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Shanjian Tang

Publications and source records attributed to Shanjian Tang.

At least 19 recordsLinked to original sources
arXiv2026

Optimal Control of Unbounded Stochastic Evolution Systems in Hilbert Spaces

Optimal control and the associated second-order Hamilton-Jacobi-Bellman (HJB) equation are studied for unbounded stochastic evolution systems in Hilbert spaces. A new notion of viscosity solution, featured by absence of B-continuity, is introduced for the second-order HJB equation in the sense of Crandall and Lions, and is shown to coincide with the classical solutions and to satisfy a stability property. The value functional is proved to be the unique continuous viscosity solution to the second-order HJB equation, with the coefficients being not necessarily B-continuous. Our result provides a new theory of viscosity solutions to the HJB equation for optimal control of stochastic evolutionary equations-driven by a linear unbounded operator-in a Hilbert space, and removes the B-continuity assumption on the coefficients which is used in the existing literature.

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arXiv2025

Uniqueness of adapted solutions to scalar BSDEs with Peano-type generators

A Backward Stochastic Differential Equation (BSDE) with a Peano-type generator, is known to have infinitely many solutions when the terminal value is vanishing, and is shown to have possibly multiple solutions even when the terminal value is not vanishing but nonnegative. In this paper, we study the uniqueness of adapted solutions of such a BSDE when the terminal value is almost surely positive. Two methods are developed. The first one is to connect the BSDE to an optimal stochastic control problem: under suitable integrability of the terminal values, with a verification argument, we prove that the first component of the adapted solution pair is the value process for the optimal stochastic control problem. The second one appeals to a change of variables, and is more inclined to analysis: by a change of variables, the original BSDE is reduced to a convex quadratic BSDE, and then using the $\theta$-difference method, we give a sharp result in some special case, which includes the BSDE governing the well-known Kreps-Porteus utility.

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arXiv2025

Mean Field Type Control Problems Driven by Jump-diffusions

In this article, we apply a probabilistic approach to study general mean field type control (MFTC) problems with jump-diffusions, and give the first global-in-time solution. We allow the drift coefficient $b$ and the diffusion coefficient $\sigma$ to nonlinearly depend on the state, distribution and control variables, and both can be unbounded and possibly degenerate; besides, the jump coefficient $\gamma$ is allowed to be non-constant. To tackle the non-linear and control-dependent diffusion $\sigma$, we further formulate a joint cone property and estimates for both processes $P$ and $Q$ of the corresponding adjoint process (where $(P,Q,R)$ is the solution triple of the associated adjoint process as a backward stochastic differential equation with jump), in contrast to our previous single cone property of the only process $P$. We study first the system of forward-backward stochastic differential equations (FBSDEs) with jumps arising from the maximum principle, and then the related Jacobian flows, which altogether yield the classical regularity of the value function and thus allow us to show that the value function is the unique classical solution of the HJB integro-partial differential equation. Most importantly, our proposed probabilistic approach can apparently handle the MFTC problem driven by a fairly general process far beyond Brownian motion, in a relatively easier manner than the existing analytic approach.

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arXiv2025

Relationship between maximum principle and dynamic programming principle for recursive optimal control problem of stochastic evolution equations

This paper aims to study the relationship between the maximum principle and the dynamic programming principle for recursive optimal control problem of stochastic evolution equations, where the control domain is not necessarily convex and the value function may be nonsmooth. By making use of the notion of conditionally expected operator-valued backward stochastic integral equations, we establish a connection between the first and second-order adjoint processes in MP and the general derivatives of the value function. Under certain additional assumptions, the value function is shown to be $C^{1,1}$-regular. Furthermore, we discuss the smooth case and present several applications of our results.

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arXiv2025

Second Order Fully Nonlinear Mean Field Games with Degenerate Diffusions

In this article, we study the global-in-time well-posedness of second order mean field games (MFGs) with both nonlinear drift functions simultaneously depending on the state, distribution and control variables, and the diffusion term depending on both state and distribution. Besides, the diffusion term is allowed to be degenerate, unbounded and even nonlinear in the distribution, but it does not depend on the control. First, we establish the global well-posedness of the corresponding forward-backward stochastic differential equations (FBSDEs), which arise from the maximum principle under a so-called $\beta$-monotonicity commonly used in the optimal control theory. The $\beta$-monotonicity admits more interesting cases, as representative examples including but not limited to the displacement monotonicity, the small mean field effect condition or the Lasry-Lions monotonicity; and ensures the well-posedness result in diverse non-convex examples. In our settings, we pose assumptions directly on the drift and diffusion coefficients and the cost functionals, rather than indirectly on the Hamiltonian, to make the conditions more visible. Our probabilistic method tackles the nonlinear dynamics with a linear but infinite dimensional version, and together with our recently proposed cone property for the adjoint processes, following in an almost straightforward way the conventional approach to the classical stochastic control problem, we derive a sufficiently good regularity of the value functional, and finally show that it is the unique classical solution to the MFG master equation. Our results require fairly few conditions on the functional coefficients for solution of the MFG, and a bit more conditions -- which are least stringent in the contemporary literature -- for classical solution of the MFG master equation.

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arXiv2025

A quadratic BSDE approach to normalization for the finite volume 2D sine-Gordon model in the finite ultraviolet regime

This paper is devoted to a new construction of the two-dimensional sine-Gordon model on bounded domains by a novel normalization technique in the finite ultraviolet regime. Our methodology involves a family of backward stochastic differential equations (BSDEs for short) driven by a cylindrical Wiener process, whose generators are purely quadratic functions of the second unknown variable. The terminal conditions of the quadratic BSDEs are uniformly bounded and converge in probability to the real part of imaginary multiplicative chaos tested against an arbitrarily given test function, which helps us describe our sine-Gordon measure through some delicate estimates concerning bounded mean oscillation martingales. As the ultraviolet cutoffs are vanishing, the quadratic BSDEs converge to a quadratic BSDE that completely characterizes the absolute continuity of our sine-Gordon measure with respect to the law of Gaussian free fields. Our approach can also be used effectively to establish the connection between our sine-Gordon measure and the scaling limit of correlation functions of the critical planar XOR-Ising model and to prove the weak convergence of the normalized charge distributions of two-dimensional log-gases.

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arXiv2024

On Mean Field Monotonicity Conditions from Control Theoretical Perspective

In this article, from the viewpoint of control theory, we discuss the relationships among the commonly used monotonicity conditions that ensure the well-posedness of the solutions arising from problems of mean field games (MFGs) and mean field type control (MFTC). We first introduce the well-posedness of general forward-backward stochastic differential equations (FBSDEs) defined on some suitably chosen Hilbert spaces under the $\beta$-monotonicity. We then propose a monotonicity condition for the MFG, namely partitioning the running cost functional into two parts, so that both parts still depend on the control and the state distribution, yet one satisfies a strong convexity and a small mean field effect condition, while the other has a newly introduced displacement quasi-monotonicity. To the best of our knowledge, the latter quasi type condition has not yet been discussed in the contemporary literature, and it can be considered as a bit more general monotonicity condition than those commonly used. Besides, for the MFG, we show that convexity and small mean field effect condition for the first part of running cost functional and the quasi-monotonicity condition for the second part together imply the $\beta$-monotonicity and thus the well-posedness for the associated FBSDEs. For the MFTC problem, we show that the $\beta$-monotonicity for the corresponding FBSDEs is simply the convexity assumption on the cost functional. Finally, we consider a more general setting where the drift functional is allowed to be non-linear for both MFG and MFTC problems.

math.OC↗
arXiv2024

A Class of Degenerate Mean Field Games, Associated FBSDEs and Master Equations

In this paper, we study a class of degenerate mean field games (MFGs) with state-distribution dependent and unbounded functional diffusion coefficients. With a probabilistic method, we study the well-posedness of the forward-backward stochastic differential equations (FBSDEs) associated with the MFG and arising from the maximum principle, and estimate the corresponding Jacobian and Hessian flows. We further establish the classical regularity of the value functional $V$; in particular, we show that when the cost function is $C^3$ in the spatial and control variables and $C^2$ in the distribution argument, then the value functional is $C^1$ in time and $C^2$ in the spatial and distribution variables. As a consequence, the value functional $V$ is the unique classical solution of the degenerate MFG master equation.

math.OC↗
arXiv2024

Multi-dimensional non-Markovian backward stochastic differential equations of interactively quadratic generators

This paper is devoted to a general solvability of multi-dimensional non-Markovian backward stochastic differential equations (BSDEs) with interactively quadratic generators. Some general structures of the generator $g$ are posed for both local and global existence and uniqueness results on BSDEs, which admit a general growth of the generator $g$ in the state variable $y$, and a quadratic growth of the $i$th component $g^i$ both in the $j$th row $z^j$ of the state variable $z$ for $j\neq i$ (by which we mean the ``{\it interactively quadratic}" growth) and in the $i$th row $z^i$ of $z$. We first establish an existence and uniqueness result on local bounded solutions and then several existence and uniqueness results on global bounded and unbounded solutions. They improve several existing works in the non-Markovian setting, and also incorporate some interesting examples, one of which is a partial answer to the problem posed in \citet{Jackson2023SPA}. A comprehensive study on the bounded solution of one-dimensional quadratic BSDEs with unbounded stochastic parameters is provided for deriving our main results.

math.PR↗
arXiv2024

Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by Lévy Processes

In this paper, we study the Cauchy problem for backward stochastic partial differential equations (BSPDEs) involving fractional Laplacian operator. Firstly, by employing the martingale representation theorem and the fractional heat kernel, we construct an explicit form of the solution for fractional BSPDEs with space invariant coefficients, thereby demonstrating the existence and uniqueness of strong solution. Then utilizing the freezing coefficients method as well as the continuation method, we establish Hölder estimates and well-posedness for general fractional BSPDEs with coefficients dependent on space-time variables. As an application, we use the fractional adjoint BSPDEs to investigate stochastic optimal control of the partially observed systems driven by $α$-stable Lévy processes.

math.PR↗
arXiv2024

Classical solution of path-dependent mean-field semilinear PDEs

The paper concerns classical solution of path-dependent partial differential equations (PPDEs) with coefficients depending on both variables of path and path-valued measure, which are crucial to understanding large-scale mean-field interacting systems in a non-Markovian setting. We construct classical solutions of the PPDEs via solution of the forward and backward stochastic differential equations. To accommodate the intricacies introduced by the appearance of the path in the coefficients, we develop a novel technique known as the ``parameter frozen'' approach to the PPDEs.

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arXiv2024

Unbounded Dynamic Concave Utilities via BSDEs

The dynamic concave utility (or the dynamic convex risk measure) of an unbounded endowment is studied and represented as the value process in the unique solution of a backward stochastic differential equation (BSDE) with an unbounded terminal value, with the help of our recent existence and uniqueness results on unbounded solutions of scalar BSDEs whose generators have a linear, super-linear, sub-quadratic or quadratic growth. Moreover, the infimum in the dynamic concave utility is proved to be attainable. The Fenchel-Legendre transform (dual representation) of convex functions, the de la Vall\'{e}e-Poussin theorem, and Young's and Gronwall's inequalities constitute the main ingredients of the dual representation.

math.PR↗
arXiv2024

Optimal Control of Unbounded Functional Stochastic Evolution Systems in Hilbert Spaces: Second-Order Path-dependent HJB Equation

Optimal control and the associated second-order path-dependent Hamilton-Jacobi-Bellman (PHJB) equation are studied for unbounded functional stochastic evolution systems in Hilbert spaces. The notion of viscosity solution without B-continuity is introduced in the sense of Crandall and Lions, and is shown to coincide with the classical solutions and to satisfy a stability property. The value functional is proved to be the unique continuous viscosity solution to the associated PHJB equation, without assuming any B-continuity on the coefficients. In particular, in the Markovian case, our result provides a new theory of viscosity solutions to the Hamilton-Jacobi-Bellman equation for optimal control of stochastic evolutionary equations -- driven by a linear unbounded operator -- in a Hilbert space, and removes the B-continuity assumption on the coefficients, which was initially introduced for first-order equations by Crandall and Lions (see J. Func. Anal. 90 (1990), 237-283; 97 (1991), 417-465), and was subsequently used by Swiech (Comm. Partial Differential Equations 19 (1994), 1999-2036) and Fabbri, Gozzi, and Swiech (Probability Theory and Stochastic Modelling 82, 2017, Springer, Berlin).

math.OC↗
arXiv2024

Maximum principle for optimal control of stochastic evolution equations with recursive utilities

We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain not to be convex and the generator of the BSDE to vary with the second unknown variable $z$. The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation.

math.OC↗
arXiv2024

Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth

In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator $g\big(t, Y, Z, \mathbb{P}_{Y}, \mathbb{P}_{Z}\big)$ has a quadratic growth in $Z$ and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth, and the convergence rate is also given. Finally, in this framework, we use the mean-field BSDE to provide a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE.

math.PR↗
arXiv2024

Mild Solution of Semilinear Rough Stochastic Evolution Equations

In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $du_{t}=\left[L_{t}u_{t}+f\left(t, u_{t}\right)\right]dt+\left(G_{t}u_{t}+g_{t}\right)d\mathbf{X}_{t}+h\left(t, u_{t}\right)dW_{t}$, where $\left(L_{t}\right)_{t \in \left[0, T\right]}$ is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, $\mathbf{X}$ is a two-step $α$-Hölder rough path with $α\in \left(1/3, 1/2\right]$ and $W$ is a Brownian motion. Existence and uniqueness of the mild solution are given through the stochastic controlled rough path approach and fixed-point argument. As a technical tool to define rough stochastic convolutions, we also develop a general mild stochastic sewing lemma, which is applicable for processes according to a monotone family.

math.PR↗
arXiv2024

Multidimensional Backward Stochastic Differential Equations with Rough Drifts

In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with an additional rough drift (rough BSDE), and give the existence and uniqueness of the adapted solution, either when the terminal value and the geometric rough path are small, or when each component of the rough drift only depends on the corresponding component of the first unknown variable (but dropped is the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), 1715-1758]). We also introduce a new notion of the $p$-rough stochastic integral for $p \in \left[2, 3\right)$, and then succeed in giving -- through a fixed-point argument -- a general existence and uniqueness result on a multidimensional rough BSDE with a general square-integrable terminal value, allowing the rough drift to be random and time-varying but having to be linear; furthermore, we connect it to a system of rough partial differential equations.

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