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Zhun Gou

Publications and source records attributed to Zhun Gou.

14 recordsLinked to original sources
arXiv2026

Time-inconsistent reinsurance and investment optimization problem with delay under random risk aversion

This paper considers a newly delayed reinsurance and investment optimization problem incorporating random risk aversion, in which an insurer pursues maximization of the expected certainty equivalent of her/his terminal wealth and the cumulative delayed information of the wealth over a period. Specially, the insurer's surplus dynamics are approximated using a drifted Brownian motion, while the financial market is described by the constant elasticity of variance (CEV) model. Moreover, the performance-linked capital flow feature is incorporated and the wealth process is formulated via a stochastic delay differential equation (SDDE). By adopting a game-theoretic approach, a verification theorem with rigorous proofs is established to capture the equilibrium reinsurance and investment strategy along with the equilibrium value function. Furthermore, analytical or semi-analytical equilibrium reinsurance and investment strategies, together with their equilibrium value functions, are obtained under the CEV model for the exponential utility and derived under the Black-Scholes model for both exponential and power utilities. Finally, several numerical experiments are conducted to analyze the behavioral characteristics of the freshly-derived equilibrium reinsurance and investment strategy.

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arXiv2025

Nash Equilibria of Noncooperative/Mixed Differential Games with Density Constraints in Infinite Dimensions

Motivated by Cournot models, this paper proposes novel models of the noncooperative and cooperative differential games with density constraints in infinite dimensions, where markets consist of infinite firms and demand dynamics are governed by controlled differential equations. Markets engage in noncooperative competition with each other, while firms within each market engage in noncooperative or cooperative games. The main problems are to find the noncooperative Nash equilibrium (NNE) of the noncooperative differential game and the mixed Nash equilibrium (MNE) of the mixed noncooperative and cooperative differential game. Moreover, fundamental relationship is established between noncooperative/mixed differential game with density constraints and infinite-dimensional differential variational inequalities with density constraints. By variational analysis, it is proved under two conditions with certain symmetry that both of the two equilibrium problems can be reduced to solving systems of finite-dimensional projection equations with integral constraints by iterative computational methods. Crucially, the two conditions with certain symmetry, ensuring the uniqueness of the NNE and the MNE, provide theoretical foundations for strategic decision making regarding competitive versus cooperative market behaviors. Finally, the theoretical framework is validated through numerical simulations demonstrating the efficacy of our results.

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arXiv2025

Linear-Quadratic Graphon Mean Field Games with Common Noise

This paper studies linear quadratic graphon mean field games (LQ-GMFGs) with common noise, in which a large number of agents are coupled via a weighted undirected graph. One special feature, compared with the well-studied graphon mean field games, is that the states of agents are described by the dynamic systems with the idiosyncratic noises and common noise. The limit LQ-GMFGs with common noise are formulated based on the assumption that these graphs lie in a sequence converging to a limit graphon. By applying the spectral decomposition method, the existence of solution for the formulated limit LQ-GMFGs is derived. Moreover, based on the adequate convergence assumptions, a set of $ε$-Nash equilibrium strategies for the finite large population problem is constructed.

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arXiv2025

Linear-quadratic stochastic nonzero-sum differential games between graphon teams

We study a class of nonzero-sum stochastic differential games between two teams with agents in each team interacting through graphon aggregates. On the one hand, in each large population group, agents act together to optimize a common social cost function. On the other hand, these two groups compete with each other, forming a Nash game between two graphon teams. We note that the original problem can be equivalently formulated as an infinite-dimensional two-agent Nash game. Applying the dynamic programming approach, we obtain a set of coupled operator-valued Riccati-type equations. By proving the existence of solutions to the equations mentioned above, we obtain a Nash equilibrium for the two teams.

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arXiv2025

Equilibrium reinsurance and investment strategies for insurers with random risk aversion under Heston's SV model

This study employs expected certainty equivalents to explore the reinsurance and investment issue pertaining to an insurer that aims to maximize the expected utility while being subject to random risk aversion. The insurer's surplus process is modeled approximately by a drifted Brownian motion, and the financial market is comprised of a risk-free asset and a risky asset with its price depicted by Heston's stochastic volatility (SV) model. Within a game theory framework, a strict verification theorem is formulated to delineate the equilibrium reinsurance and investment strategies as well as the corresponding value function. Furthermore, through solving the pseudo Hamilton-Jacobi-Bellman (HJB) system, semi-analytical formulations for the equilibrium reinsurance and investment strategies and the associated value function are obtained under the exponential utility. Additionally, several numerical experiments are carried out to demonstrate the characteristics of the equilibrium reinsurance and investment strategies.

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arXiv2024

Consumption and portfolio optimization solvable problems with recursive preferences

This paper considers consumption and portfolio optimization problems with recursive preferences in both infinite and finite time regions. Specially, the financial market consists of a risk-free asset and a risky asset that follows a general stochastic volatility process. By using Bellman's dynamic programming principle, the Hamilton-Jacobi-Bellman (HJB) equation is derived for characterizing the optimal consumption-investment strategy and the corresponding value function. Based on the conjecture of the exponential-polynomial form of the value function, we prove that, when the order of the polynomial $n\leq2$, the HJB equation has an analytical solution if the investor with unit elasticity of intertemporal substitution (EIS) and an approximate solution otherwise.

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arXiv2024

Social Optima in Linear Quadratic Graphon Field Control: Analysis via Infinite Dimensional Approach

This paper is concerned with linear quadratic graphon field social control problem where the noises of individual agents are correlated. Compared with the well-studied mean field system, the graphon field system consists of a large number of agents coupled weakly via a weighted undirected graph where each node represents an individual agent. Another notable feature of this paper is that the dynamics of states of agents are driven by Brownian motions with a correlation matrix. The infinite dimensional approach is adopted to design the centralized and decentralized controls for our large population system. By graphon theory, we prove that the linear quadratic (LQ) social optimum control problem under the centralized information pattern is equivalent to an LQ optimal control problem concerned with a stochastic evolution equation, and the feedback-type optimal centralized control is obtained. Then, by designing an auxiliary infinite dimensional optimal control problem through agent number $N\rightarrow\infty$, a set of decentralized strategies are constructed, which are further shown to be asymptotically social optimal.

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arXiv2024

Linear-quadratic Stochastic Stackelberg Differential Games with Affine Constraints

This paper investigates the non-zero-sum linear-quadratic stochastic Stackelberg differential games with affine constraints, which depend on both the follower's response and the leader's strategy. With the help of the stochastic Riccati equations and the Lagrangian duality theory, the feedback expressions of optimal strategies of the follower and the leader are obtained and the dual problem of the leader's problem is established. Under the Slater condition, the equivalence is proved between the solutions to the dual problem and the leader's problem, and the KKT condition is also provided for solving the dual problem. Then, the feedback Stackelberg equilibrium is provided for the linear-quadratic stochastic Stackelberg differential games with affine constraints, and a new positive definite condition is proposed for ensuring the uniqueness of solutions to the dual problem. Finally, two non-degenerate examples with indefinite coefficients are provided to illustrate and to support our main results.

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arXiv2024

Stochastic Linear-quadratic Control Problems with Affine Constraints

This paper investigates the stochastic linear-quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality theory, the dual problem of original problem is established and the state feedback form of the solution to the optimal control problem is obtained. Under the Slater condition, the equivalence is proved between the solutions to the original problem and the ones of the dual problem, and the KKT condition is also provided for solving original problem. Especially, a new sufficient condition is given for the invertibility assumption, which ensures the uniqueness of the solutions to the dual problem.

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arXiv2024

Optimal Control Problems Governed by MFSDEs with multi-defaults

In this paper, we solve an optimal control problem governed by a system of mean-field stochastic differential equations with multiple defaults (MMFSDEs). We transform the global optimal control problem into several optimal control subproblems governed by a system of mean-field stochastic differential equations with single default (SMFSDEs) and derive both the sufficient and necessary maximum principles for these subproblems. We also give the existence and uniqueness of solutions to the MMFSDEs and the mean-field backward stochastic differential equations with multiple defaults (MMFBSDEs), respectively. Finally, as an example, our results are applied to obtain the explicit solution for an optimal control problem whose cost function is considered as a recursive utility process with multiple defaults.

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arXiv2021

A Linear-quadratic Mean-Field Stochastic Stackelberg Differential Game with Random Exit Time

In this paper, we investigate a new model of a linear-quadratic mean-field stochastic Stackelberg differential game with one leader and two followers, in which the leader is allowed to stop her strategy at a random time. Our overarching goal is to find the Stackelberg solution of the leader and followers for such a model. By employing the backward induction method, the state equation is divided into two-stage equations. Moreover, by using the maximum principle and the verification theorem, the Stackelberg solution is obtained for such a model.

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arXiv2021

Set-valued Ito's formula with an application to the general set-valued backward stochastic differential equation

The overarching goal of this paper is to establish a set-valued Itô's formula. As an application, we obtain the existence and uniqueness of solutions for the general set-valued backward stochastic differential equation which gives an answer to an open question proposed by Ararat et al. (C. Ararat, J. Ma and W.Q. Wu, Set-valued backward stochastic differential equation, arXiv:2007.15073).

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arXiv2020

A stochastic optimal control problem governed by SPDEs via a spatial-temporal interaction operator

In this paper, we first introduce a new spatial-temporal interaction operator to describe the space-time dependent phenomena. Then we consider the stochastic optimal control of a new system governed by a stochastic partial differential equation with the spatial-temporal interaction operator. To solve such a stochastic optimal control problem, we derive an adjoint backward stochastic partial differential equation with spatial-temporal dependence by defining a Hamiltonian functional, and give both the sufficient and necessary (Pontryagin-Bismut-Bensoussan type) maximum principles. Moreover, the existence and uniqueness of solutions are proved for the corresponding adjoint backward stochastic partial differential equations. Finally, our results are applied to study the population growth problems with the space-time dependent phenomena.

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