arXiv subjects

Tianxiao Wang

Publications and source records attributed to Tianxiao Wang.

At least 19 recordsLinked to original sources

arXiv2026

Closed-loop solvability of delayed control problems: A stochastic Volterra system approach

A general and new stochastic linear quadratic optimal control problem is studied, where the coefficients are allowed to be time-varying, and both state delay and control delay can appear simultaneously in the state equation and the cost functional. The closed-loop outcome control of this delayed problem is given by a new Riccati system whose solvability is carefully established. To this end, a novel method is introduced to transform the delayed problem into a control problem driven by a stochastic Volterra integral system without delay. This method offers several advantages: it bypasses the difficulty of decoupling the forward delayed state equation and the backward anticipated adjoint equation, avoids the introduction of infinite-dimensional spaces and unbounded control operators, and ensures that the closed-loop outcome control depends only on past state and control, without relying on future state or complex conditional expectation calculations. Finally, several particular important stochastic systems are discussed. It is found that the model can cover a class of stochastic integro-differential systems, whose closed-loop solvability has not been available before.

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arXiv2024

Causal feedback strategies for controlled stochastic Volterra systems: a unified treatment

This paper is concerned with a unified treatment of linear quadratic control problem for stochastic Volterra integral equations (SVIEs), motivated by the various approaches and scattered results in the existing literature. A novel class of optimal causal feedback strategy is introduced and characterized by means of a new Riccati system. To this end, a fundamental function space and an appropriate multiplicative rule among functions are defined for the first time. In contrast with the existing works, our unified treatment not only provides a new approach, but also extends or improves the known conclusions in stochastic differential equations, convolution SVIEs, stochastic Volterra integro-differential equations (VIDEs), deterministic VIEs, deterministic VIDEs. In addition, an interesting phenomenon is reveal by the current study: for SVIEs the conventional structure of state feedback is replaced by a suitable causal form, and the original state process no longer plays indispensable role in the feedbacks while an auxiliary state process does.

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arXiv2023

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

In this paper, the notion of singular backward stochastic Volterra integral equations (singular BSVIEs for short) in infinite dimensional space is introduced, and the corresponding well-posedness is carefully established. A class of singularity conditions are proposed, which not only cover that of fractional kernel, Volterra Heston model kernel, completely monotone kernels, to mention a few, but also happen to be used in the forward stochastic Volterra integral with new conclusions arising. Motivated by mathematical physics problem such as the viscoelasticity/thermoviscoelasticity of materials, heat conduction in materials with memory, optimal control problems of abstract stochastic Volterra integral equations (including fractional stochastic evolution equations and stochastic evolutionary integral equations) are presented. At last, our BSVIEs are surprisingly used in maximum principle of controlled stochastic delay evolution equations. One advantage of this new standpoint is that the final cost functional can naturally depend on the past state for the first time.

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arXiv2023

Linear-quadratic stochastic Volterra controls II: Optimal strategies and Riccati--Volterra equations

In this paper, we study linear-quadratic control problems for stochastic Volterra integral equations with singular and non-convolution-type coefficients. The weighting matrices in the cost functional are not assumed to be non-negative definite. From a new viewpoint, we formulate a framework of causal feedback strategies. The existence and the uniqueness of a causal feedback optimal strategy are characterized by means of the corresponding Riccati--Volterra equation.

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arXiv2023

Linear-quadratic stochastic Volterra controls I: Causal feedback strategies

In this paper, we formulate and investigate the notion of causal feedback strategies arising in linear-quadratic control problems for stochastic Volterra integral equations (SVIEs) with singular and non-convolution-type coefficients. We show that there exists a unique solution, which we call the causal feedback solution, to the closed-loop system of a controlled SVIE associated with a causal feedback strategy. Furthermore, introducing two novel equations named a Type-II extended backward stochastic Volterra integral equation and a Lyapunov--Volterra equation, we prove a duality principle and a representation formula for a quadratic functional of controlled SVIEs in the framework of causal feedback strategies.

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arXiv2023

A general maximum principle for optimal control of stochastic differential delay systems

In this paper, we solve an open problem and obtain a general maximum principle for a stochastic optimal control problem where the control domain is an arbitrary non-empty set and all the coefficients (especially the diffusion term and the terminal cost) contain the control and state delay. In order to overcome the difficulty of dealing with the cross term of state and its delay in the variational inequality, we propose a new method: transform a delayed variational equation into a Volterra integral equation without delay, and introduce novel first-order, second-order adjoint equations via the backward stochastic Volterra integral equation theory. Finally we express these two kinds of adjoint equations in more compact anticipated backward stochastic differential equation types for several special yet typical control systems.

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arXiv2022

Multi-Dimensional Super-Linear Backward Stochastic Volterra Integral Equations

In this paper, a systematic investigation is carried out for the general solvability of multi-dimensional backward stochastic Volterra integral equations (BSVIEs) with the generators being super-linear in the adjustment variable $Z$. Two major situations are discussed: (i) When the free term is bounded with the dependence of the generator on $Z$ being of ``diagonally strictly'' quadratic growth and being sub-quadratically coupled with off-diagonal components; (ii) When the free term is unbounded having exponential moments of arbitrary order with the dependence of the generator on $Z$ being diagonally no more than quadratic and being independent of off-diagonal components. Besides, for the case that the generator is super-quadratic in $Z$, some negative results are presented.

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arXiv2022

Spike Variations for Stochastic Volterra Integral Equations

Spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for differential equations of several types, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differentia equations (SDEs), when the control domains are not assumed to be convex. This technique also applies to (deterministic forward) Volterra intrgral equations (FVIEs). It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome the difficulty, we introduce an auxiliary process for which one can use Itô's formula, and adopt a trick used in linear-quadratic stochastic optimal control problems. Then a suitable representation of the above-mentioned quadratic form is obtained, and the second order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well.

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arXiv2022

Optimal Feedback Controls of Stochastic Linear Quadratic Control Problems in Infinite Dimensions with Random Coefficients

It is a longstanding unsolved problem to characterize the optimal feedback controls for general linear quadratic optimal control problem of stochastic evolution equation with random coefficients. A solution to this problem is given in [21] under some assumptions which can be verified for interesting concrete models, such as controlled stochastic wave equations, controlled stochastic Schrödinger equations, etc. More precisely, the authors establish the equivalence between the existence of optimal feedback operator and the solvability of the corresponding operator-valued, backward stochastic Riccati equations. However, their result cannot cover some important stochastic partial differential equations, such as stochastic heat equations, stochastic stokes equations, etc. A key contribution of the current work is to relax the $C_0$-group assumption of unbounded linear operator $A$ in [21] and using contraction semigroup assumption instead. Therefore, our result can be well applicable in the linear quadratic problem of stochastic parabolic partial differential equations. To this end, we introduce a suitable notion to the aforementioned Riccati equation, and some delicate techniques which are even new in the finite dimensional case.

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arXiv2018

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward-backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.

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arXiv2018

Mean-variance portfolio selection and variance hedging with random coefficients: closed-loop equilibrium strategy

In this paper, both dynamic mean-variance portfolio selection problems and dynamic variance hedging problems are discussed under non-Markovian framework. Explicit closed-loop equilibrium strategies of these problems are respectively obtained via a unified approach for the first time. Several new interesting facts arise in mean-variance problems with constant risk aversion. For example, it is shown that equilibrium strategies are still allowed to rely on initial wealth as long as risk-free return rate is random. In addition, the closed-loop equilibrium strategy and open-loop equilibrium strategy have the same connection with initial wealth in non-Markovian setting, and they happen to equal to each other if only risk-free return rate is deterministic.

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arXiv2018

Equilibrium controls in time inconsistent stochastic linear quadratic problems

This paper deals with a class of time inconsistent stochastic linear quadratic (SLQ) optimal control problems in Markovian framework. Three notions, i.e., closed-loop equilibrium controls/strategies, open-loop equilibrium controls and their closed-loop representations, are characterized in unified manners. These results indicate clearer and deeper distinctions among these notions. For example, in particular time consistent setting, the open-loop equilibrium controls are fully characterized by first-order, second-order necessary optimality conditions, and become needlessly optimal, while the closed-loop equilibrium controls naturally reduce into closed-loop optimal controls.

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arXiv2018

Uniqueness of equilibrium strategies in dynamic mean-variance problems with random coefficients

This paper is concerned with the uniqueness issue of open-loop equilibrium investment strategies of dynamic mean-variance portfolio selection problems with random coefficients. A unified method is developed to treat both the problems with deterministic risk-free return rate, state-dependent risk aversion, and that with full random coefficients, constant risk aversion. To do so, some new necessity conditions for the existence of equilibrium investment strategies are established, which considerably extends the analogue in [9] with distinctive ideas. Some new interesting facts are revealed. For example, if risk-free return rate is random, it is shown that there exists a unique open-loop equilibrium investment strategy relying on initial wealth, even when risk aversion is merely a constant but not state-dependent.

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arXiv2018

General maximum principles for optimal control problems of stochastic Volterra integral equations

Optimal control problems of forward stochastic Volterra integral equations (SVIEs) are formulated and studied. When control region is arbitrary subset of Euclidean space and control enters into the diffusion, necessary conditions of Pontryagin's type for optimal controls are established via spike variation. Our conclusions naturally cover the analogue of stochastic differential equations (SDEs), and our developed methodology drops the reliance on Itô formula and second-order adjoint equations. Some new features, that are concealed in the SDEs framework, are revealed in our situation. For example, instead of using second-order adjoint equations, it is more appropriate to introduce second-order adjoint processes. Moreover, the conventional way of using one second-order adjoint equation is inadequate here. In other words, two adjoint processes, which just merge into the solution of second-order adjoint equation in SDEs situation, are actually required and proposed in our setting.

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arXiv2018

Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I

In this paper, a class of time inconsistent linear quadratic optimal control problems of mean-field stochastic differential equations (SDEs) is considered under Markovian framework. Open-loop equilibrium controls and their particular closed-loop representations are introduced and characterized via variational ideas. Several interesting features are revealed and a system of coupled Riccati equations is derived. In contrast with the analogue optimal control problems of SDEs, the mean-field terms in state equation, which is another reason of time inconsistency, prompts us to define above two notions in new manners. An interesting result, which is almost trivial in the counterpart problems of SDEs, is given and plays significant role in the previous characterizations. As application, the uniqueness of open-loop equilibrium controls is discussed.

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arXiv2016

Characterization of Optimal Feedback for Stochastic Linear Quadratic Control Problems

One of the fundamental issues in Control Theory is to design feedback controls. It is well-known that, the purpose of introducing Riccati equations in the deterministic case is to provide the desired feedback controls for linear quadratic control problems. To date, the same problem in the stochastic setting is only partially well-understood. In this paper, we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a suitable sense. We also give a counterexample showing the nonexistence of feedback controls to a solvable stochastic linear quadratic control problem. This is a new phenomenon in the stochastic setting, significantly different from its deterministic counterpart.

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arXiv2016

Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) with closed control regions are formulated and studied. Instead of using spike variation method as one may imagine, here we turn to treat the non-convexity of the control regions by borrowing some tools in set-valued analysis and adapting them into our stochastic control systems. A duality principle between linear backward stochastic Volterra integral equations and linear stochastic Fredholm-Volterra integral equations with conditional expectation are derived, which extends and improves the corresponding results in [25], [30]. Some first order necessary optimality conditions for optimal controls of FBSVIEs are established. In contrast with existed common routines to treat the non-convexity of stochastic control problems, here only one adjoint system and one-order differentiability requirements of the coefficients are needed.

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arXiv2014

Optimal Control Problems of Forward-Backward Stochastic Volterra Integral Equations

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs in short) are formulated and studied. A general duality principle is established for linear backward stochastic integral equation and linear stochastic Fredholm-Volterra integral equation with mean-field. With the help of such a duality principle, together with some other new delicate and subtle skills, Pontryagin type maximum principles are proved for two optimal control problems of FBSVIEs.

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