arXiv subjects

Qingmeng Wei

Publications and source records attributed to Qingmeng Wei.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic Representations of Stationary HJBI-Type Variational Inequalities with Bilateral Constraints

In this paper, we study probabilistic representations for stationary HJBI-type variational inequalities with bilateral constraints. We provide two complementary stochastic representations.The first representation is obtained through an augmented infinite-horizon two-player zero-sum stochastic differential game (SDG). By enlarging the control spaces with two additional stopping symbols, the obstacle terms are incorporated into the running payoff. Using the framework of infinite-horizon stochastic recursive differential games, we show that the resulting lower and upper value functions are the unique bounded viscosity solutions of the corresponding HJBI variational inequalities. The second representation is given by a two-player zero-sum mixed control--stopping SDG. In this formulation, each player chooses both a continuous control and a stopping decision, and the payoff is defined by a BSDE with a random terminal time. To make the stopping component compatible with the Elliott--Kalton strategy framework, we introduce nonanticipative stopping strategies depending on the opponent's control process. The proof is based on penalized infinite-horizon SDGs coupled with their own value functions, together with dynamic programming arguments and stability estimates for backward semigroups. We prove that the value functions of the mixed control--stopping game coincide with the unique bounded viscosity solutions of the bilateral HJBI variational inequalities.

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arXiv2026

Infinite-Horizon Non-Autonomous Zero-Sum Stochastic Recursive Differential Games and HJBI Equations

In this paper, we study an infinite horizon non-autonomous stochastic recursive differential game. To this end, we first establish well-posedness and stability results for BSDEs with a time-dependent discount factor and a possibly unbounded random terminal time. The generator $f$ is allowed to be non-uniformly bounded at the origin, namely, $|f(t,0,0)|\les \beta_1(t)+\beta_2,$ $t\in[0,\infty),$ $\dbP\text{-a.s.},$ with $\beta_1\in L^1(0,\infty)$ and $\beta_2\ges0$. We then formulate a two-person zero-sum stochastic recursive differential game on the infinite horizon, where the drift, diffusion, generator and discount factor may depend explicitly on time. The lower and upper value functions are defined through Elliott--Kalton nonanticipative strategies and BSDE recursive payoffs. By finite horizon approximation, BSDE stability estimates and viscosity solution arguments, we prove that both the lower and upper value functions are deterministic and are the unique bounded viscosity solutions of their corresponding non-autonomous HJBI equations. Finally, the time-homogeneous case is recovered as a special case. Using the uniqueness of the non-autonomous HJBI equation, rather than a probabilistic shift argument, we show from the PDE viewpoint that the value functions of the autonomous system are independent of the initial time and solve the corresponding stationary HJBI equations in the viscosity sense.

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arXiv2025

Infinite Horizon Linear Quadratic Mean Field Problems with Common Noise and Regime Switching via Conditional McKean-Vlasov FBSDEs

This paper studies infinite horizon linear quadratic (LQ) mean field problems with common noise and regime switching, covering both control and game formulations. To establish a theoretical foundation for the LQ framework, we first analyze fully coupled forward-backward stochastic differential equations (FBSDEs) of conditional McKean-Vlasov type with Markovian switching and establish its well-posedness under a generalized domination-monotonicity condition. Building upon this solvability result, we then derive necessary and sufficient conditions for both the open-loop optimal control in the control problem and the mean-field Nash equilibria in the game problem.

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arXiv2025

A Time-Inconsistent Stochastic Optimal Control Problem in an Infinite Time Horizon

This paper is concerned with a time-inconsistent stochastic optimal control problem in an infinite time horizon with a non-degenerate diffusion in the state equation. A major assumption is that people become rational after a large time. Under such a condition, the problem in an infinite time horizon can be decomposed into two parts: a non-autonomous time-consistent problem in an infinite time horizon and a time-inconsistent problem in a finite time horizon. Then an equilibrium strategy will be constructed. Both Bolza type problem and recursive cost problem are considered.

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arXiv2025

Optimal control of SDEs with merely measurable drift: an HJB approach

We investigate an optimal control problem for a diffusion whose drift and running cost are merely measurable in the state variable. Such low regularity rules out the use of Pontryagin's maximum principle and also invalidates the standard proof of the Bellman principle of optimality. We address these difficulties by analyzing the associated Hamilton-Jacobi-Bellman (HJB) equation. Working in a weak formulation of admissible controls, we first establish the state-equation solvability and Krylov estimates needed to make the control problem well defined. Using PDE techniques together with a policy iteration scheme, we prove that the HJB equation admits a unique strong solution, and this solution coincides with the value function of the control problem. Based on this identification, we establish a verification theorem and recover the Bellman optimality principle without imposing any additional smoothness assumptions. We further investigate a mollification scheme depending on a parameter $\varepsilon > 0$. It turns out that the smoothed value functions $V_{\varepsilon}$ may fail to converge to the original value function $V$ as $\varepsilon \to 0$, and we provide an explicit counterexample. To resolve this, we identify a structural condition on the control set. When the control set is countable, convergence $V_{\varepsilon} \to V$ holds locally uniformly.

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arXiv2025

Reflected stochastic recursive control problems with jumps: dynamic programming and stochastic verification theorems

This paper mainly investigates reflected stochastic recursive control problems governed by jump-diffusion dynamics. The system's state evolution is described by a stochastic differential equation driven by both Brownian motion and Poisson random measures, while the recursive cost functional is formulated via the solution process Y of a reflected backward stochastic differential equation driven by the same dual stochastic sources. By establishing the dynamic programming principle, we provide the probabilistic interpretation of an obstacle problem for partial integro-differential equations of Hamilton-Jacobi-Bellman type in the viscosity solution sense through our control problem's value function. Furthermore, the value function is proved to inherit the semi-concavity and joint Lipschitz continuity in state and time coordinates, which play key roles in deriving stochastic verification theorems of control problem within the framework of viscosity solutions. We remark that some restrictions in previous study are eliminated, such as the frozen of the reflected processes in time and state, and the independence of the driver from diffusion variables.

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arXiv2025

Infinite Horizon Mean-Field Linear-Quadratic Optimal Control Problems with Switching and Indefinite-Weighted Costs

This paper is concerned with an infinite horizon stochastic linear quadratic (LQ, for short) optimal control problems with conditional mean-field terms in a switching environment. Different from [17], the cost functionals do not have positive-definite weights here. When the problems are merely finite, we construct a sequence of asymptotic optimal controls and derive their closed-loop representations. For the solvability, an equivalence result between the open-loop and closed-loop cases is established through algebraic Riccati equations and infinite horizon backward stochastic differential equations. It can be seen that the research in [17] with positive-definite weights is a special case of the current paper.

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arXiv2025

Linear-Quadratic Optimal Control for Mean-Field Stochastic Differential Equations in Infinite-Horizon with Regime Switching

This paper is concerned with stochastic linear quadratic (LQ, for short) optimal control problems in an infinite horizon with conditional mean-field term in a switching regime environment. The orthogonal decomposition introduced in [21] has been adopted. Desired algebraic Riccati equations (AREs, for short) and a system of backward stochastic differential equations (BSDEs, for short) in infinite time horizon with the coefficients depending on the Markov chain have been derived. The determination of closed-loop optimal strategy follows from the solvability of ARE and BSDE. Moreover, the solvability of BSDEs leads to a characterization of open-loop solvability of the optimal control problem.

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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.

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arXiv2023

Infinite Horizon Mean-Field Linear Quadratic Optimal Control Problems with Jumps and the related Hamiltonian Systems

In this work, we focus on an infinite horizon mean-field linear-quadratic stochastic control problem with jumps. Firstly, the infinite horizon linear mean-field stochastic differential equations and backward stochastic differential equations with jumps are studied to support the research of the control problem. The global integrability properties of their solution processes are studied by introducing a kind of so-called dissipation conditions suitable for the systems involving the mean-field terms and jumps. For the control problem, we conclude a sufficient and necessary condition of open-loop optimal control by the variational approach. Besides, a kind of infinite horizon fully coupled linear mean-field forward-backward stochastic differential equations with jumps is studied by using the method of continuation. Such a research makes the characterization of the open-loop optimal controls more straightforward and complete.

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arXiv2023

General mean-field BSDEs with diagonally quadratic generators in multi-dimension

The purpose of this paper is to investigate general mean-field backward stochastic differential equations (MFBSDEs) in multi-dimension with diagonally quadratic generators $f(ω,t,y,z,μ)$, that is, the coefficients depend not only on the solution processes $(Y,Z)$, but also on their law $\mathbb{P}_{(Y,Z)}$, as well as have a diagonally quadratic growth in $Z$ and super-linear growth (or even a quadratic growth) in the law of $Z$ which is totally new. We start by establishing through a fixed point theorem the existence and the uniqueness of local solutions in the ``Markovian case'' $f(t,Y_{t},Z_{t},\mathbb{P}_{(Y_{t},Z_{t})})$ when the terminal value is bounded. Afterwards, global solutions are constructed by stitching local solutions. Finally, employing the $θ$-method, we explore the existence and the uniqueness of global solutions for diagonally quadratic mean-field BSDEs with convex generators, even in the case of unbounded terminal values that have exponential moments of all orders. These results are extended to a Volterra-type case where the coefficients can even be of quadratic growth with respect to the law of $Z$.

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arXiv2023

Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations with a Type of Random Coefficients

Motivated by linear-quadratic optimal control problems (LQ problems, for short) for mean-field stochastic differential equations (SDEs, for short) with the coefficients containing regime switching governed by a Markov chain, we consider an LQ problem for an SDE with the coefficients being adapted to a filtration independent of the Brownian motion driving the control system. Classical approach of completing the square is applied to the current problem and obvious shortcomings are indicated. Open-loop and closed-loop solvability are introduced and characterized.

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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.

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arXiv2023

Stochastic Verification Theorems for Stochastic Control Problems of Reflected FBSDEs

In this paper, the stochastic verification theorems for stochastic control problems of reflected forward-backward stochastic differential equations are studied. We carry out the work within the frameworks of classical and viscosity solutions. The sufficient conditions of verifying the controls to be optimal are given. We also construct the feedback optimal control laws from the classical and viscosity solutions of the associated Hamilton-Jacobi-Bellman equations with obstacles. Finally, we apply the theoretical results in two concrete examples. One is for the case of the classical solution, and the other is for the case of the viscosity solution.

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arXiv2020

Optimal Ergodic Control of Linear Stochastic Differential Equations with Quadratic Cost Functionals Having Indefinite Weights

An optimal ergodic control problem (EC problem, for short) is investigated for a linear stochastic differential equation with quadratic cost functional. Constant nonhomogeneous terms, not all zero, appear in the state equation, which lead to the asymptotic limit of the state non-zero. Under the stabilizability condition, for any (admissible) closed-loop strategy, an invariant measure is proved to exist, which makes the ergodic cost functional well-defined and the EC problem well-formulated. Sufficient conditions, including those allowing the weighting matrices of cost functional to be indefinite, are introduced for finiteness and solvability for the EC problem. Some comparisons are made between the solvability of EC problem and the closed-loop solvability of stochastic linear quadratic optimal control problem in the infinite horizon. Regularized EC problem is introduced to be used to obtain the optimal value of the EC problem.

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arXiv2019

Linear Quadratic Stochastic Optimal Control Problems with Operator Coefficients: Open-Loop Solutions

An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is investigated, which is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients. Under proper conditions, the well-posedness of such an FBSDE is established, which leads to the existence of an open-loop optimal control. Finally, as an application of our main results, a general mean-field linear quadratic control problem in the open-loop case is solved.

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arXiv2016

Time-Inconsistent Recursive Stochastic Optimal Control Problems

In this paper, we study a time-inconsistent stochastic optimal control problem with a recursive cost functional by a multi-person hierarchical differential game approach. An equilibrium strategy of this problem is constructed and a corresponding equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation is established to characterize the associated equilibrium value function. Moreover, a well-posedness result of the equilibrium HJB equation is established under certain conditions.

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arXiv2013

An optimal control problem of forward-backward stochastic Volterra integral equations with state constraints

This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequality. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.

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