Special Session 87: Large Population Optimization, Stochastic Filtering and Mathematical Finance

On Poles and Zeros of Linear Quantum Systems

Zhiyuan Dong Harbin Institute of Technology, Shenzhen Peoples Rep of China Co-Author(s):    Guofeng Zhang, Heung-wing Joseph Lee

Abstract: The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which in the linear realm are manifested by the physical realizability conditions on system matrices. These restrictions endow system matrices with special structure. The purpose of this paper is to study such structure by investigating zeros and poses of linear quantum systems. In particular, we show that $-s_0^\ast$ is a transmission zero if and only if $s_0$ is a pole, and which is further generalized to the relationship between system eigenvalues and invariant zeros. Additionally, we study left-invertibility and fundamental tradeoff for linear quantum systems in terms of their zeros and poles.

Pareto game of stochastic differential system with terminal state constraint

Pengyan Huang Shandong University of Finance and Economics Peoples Rep of China Co-Author(s):    Guangchen Wang and Shujun Wang

Abstract: In this work, we focus on a type of Pareto game of stochastic differential equation with terminal state constraint. Firstly, we transform equivalently a nonlinear Pareto game problem with convex control space and terminal state constraint into a constrained stochastic optimal control problem. By virtue of duality theory and stochastic maximum principle, a necessary condition for Pareto efficient strategy is established. With some convex assumptions, we also give a sufficient condition for Pareto efficient strategy. Secondly, we consider a linear-quadratic (LQ) Pareto game with terminal state constraint, and a feedback representation for Pareto efficient strategy is derived. Finally, as an application, we solve a government debt stabilization problem and give some numerical results.

Mean Field Games of Major-Minor Agents with Recursive Functionals

Huang Jianhui Hong Kong PolyU Hong Kong Co-Author(s):

Abstract: This paper studies a general class of mean field games involving a major agent and sufficiently many minor agents, whose payoff functionals are of the recursive types in terms of backward stochastic differential equations (BSDEs).

Cyber Risk Management Through Investment in Cybersecurity Technology

Zhuo Jin Macquarie University Australia Co-Author(s):    Jiannan Zhang, Sizhe Chen, Hailiang Yang

Abstract: Investment in cybersecurity measures has become a significant aspect of our modern society, with the cost of cybersecurity technology fluctuating due to technological innovations and changes in market dynamics. In this paper, we investigate the optimal timing for a company to invest in cybersecurity technology to reduce cyberattack losses.

Zero-Sum Differential Games in the Wasserstein Space

Jun Moon Hanyang University Korea Co-Author(s):    Tamer Basar

Abstract: We consider two-player zero-sum differential games (ZSDGs), where the state process (dynamical system) depends on the random initial condition and the state process`s distribution, and the objective functional includes the state process`s distribution and the random target variable. Unlike ZSDGs studied in the existing literature, the ZSDG of this paper introduces a new technical challenge, since the corresponding (lower and upper) value functions are defined on $\mathcal{P}_2$ (the set of probability measures with finite second moments) or $\mathcal{L}_2$ (the set of random variables with finite second moments), both of which are infinite-dimensional spaces. We show that the (lower and upper) value functions on $\mathcal{P}_2$ and $\mathcal{L}_2$ are equivalent (law invariant) and continuous, satisfying dynamic programming principles. We use the notion of derivative of a function of probability measures in $\mathcal{P}_2$ and its lifted version in $\mathcal{L}_2$ to show that the (lower and upper) value functions are unique viscosity solutions to the associated (lower and upper) Hamilton-Jacobi-Isaacs equations, which are (infinite-dimensional) first-order PDEs on $\mathcal{P}_2$ and $\mathcal{L}_2$, where the uniqueness is obtained via the comparison principle. Under the Isaacs condition, we show that the ZSDG has a value.

On Well-posedness of Mean Field Game Master Equations: a Unified Approach

Chenchen Mou City University of Hong Kong Peoples Rep of China Co-Author(s):    Jianfeng Zhang, Jianjun Zhou

Abstract: There have been many serious studies on mean field game master equations in the literature. It is well known that the a priori Lipschitz estimates of solutions to master equations with respect to the probability measure variable are essentially necessary for their global well-posedness. The Lasry-Lions monotonicity, displacement monotonicity and anti-monotonicity conditions are found to be sufficient conditions to these Lipschitz estimates. However, whether these monotonicity conditions are necessary is unknown. In this talk, an essentially necessary and sufficient condition to these a priori Lipschitz estimates will be discussed and a unified approach to uniquely solve master equations will be established. The talk is based on a joint work with Jianfeng Zhang and Jianjun Zhou.

Diffusion Approximation and Stability of Stochastic Differential Equations with Singular Perturbation

Fuke Wu Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):    Huagui Liu, Shujun Liu

Abstract: This paper investigates diffusion approximation of non-homogeneous singularly perturbed stochastic differential equations with locally Lipschitz continuous coefficients by using the first-order perturbation test function method and formulation of the martingale problem. Under appropriate conditions, if the averaging system is exponential stable, the slow component is also uniformly asymptotically stable. Since the averaging system is often simpler than the original system, this stability result is interesting.

Synchronous Stability Analysis of Power Systems Under Stochastic Disturbances

Kaihua Xi Shandong University Peoples Rep of China Co-Author(s):

Abstract: For the enhancement of the transient stability of power systems, the key is to define a quantitative optimization formulation with system parameters as decision variables. In this paper, we model the disturbances by Gaussian noise and define a metric named Critical Escape Probability (CREP) based on the invariant probability measure of a linearised stochastic processes. CREP characterizes the probability of the state escaping from a critical set. CREP involves all the system parameters and reflects the size of the basin of attraction of the nonlinear systems. An optimization framework that minimizes CREP with the system parameters as decision variables is presented. Simulations show that the mean of the first hitting time when the state hits the boundary of the critical set, that is often used to describe the stability of nonlinear systems, is dramatically increased by minimizing CREP. This indicates that the transient stability of the system is effectively enhanced. It also shown that suppressing the state fluctuations only is insufficient for enhancing the transient stability. This new metric opens a new avenue for the transient stability analysis of future power systems integrated with large amounts of renewable energy.

A Mean-Field Game for a Forward-Backward Stochastic System With Partial Observation and Common Noise

Hua Xiao Shandong University Peoples Rep of China Co-Author(s):    Pengyan Huang, Guangchen Wang, Shujun Wang

Abstract: This paper considers a linear-quadratic (LQ) mean-field game governed by a forward-backward stochastic system with partial observation and common noise, where a coupling structure enters state equations, cost functionals and observation equations. Firstly, to reduce the complexity of solving the mean-field game, a limiting control problem is introduced. By virtue of the decomposition approach, an admissible control set is proposed. Applying a filter technique and dimensional-expansion technique, a decentralized control strategy and a consistency condition system are derived, and the related solvability is also addressed. Secondly, we discuss an approximate Nash equilibrium property of the decentralized control strategy. Finally, we work out a financial concern with some numerical simulations.

Recursive stochastic differential games with non-Lipschitzian generators and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations

Zhuangzhuang Xing Henan Normal University Peoples Rep of China Co-Author(s):    Guangchen Wang

Abstract: This investigation is dedicated to a two-player zero-sum stochastic differential game (SDG), where a cost function is characterized by a backward stochastic differential equation (BSDE) with a continuous and monotonic generator regarding the first unknown variable, which possesses immense applicability in financial engineering. A verification theorem by virtue of classical solution of derived Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation is given. The dynamic programming principle (DPP) and unique weak (viscosity) solvability of the HJBI equation are formulated through a comparison theorem for BSDEs with monotonic generators and stability of viscosity solution. Some new regularity properties of the value function are presented. Finally, we propose three concrete examples, which are concerned with, resp., classical and viscosity solutions of the HJBI equation, as well as a financial application where an investor with a non-Lipschitzian Epstein-Zin utility deals with market friction to maximize her utility preference.

On optimal carbon tax in China: implications for net-zero emissions and development

Detao Zhang School of Economics, Shandong University Peoples Rep of China Co-Author(s):    Jingjing Zhang, Pan Chen, Mondher Bellalah, Detao Zhang

Abstract: The carbon pricing is recognized as an effective tool of carbon reduction. Hence, China urgently needs to improve the carbon pricing system. This paper constructs a dynamic stochastic general equilibrium model with the inter-temporal accumulation of carbon emissions and climate damage function to optimize the estimation of the carbon tax rate in China. It develops a four-sector dynamic general equilibrium model to investigate the effects of a carbon tax. We find that China`s optimal carbon tax rate is increasing with economic growth, and levying a carbon tax can achieve the double bonus of decreasing carbon and expanding the economy. Moreover, the synergy of the carbon tax with related green policies, such as green finance, enhancing green industries competitive advantage, government environmental regulation, and green transformation goals, further amplifies the green effects of a carbon tax. Our conclusions provide a reference for implementing carbon tax system when facing the pressure of carbon reduction, thus promoting global low-carbon development.

Viscosity Solutions for HJB Equations on the Process Space: Application to Mean Field Control with Common Noise

Jianjun Zhou Northwest A&F University Peoples Rep of China Co-Author(s):    Nizar Touzi; Jianfeng Zhang

Abstract: In this talk, we investigate a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second order path dependent HJB equation on the process space, which is by nature infinite dimensional. In particular, our model covers mean field control problems with common noise as a special case. We shall introduce a new notion of viscosity solutions and establish both existence and comparison principle, under merely Lipschitz continuity assumptions. The main feature of our notion is that, besides the standard smooth part, the test function consists of an extra singular component which allows us to handle the second order derivatives of the smooth test functions without invoking the Ishii`s lemma. We shall use the doubling variable arguments, combined with the Ekeland-Borwein-Preiss Variational Principle in order to overcome the noncompactness of the state space. A smooth gauge-type function on the path space is crucial for our estimates.

Optimal control of LQ problem with anticipative partial observations

Yonghui Zhou Guizhou Normal University School of Big Data and Computer Science Peoples Rep of China Co-Author(s):

Abstract: A stochastic linear-quadratic (LQ for short) problem with anticipative (i.e., not adapted) partial observations is studied. With the help of enlargement of filtration, we turn the anticipative signal observation system into a higher-dimensional adapted one, and obtain a linear filtering equation of the latter by the martingale representation theorem and a related equivalent control problem. By introducing a Riccati equation and an ordinary differential equation, we provide a unique optimal feedback control for another equivalent optimal control problem with the controlled state being the linear filtering equation. Finally, the optimal cost function for the original anticipative LQ problem is obtained, which is represented by the filtering of the extended adaptive system and some modified coefficients. Our result covers that of the classical stochastic LQ problem with adapted partial observations. As an application, the optimal control of an interception problem with anticipative radar tracking is explicitly given.