arXiv subjects

Jingtao Shi

Publications and source records attributed to Jingtao Shi.

At least 19 recordsLinked to original sources

arXiv2026

Relationship between MP and DPP for Risk-Sensitive Stochastic Optimal Control Problems: Viscosity Solution Framework

In this paper, we study the relationship between general maximum principle and dynamic programming principle for risk-sensitive stochastic optimal control problems, where the control domain is not necessarily convex. The original problem is equivalent to a stochastic recursive optimal control problem of a forward-backward system with quadratic generators. Relations among the adjoint processes, the generalized Hamiltonian function and the value function are proved under the framework of viscosity solutions. Some examples are given to illustrate the theoretical results.

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arXiv2026

Stochastic Optimal Control Problem under Inside Information

This paper is concerned with a stochastic optimal control problem under inside information. The control process depends on an $\mathcal{F}_{T_0}$-measurable random variable $Y$, representing the static inside information, and is adapted to the enlarged filtration generated by the underlying Brownian motion and the random variable $Y$. Accordingly, the traditional stochastic integral fails to be well-defined in this non-adapted setting; we adopt forward integrals to formulate the stochastic integral terms in the system. By means of the Donsker delta function, the original controlled system is transformed into a $y$-parameterized system. We further establish the existence of solutions to forward \textit{stochastic differential equations} (SDEs), and prove the uniqueness of solutions via flow transformation techniques. Under a Gaussian assumption on $Y$, we derive both necessary and sufficient optimality conditions for the aforementioned control problem. Subsequently, we formulate the \textit{linear-quadratic} (LQ) optimal control problem under inside information. Through the $y$-parameterized transformation, the original problem is converted into an LQ control problem with random coefficients. A numerical example for the LQ case is provided at the end to validate our theoretical findings.

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arXiv2026

An $α$-Potential Game Approach to $N$-Player Stochastic Linear-Quadratic Differential Games

This paper studies $N$-player stochastic linear-quadratic (LQ) differential games from the perspective of $α$-potential games. We first consider a closed-loop LQ game with multiplicative noise, where both the drift and the diffusion coefficients depend linearly on the state and the full control vector. For this model, we derive probabilistic and partial differential equation (PDE) representations for the first- and second-order linear derivatives of the players' cost function and prove the equivalence between them. We then develop an open-loop stochastic LQ $α$-potential game framework. Using the linear derivative construction, we build an $α$-potential function and derive an explicit upper bound for the approximation parameter $α$ in terms of the model coefficients and the admissible control radius. Moreover, the minimization of the $α$-potential function is reduced to a finite-dimensional stochastic control problem by augmenting the state with the variational process, which yields an open-loop $α$-Nash equilibrium. As an application, we revisit a network LQ game considered in \cite{GuoLiZhang2025} and show that the feedback representation obtained from our approach coincides with the feedback in the existing conditional McKean--Vlasov approach, while our characterization follows directly from a standard finite-dimensional LQ control problem.

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arXiv2026

A linear-quadratic partially observed Stackelberg stochastic differential game with multiple followers and its application to multi-agent formation control

In this paper, we study a linear-quadratic partially observed Stackelberg stochastic differential game problem in which a single leader and multiple followers are involved. We consider more practical formulation for partial information that none of them can observed the complete information and the followers know more than the leader. Some completely different methods including a novel state decomposition and orthogonal decomposition are applied to overcome the difficulties caused by partially observability which improves the tools and relaxes the constraint condition imposed on admissible control in the existing literature. More precisely, the followers encounter the standard linear-quadratic partially observed optimal control problems, however, a kind of forward-backward indefinite linear-quadratic partially observed optimal control problem is considered by the leader. Instead of maximum principle of forward-backward control systems, inspired by the existing work related to definite case and classical forward control system, some distinct forward-backward linear-quadratic decoupling techniques including the method of completion of squares are applied to solve the leader's problem. More interestingly, we develop the deterministic formation control in multi-agent system with a framework of Stackelberg differential game and extend it to the stochastic case. The optimal strategies are obtained by our theoretical result suitably.

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arXiv2026

Robust Incentive Stackelberg Mean Field Stochastic Linear-Quadratic Differential Game with Model Uncertainty

This paper investigates a robust incentive Stackelberg stochastic differential game problem for a linear-quadratic mean field system, where the model uncertainty appears in the drift term of the leader's state equation. Moreover, both the state average and control averages enter into the leader's dynamics and cost functional. Based on the zero-sum game approach, mean field approximation and duality theory, firstly the representation of the leader's limiting cost functional and the closed-loop representation of decentralized open-loop saddle points are given, via decoupling methods. Then by convex analysis and the variational method, the decentralized strategies of the followers' auxiliary limiting problems and the corresponding consistency condition system are derived. Finally, applying decoupling technique, the leader's approximate incentive strategy set is obtained, under which the asymptotical robust incentive optimality of the decentralized mean field strategy is verified. A numerical example is given to illustrate the theoretical results.

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arXiv2026

Leader-Follower Linear-Quadratic Stochastic Graphon Games

This paper investigates leader-follower linear-quadratic stochastic graphon games, which consist of a single leader and a continuum of followers. The state equations of the followers interact through graphon coupling terms, with their diffusion coefficients depending on the state, the graphon aggregation term, and the control variables. The diffusion term of the leader's state equation depends on its state and control variables. Within this framework, a hierarchical decision-making structure is established: for any strategy adopted by the leader, the followers compete to attain a Nash equilibrium, while the leader optimizes its own cost functional by anticipating the followers' equilibrium response. This work develops a rigorous mathematical model for the game, proves the existence and uniqueness of solutions to the system's state equations under admissible control sets, and constructs a Stackelberg-Nash equilibrium for the continuum follower game. By employing the continuity method, we establish the existence, uniqueness, and stability of solutions to the associated forward-backward stochastic differential equation with a graphon aggregation term.

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arXiv2026

A Partially Observed Stochastic Linear Stackelberg Differential Game with Poisson Jumps under Mean-Variance Criteria

In this paper, a partially observed stochastic linear Stackelberg differential game with mean-variance criteria is studied. Randomness comes from Brownian motions and Poisson random measures. which leads to a circular dependency. We follow the orthogonal decomposition method to overcome the circular dependency of the control and state processes. Both original problems of the follower and leader are decomposed into several fully observed problems with mean-variance criteria. During these processes, non-linear stochastic filtering with Poisson random measures, developed in this paper, plays an important role. Besides the follower's problem is embedded into a class of auxiliary stochastic linear-quadratic optimal control problem of stochastic differential equations with Poisson jumps, the leader's problem is also embedded into a class of auxiliary stochastic linear-quadratic optimal control problem of forward-backward stochastic differential equations with Poisson jumps. Observable state feedback Stackelberg equilibria are obtained, via some Riccati equations.

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arXiv2026

The Optimal Control Problem of Stochastic Differential System with Extended Mixed Delays and Applications

This paper investigates an optimal control problem where the system is described by a stochastic differential equation with extended mixed delays that contain point delay, extended distributed delay, and extended noisy memory. The model is general in that the extended mixed delays of the state variable and control variable are components of all the coefficients, in particular, the diffusion term and the terminal cost. To address the difficulties induced by the extended noisy memory, by stochastic Fubini theorem, we transform the delay variational equation into a Volterra integral equation without delay, and then a kind of backward stochastic Volterra integral equation with Malliavin derivatives is introduced by the developed coefficient decomposition method and the generalized duality principle. Therefore, the stochastic maximum principle and the verification theorem are established. Subsequently, with Clark-Ocone formula, the adjoint equation is expressed as a set of anticipated backward stochastic differential equations. Finally, a nonzero-sum stochastic differential game with extended mixed delays and a linear-quadratic solvable example are discussed, as applications.

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arXiv2026

Stochastic Linear-Quadratic Optimal Control Problems with Markovian Regime Switching and $H_\infty$ Constraint under Partial Information

This paper is concerned with a stochastic linear-quadratic optimal control problem of Markovian regime switching system with model uncertainty and partial information, where the information available to the control is based on a sub-$σ$-algebra of the filtration generated by the underlying Brownian motion and the Markov chain. Based on $H_\infty$ control theory, we turn to deal with a soft-constrained zero-sum linear-quadratic stochastic differential game with Markov chain and partial information. By virtue of the filtering technique, the Riccati equation approach, the method of orthogonal decomposition, and the completion-of-squares method, we obtain the closed-loop saddle point of the zero-sum game via the optimal feedback control-strategy pair. Subsequently, we prove that the corresponding outcome of the closed-loop saddle point satisfies the $H_\infty$ performance criterion. Finally, the obtained theoretical results are applied to a stock market investment problem to further illustrate the practical significance and effectiveness.

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arXiv2025

Linear-Quadratic Partially Observed Mean Field Stackelberg Stochastic Differential Game with Applications

This paper is concerned with a linear-quadratic partially observed mean field Stackelberg stochastic differential game, which contains a leader and a large number of followers. Specifically, the followers confront a large-population Nash game subsequent to the leader's initial announcement of his strategy. In turn, the leader optimizes his own cost functional, taking into account the anticipated reactions of the followers. The state equations of both the leader and the followers are general stochastic differential equations, where the drift terms contain both the state average term and the state expectation term. However, the followers' state average terms enter into the drift term of the leader's state equation and the state expectation term of the leader enters into the state equation of the follower, reflecting the mutual influence between the leader and the followers. By utilizing the techniques of state decomposition and backward separation principle, we deduce the open-loop adapted decentralized strategies and feedback decentralized strategies of this leader-followers system, and demonstrate that the decentralized strategies are the corresponding $\varepsilon$-Stackelberg-Nash equilibrium. Finally, we apply the theoretical result to a product planning problem with sticky prices.

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arXiv2025

MP and DPP for Mean-Variance Portfolio Selection Problem with Poisson Jumps, Recursive Utility and Their Relationship

In this paper, the mean-variance portfolio selection problem with Poisson jumps are studied, where the recursive utility is given by the solution to a backward stochastic differential equation with Poisson jumps. Both the maximum principle and dynamic programming principle are applied to solve this problem, and their relationship is also investigated. The optimal portfolio and efficient frontier of Markowitz's type are derived using both methods. A comparison of efficient frontiers obtained in this paper and in the framework without jumps is conducted.

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arXiv2025

Linear-Quadratic Non-zero Sum Differential Game with Asymmetric Delayed Information

This paper is concerned with a linear-quadratic non-zero sum differential game with asymmetric delayed information. To be specific, two players exist time delays simultaneously which are different, leading the dynamical system being an asymmetric information structure. By virtue of stochastic maximum principle, the stochastic Hamiltonian system is given which is a delayed forward-backward stochastic differential equation. Utilizing discretisation approach and backward iteration technique, we establish the relationship between forward and backward processes under asymmetric delayed information structure and obtain the state-estimate feedback Nash equilibrium of our problem.

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arXiv2025

A Linear-Quadratic Stackelberg Differential Game with Mixed Deterministic and Stochastic Controls

This paper is concerned with a linear-quadratic (LQ) leader-follower differential game with mixed deterministic and stochastic controls. In the game, the follower is a random controller which means that the follower can choose adapted stochastic processes, while the leader is a deterministic controller which means that the leader can choose only deterministic time functions. Such problem is motivated by a pension fund insurance problem, with government, supervisory or employer being a deterministic leader and individual producer or retail investor being a random follower. An open-loop Stackelberg equilibrium solution is considered. First, an optimal control process of the follower is characterized by a stationary condition of forward-backward stochastic differential equation (FBSDE) and a convexity condition of SDE. Then it is represented as a linear functional of optimal state variable of the follower and the leader's control variable, via a classical Riccati equation. Then an optimal control function of the leader is first characterized by a convexity condition of FBSDE and a stationary condition of mean-field type FBSDE. And it is represented as a functional of expectation of optimal state variable of the leader, with the help of a system consisting of two cross-coupled Riccati equations and a two-point boundary value problem of ordinary differential equations (ODEs). The solvabilities of this new system of Riccati equations and two-point boundary value problem and investigated.

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arXiv2025

General Linear-Quadratic Mean Field Stochastic Differential Game with Common Noise: A Direct Method

This paper investigates a class of general linear-quadratic mean field games with common noise, where the diffusion terms of the system contain the state variables, control variables, and the average state terms. We solve the problem using both the direct method and the fixed-point method in the paper. First, by using the variational method to solve a finite $N$-players game problem, we obtain the necessary and sufficient conditions for the centralized open-loop Nash equilibrium strategy. Subsequently, by employing the decoupling technique, we derive the state feedback representation of the centralized open-loop Nash equilibrium strategy in terms of Riccati equations. Next, by studying the asymptotic solvability of the Riccati equations, we construct the decentralized open-loop asymptotic Nash equilibrium strategies. Finally, through some estimates, we prove the asymptotic optimality of the decentralized open-loop Nash equilibrium strategies. Moreover, we find that the decentralized open-loop asymptotic Nash equilibrium strategies obtained via the fixed-point method are identical to those derived using the direct method. Finally, we apply the main results of this paper to a concrete production planning example.

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arXiv2025

Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach

This paper investigates a linear-quadratic mean field games problem with common noise, where the drift term and diffusion term of individual state equations are coupled with both the state, control, and mean field terms of the state, and we adopt the direct approach to tackle this problem. Compared with addressing the corresponding mean field teams problem, the mean field games problem with state coupling presents greater challenges. This is not only reflected in the explosive increase in the number of adjoint equations when applying variational analysis but also in the need for more Riccati equations during decoupling the high-dimensional forward-backward stochastic differential equations system. We take a different set of steps and ingeniously utilize the inherent properties of the equations to address this challenge. First, we solve an $N$-player games problem within a vast and finite population setting, and obtain a set of forward-backward stochastic differential equations by variational analysis. Then, we derive the limiting forward-backward stochastic differential equations by taking the limit as $N$ approaches infinity and applying the law of large numbers. Based on the existence and uniqueness of solutions to backward stochastic differential equations, some variables in the equations are identically zero, which significantly reduces the complexity of the analysis. This allows us to introduce just two Riccati equations to explicitly construct decentralized strategies for all participants. Moreover, we demonstrate that the constructed decentralized strategies constitute an $ε$-Nash equilibrium strategy for the original problem. We also extend the results to the infinite-horizon case and analyze the solvability of algebraic Riccati equations. Finally, numerical simulations are provided to illustrate the preceding conclusions.

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arXiv2025

Two Stochastic Control Methods for Mean-Variance Portfolio Selection of Jump Diffusions and Their Relationship

This paper is concerned with the maximum principle and dynamic programming principle for mean-variance portfolio selection of jump diffusions and their relationship. First, the optimal portfolio and efficient frontier of the problem are obtained using both methods. Furthermore, the relationship between these two methods is investigated. Specially, the connections between the adjoint processes and value function are given.

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arXiv2025

Relationship between Maximum Principle and Dynamic Programming Principle for Risk-Sensitive Stochastic Optimal Control Problems with Applications

This paper is concerned with the relationship between maximum principle and dynamic programming principle for risk-sensitive stochastic optimal control problems. Under the smooth assumption of the value function, relations among the adjoint processes, the generalized Hamiltonian function, and the value function are given. As an application, a linear-quadratic risk-sensitive portfolio optimization problem in the financial market is discussed.

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arXiv2025

Global Maximum Principle for Partially Observed Risk-Sensitive Progressive Optimal Control of FBSDE with Poisson Jumps

This paper is concerned with one kind of partially observed progressive optimal control problems of coupled forward-backward stochastic systems driven by both Brownian motion and Poisson random measure with risk-sensitive criteria. The control domain is not necessarily convex, and the control variable can enters into all the coefficients. The observation equation also has correlated noises with the state equation. Under the Poisson jump setting, the original problem is equivalent to a complete information stochastic recursive optimal control problem of a forward-backward system with quadratic-exponential generator. In order to establish the first- and second-order variations, some new techniques are introduced to overcome difficulties caused by the quadratic-exponential feature. A new global stochastic maximum principle is deduced. As an application, a risk-sensitive optimal investment problem with factor model is studied. Moreover, the risk-sensitive stochastic filtering problem is also studied, which involves both Brownian and Poissonian correlated noises. A modified Zakai equation is obtained.

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