arXiv subjects

Jingtao Shi

Publications and source records attributed to Jingtao Shi.

At least 55 records · Page 3Linked to original sources

arXiv2022

A Three-level Stochastic Linear-quadratic Stackelberg Differential Game with Asymmetric Information

This paper is concerned with a three-level stochastic linear-quadratic Stackelberg differential game with asymmetric information, in which three players participate credited as Player 1, Player 2 and Player 3. Player 3 acts as the leader of Player 2 and Player 1, Player 2 acts as the leader of Player 1 and Player 1 acts as the follower. The asymmetric information considered is: the information available to Player 1 is based on the sub-$σ$-algebra of the information available to Player 2, and the information available to Player 2 is based on the sub-$σ$-algebra of the information available to Player 3. By maximum principle of forward-backward stochastic differential equations and optimal filtering, feedback Stackelberg equilibrium of the game is given with the help of a new system consisting of three Riccati equations.

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arXiv2022

Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps

This paper is concerned with the stochastic linear-quadratic optimal control problem with Poisson jumps. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed indefinite. The notion of closed-loop strategies is introduced, and the optimal closed-loop strategy is characterized by a Riccati integral-differential equation and a backward stochastic differential equation with Poisson jumps.

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arXiv2022

The Global Maximum Principle for Progressive Optimal Control of Partially Observed Forward-Backward Stochastic Systems with Random Jumps

IIn this paper, we study a partially observed progressive optimal control problem of forward-backward stochastic differential equations with random jumps, where the control domain is not necessarily convex, and the control variable enter into all the coefficients. In our model, the observation equation is not only driven by a Brownian motion but also a Poisson random measure, which also have correlated noises with the state equation. For preparation, we first derive the existence and uniqueness of the solutions to the fully coupled forward-backward stochastic system with random jumps and its estimation in $L^β(β\geq2)$-space under some assumptions, and the non-linear filtering equation of partially observed stochastic system with random jumps. Then we derive the partially observed global maximum principle with random jumps. To show its applications, a partially observed linear quadratic progressive optimal control problem with random jumps is investigated, by the maximum principle and stochastic filtering. State estimate feedback representation of the optimal control is given in a more explicit form by introducing some ordinary differential equations.

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arXiv2022

The Maximum Principle for Discounted Optimal Control of Partially Observed Forward-Backward Stochastic Systems with Jumps on Infinite Horizon

This paper is concerned with a discounted optimal control problem of partially observed forward-backward stochastic systems with jumps on infinite horizon. The control domain is convex and a kind of infinite horizon observation equation is introduced. The uniquely solvability of infinite horizon forward (backward) stochastic differential equation with jumps is obtained and more extended analysis, especially for the backward case, is made. Some new estimates are first given and proved for the critical variational inequality. Then an ergodic maximum principle is obtained by introducing some infinite horizon adjoint equations whose uniquely solvabilities are guaranteed necessarily. Finally, some comparison are made with two kinds of representative infinite horizon stochastic systems and their related optimal controls.

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arXiv2021

Linear Quadratic Leader-follower Stochastic Differential Games: Closed-Loop Solvability

In this paper, a leader-follower stochastic differential game is studied for a linear stochastic differential equation with a quadratic cost functional. The coefficients in the state equation and the weighting matrices in the cost functionals are all deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. The follower first solves a stochastic linear quadratic optimal control problem, and his optimal closed-loop strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem of a forward-backward stochastic differential equation, necessary conditions for the existence of optimal closed-loop strategies for the leader is given by the existence of a Riccati equation. Some examples are also given.

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arXiv2021

A Linear Quadratic Partially Observed Stackelberg Stochastic Differential Game with Applications

This paper is concerned with a linear-quadratic partially observed Stackelberg stochastic differential game with correlated state and observation noises, where the diffusion coefficient does not contain the control variable and the control set is not necessarily convex. Both the leader and the follower have their own observation equations, and the information filtration available to the leader is contained in that to the follower. By spike variational, state decomposition and backward separation techniques, necessary and sufficient conditions of the Stackelberg equilibrium points are derived. In the follower's problem, the state estimation feedback of optimal control can be represented by a forward-backward stochastic differential filtering equation and some Riccati equation. In the leader's problem, via the innovation process, the state estimation feedback of optimal control is represented by a stochastic differential filtering equation, a semi-martingale process and three high-dimensional Riccati equations. At the same time, the uniqueness and existence of solutions to adjoint equations can be guaranteed by a new combined idea, and a kind of fully coupled forward-backward stochastic differential equations with filtering is studied as a by-product. Then we give explicit expressions of Stackelberg equilibrium points in a special case. As a practical application, an inspiring dynamic advertising problem with asymmetric information is studied, and the effectiveness and reasonability of the theoretical result is illustrated by numerical simulations. Moreover, the relationship between optimal control, state estimate and some practical parameters is analyzed in detail.

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arXiv2021

Mean-Field Linear-Quadratic Stochastic Differential Games in an Infinite Horizon

This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. Existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite time horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

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arXiv2020

Linear Quadratic Stackelberg Stochastic Differential Games: Closed-Loop Solvability

This paper is concerned with the closed-loop solvability of one kind of linear-quadratic Stackelberg stochastic differential game, where the coefficients are deterministic. The notion of the closed-loop solvability is introduced, which require to be independent of the initial state. The follower's problem is solved first, and the closed-loop optimal strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the necessary conditions of the existence of the leader's nonanticipating closed-loop optimal strategy is obtained via a system of cross-coupled Riccati equations. The sufficiency is open since the completion-of-square method is invalid.

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arXiv2020

A Linear Quadratic Stochastic Stackelberg Differential Game with Time Delay

This paper is concerned with a linear quadratic stochastic Stackelberg differential game with time delay. The model is general, in which the state delay and the control delay both appear in the state equation, moreover, they both enter into the diffusion term. By introducing two Pseudo-Riccati equations and a special matrix equation, the state feedback representation of the open-loop Stackelberg strategy is derived, under some assumptions. Finally, two examples are given to illustrate the applications of the theoretical results.

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arXiv2020

Linear Quadratic Optimal Control Problems of Delayed Backward Stochastic Differential Equations

This paper is concerned with a linear quadratic optimal control problem of delayed backward stochastic differential equations. An explicit representation is derived for the optimal control, which is a linear feedback of the entire past history and the future state trajectory in a short period of time. This is one of the major distinctive features of the delayed backward stochastic linear quadratic optimal control problem. To obtain the optimal feedback, a new class of delayed Riccati equations is introduced and the unique solvability of their solutions are discussed in detail.

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arXiv2020

A Global Maximum Principle for the Stochastic Optimal Control Problem with Delay

In this paper, an open problem is solved, for the stochastic optimal control problem with delay where the control domain is nonconvex and the diffusion term contains both control and its delayed term. Inspired by previous results by Øksendal and Sulem [{\it A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In J. M. Menaldi, E. Rofman, A. Sulem (Eds.), Optimal control and partial differential equations, ISO Press, Amsterdam, 64-79, 2000}] and Chen and Wu [{\it Maximum principle for the stochastic optimal control problem with delay and application, Automatica, 46, 1074-1080, 2010}], Peng's general stochastic maximum principle [{\it A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 966-979, 1990}] is generalized to the time delayed case, which is called the global maximum principle. A new backward random differential equation is introduced to deal with the cross terms, when applying the duality technique. Comparing with the classical result, the maximum condition contains an indicator function, in fact it is the characteristic of the stochastic optimal control problem with delay. The multi-dimensional case and a solvable linear-quadratic example are also discussed.

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arXiv2020

Stackelberg Stochastic Differential Game with Asymmetric Noisy Observations

This paper is concerned with a Stackelberg stochastic differential game with asymmetric noisy observation, with one follower and one leader. In our model, the follower cannot observe the state process directly, but could observe a noisy observation process, while the leader can completely observe the state process. Open-loop Stackelberg equilibrium is considered. The follower first solve an stochastic optimal control problem with partial observation, the maximum principle and verification theorem are obtained. Then the leader turns to solve an optimal control problem for a conditional mean-field forward-backward stochastic differential equation, and both maximum principle and verification theorem are proved. An linear-quadratic Stackelberg stochastic differential game with asymmetric noisy observation is discussed to illustrate the theoretical results in this paper. With the aid of some Riccati equations, the open-loop Stackelberg equilibrium admits its state estimate feedback representation.

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arXiv2020

Optimal Reinsurance and Investment Strategies under Mean-Variance Criteria: Partial and Full Information

This paper is concerned with an optimal reinsurance and investment problem for an insurance firm under the criterion of mean-variance. The driving Brownian motion and the rate in return of the risky asset price dynamic equation cannot be directly observed. And the short-selling of stocks is prohibited. The problem is formulated as a stochastic linear-quadratic (LQ) optimal control problem where the control variables are constrained. Based on the separation principle and stochastic filtering theory, the partial information problem is solved. Efficient strategies and efficient frontier are presented in closed forms via solutions to two extended stochastic Riccati equations. As a comparison, the efficient strategies and efficient frontier are given by the viscosity solution for the Hamilton-Jacobi-Bellman (HJB) equation in the full information case. Some numerical illustrations are also provided.

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arXiv2019

Stochastic Recursive Optimal Control Problem with Mixed Delay under Viscosity Solution's Framework

This paper is concerned with the stochastic recursive optimal control problem with mixed delay. The connection between Pontryagin's maximum principle and Bellman's dynamic programming principle is discussed. Without containing any derivatives of the value function, relations among the adjoint processes and the value function are investigated by employing the notions of super- and sub-jets introduced in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is really optimal.

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arXiv2019

A Stackelberg Game of Backward Stochastic Differential Equations with Partial Information

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-$σ$-algebra of that of the leader. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the partial information stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs with partial information is investigated. The state estimate feedback representation for the optimal control of the follower is first given via two Riccati equations. Then the leader's problem is formulated as an optimal control problem of FBSDE. Four high-dimensional Riccati equations are introduced to represent the state estimate feedback for the optimal control of the leader. Theoretic results are applied to a pension fund management problem of two players in the financial market.

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arXiv2019

An Optimal Investment Problem under Correlated Noises: Risk-Sensitive Stochastic Control Approach

This paper is concerned with an optimal investment problem under correlated noises in the financial market, and the expected utility functional is hyperbolic absolute risk aversion (HARA) with the exponent $γ\neq0$. The problem can be reformulated as a risk-sensitive stochastic control problem. A new stochastic maximum principle is obtained first, where the adjoint equations and maximum condition heavily depend on the risk-sensitive parameter and the correlation coefficient. The optimal investment strategy is obtained explicitly in a state feedback form via the solution to a certain Riccati equation, under the risk-seeking case. Numerical simulation and figures are given to illustrate the sensitivity for the optimal investment strategy, with respect to the risk-sensitive parameter and the correlation coefficient.

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arXiv2019

A Stackelberg Game of Backward Stochastic Differential Equations with Applications

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs), where the coefficients of the backward system and the cost functionals are deterministic, and the control domain is convex. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs is investigated under standard assumptions. The state feedback representation for the optimal control of the follower is first given via two Riccati equations. Then the leader's problem is formulated as an optimal control problem of FBSDE with the control-independent diffusion term. Two high-dimensional Riccati equations are introduced to represent the state feedback for the optimal control of the leader. The solvability of the four Riccati equations are discussed. Theoretic results are applied to an optimal consumption rate problem of two players in the financial market.

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arXiv2018

Stochastic Linear Quadratic Stackelberg Differential Game with Overlapping Information

This paper is concerned with the stochastic linear quadratic Stackelberg differential game with overlapping information, where the diffusion terms contain the control and state variables. Here the term "overlapping" means that there are common part between the follower's and the leader's information, while they have no inclusion relation. Optimal controls of the follower and the leader are obtained by the stochastic maximum principle, the direct calculation of the derivative of the cost functional and stochastic filtering. A new system of Riccati equations is introduced to represent the state estimate feedback of the Stackelberg equilibrium strategy. A special solvable case is then studied and is applied to the continuous-time principal-agent problem.

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