temperature sensitivity of Raman thermometry technique or the presence of contact thermal resistance errors in sensitive two-probe resistance thermometry measurements. Here we report advances in a multi-probe measurement of the intrinsic thermal conductivity of individual multi-walled CNT samples that are transferred from the growth substrate onto the measurement device. The sample-thermometer thermal interface resistance is directly measured by this multi-probe method and used to model the temperature distribution along the contacted sample segment. The detailed temperature profile helps to eliminate the contact thermal resistance error in the obtained thermal conductivity of the suspended sample segment. A differential electro-thermal bridge measurement method is established to enhance the signal-to-noise ratio and reduce the measurement uncertainty by over 40%. The obtained thermal resistances of multiple suspended segments of the same MWCNT sample increase linearly with increasing length, revealing diffusive phonon transport as a result of phonon-defect scattering in these MWCNTs. The measured thermal conductivity increases with temperature and reaches up to 390+- 20 W m-1 K-1 at room temperature for a 9-walled MWCNT. Theoretical analysis of the measurement results suggests submicron phonon mean free paths due to extrinsic phonon scattering by extended defects such as grain boundaries. The obtained thermal conductivity is decreased by a factor of 3 upon electron beam damage and surface contamination of the CNT sample.

physics.app-ph↗

arXiv2023

Stochastic representation for solutions of a system of coupled HJB-Isaacs equations with integral-partial operators

In this paper, we focus on the stochastic representation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs (HJB-Isaacs (HJBI), for short) equations which is in fact a system of coupled Isaacs' type integral-partial differential equation. For this, we introduce an associated zero-sum stochastic differential game, where the state process is described by a classical stochastic differential equation (SDE, for short) with jumps, and the cost functional of recursive type is defined by a new type of backward stochastic differential equation (BSDE, for short) with two Poisson random measures, whose wellposedness and a prior estimate as well as the comparison theorem are investigated for the first time. One of the Poisson random measures appearing in the SDE and the BSDE stems from the integral term of the HJBI equations; the other random measure in BSDE is introduced to link the coupling factor of the HJBI equations. We show through an extension of the dynamic programming principle that the lower value function of this game problem is the viscosity solution of the system of our coupled HJBI equations. The uniqueness of the viscosity solution is also obtained in a space of continuous functions satisfying certain growth condition. In addition, also the upper value function of the game is shown to be the solution of the associated system of coupled Issacs' type of integral-partial differential equations. As a byproduct, we obtain the existence of the value for the game problem under the well-known Isaacs' condition.

math.OC↗

arXiv2023

Thermal resistance from non-equilibrium phonons at Si-Ge interface

As nanostructured devices become prevalent, interfaces often play an important role in thermal transport phenomena. However, interfacial thermal transport remains poorly understood due to complex physics across a wide range of length scales from atomistic to microscale. Past studies on interfacial thermal resistance have focused on interface-phonon scattering at the atomistic scale but overlooked the complex interplay of phonon-interface and phonon-phonon scattering at microscale. Here, we use the Peierls-Boltzmann transport equation to show that the resistance from the phonon-phonon scattering of non-equilibrium phonons near a Si-Ge interface is much larger than that directly caused by the interface scattering. We report that non-equilibrium in phonon distribution leads to significant entropy generation and thermal resistance upon three-phonon scattering by the Boltzmann's H-theorem. The physical origin of non-equilibrium phonons in Ge is explained with the mismatch of phonon dispersion, density-of-states, and group velocity, which serve as general guidance for estimating the non-equilibrium effect on interfacial thermal resistance. Our study bridges a gap between atomistic scale and less studied microscale phenomena, providing comprehensive understanding of overall interfacial thermal transport and the significant role of phonon-phonon scattering.

cond-mat.mes-hall↗

arXiv2023

Primitive to conventional geometry projection for efficient phonon transport calculations

The primitive Wigner-Seitz cell and corresponding first Brillouin zone (FBZ) are typically used in calculations of lattice vibrational and transport properties as they contain the smallest number of degrees of freedom and thus have the cheapest computational cost. However, in complex materials, the FBZ can take on irregular shapes where lattice symmetries are not apparent. Thus, conventional cells (with more atoms and regular shapes) are often used to describe materials, though dynamical and transport calculations are more expensive. Here we discuss an efficient anharmonic lattice dynamic method that maps conventional cell dynamics to primitive cell dynamics based on translational symmetries. This leads to phase interference conditions that act like conserved quantum numbers and a conservation rule for phonon scattering that is hidden in conventional dynamics which significantly reduces computational cost. We demonstrate this method for phonon transport in a variety of materials with inputs from first-principles calculations and attribute its efficiency to reduced scattering phase space and fewer summations in scattering matrix element calculations.

cond-mat.mtrl-sci↗

arXiv2023

Reinforcement Learning-Based Optimal Control for Multiplicative-Noise Systems with Input Delay

In this paper, the reinforcement learning (RL)-based optimal control problem is studied for multiplicative-noise systems, where input delay is involved and partial system dynamics is unknown. To solve a variant of Riccati-ZXL equations, which is a counterpart of standard Riccati equation and determines the optimal controller, we first develop a necessary and sufficient stabilizing condition in form of several Lyapunov-type equations, a parallelism of the classical Lyapunov theory. Based on the condition, we provide an offline and convergent algorithm for the variant of Riccati-ZXL equations. According to the convergent algorithm, we propose a RL-based optimal control design approach for solving linear quadratic regulation problem with partially unknown system dynamics. Finally, a numerical example is used to evaluate the proposed algorithm.

math.OC↗

arXiv2022

Optimal consumption and life insurance under shortfall aversion and a drawdown constraint

This paper studies a life-cycle optimal portfolio-consumption problem when the consumption performance is measured by a shortfall aversion preference with an additional drawdown constraint on consumption rate. Meanwhile, the agent also dynamically chooses her life insurance premium to maximize the expected bequest at the death time. By using dynamic programming arguments and the dual transform, we solve the HJB variational inequality explicitly in a piecewise form across different regions and derive some thresholds of the wealth variable for the piecewise optimal feedback controls. Taking advantage of our analytical results, we are able to numerically illustrate some quantitative impacts on optimal consumption and life insurance by model parameters and discuss their financial implications.

math.OC↗