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Shanjian Tang

Publications and source records attributed to Shanjian Tang.

At least 37 records · Page 2Linked to original sources
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.

math.PR↗
arXiv2023

Degenerate Mean Field Type Control with Linear and Unbounded Diffusion, and their Associated Equations

We study the well-posedness of a system of forward-backward stochastic differential equations (FBSDEs) corresponding to a degenerate mean field type control problem, when the diffusion coefficient depends on the state together with its measure and also the control. Degenerate mean field type control problems are rarely studied in the literature. Our method is based on a lifting approach which embeds the control problem and the associated FBSDEs in Wasserstein spaces into certain Hilbert spaces. We use a continuation method to establish the solvability of the FBSDEs and that of the Gâteaux derivatives of this FBSDEs. We then explore the regularity of the value function in time and in measure argument, and we also show that it is the unique classical solution of the associated Bellman equation. We also study the higher regularity of the linear functional derivative of the value function, by then, we obtain the classical solution of the mean field type master equation.

math.PR↗
arXiv2023

Mild Solution of Semilinear SPDEs with Young Drifts

In this paper, we study a semilinear SPDE with a linear Young drift $du_{t}=Lu_{t}dt+f\left(t, u_{t}\right)dt+\left(G_{t}u_{t}+g_{t}\right)dη_{t}+h\left(t, u_{t}\right)dW_{t}$, where $L$ is the generator of an analytical semigroup, $η$ is an $α$-Hölder continuous path with $α\in \left(1/2, 1\right)$ and $W$ is a Brownian motion. After establishing through two different approaches the Young convolution integrals for stochastic integrands, we introduce the corresponding definition of mild solutions and continuous mild solutions, and give via a fixed-point argument the existence and uniqueness of the (continuous) mild solution under suitable conditions.

math.PR↗
arXiv2023

Discrete-time Approximation of Stochastic Optimal Control with Partial Observation

We consider a class of stochastic optimal control problems with partial observation, and study their approximation by discrete-time control problems. We establish a convergence result by using weak convergence technique of Kushner and Dupuis [Numerical Methods for Stochastic Control Problems in Continuous Time (2001), Springer-Verlag, New York], together with the notion of relaxed control rule introduced by El Karoui, Huu Nguyen and Jeanblanc-Picqué [SIAM J. Control Optim., 26 (1988) 1025-1061]. In particular, with a well chosen discrete-time control system, we obtain a first implementable numerical algorithm (with convergence) for the partially observed control problem. Moreover, our discrete-time approximation result would open the door to study convergence of more general numerical approximation methods, such as machine learning based methods. Finally, we illustrate our convergence result by the numerical experiments on a partially observed control problem in a linear quadratic setting.

math.OC↗
arXiv2023

Multi-dimensional Mean-field Type Backward Stochastic Differential Equations with Diagonally Quadratic Generators

In this paper, we study the multi-dimensional backward stochastic differential equations (BSDEs) whose generator depends also on the mean of both variables. When the generator is diagonally quadratic, we prove that the BSDE admits a unique local solution with a fixed point argument. When the generator has a logarithmic growth of the off-diagonal elements (i.e., for each $i$, the $i$-th component of the generator has a logarithmic growth of the $j$-th row $z^j$ of the variable $z$ for each $j \neq i$), we give a new apriori estimate and obtain the existence and uniqueness of the global solution.

math.PR↗
arXiv2022

Fully Coupled Nonlocal Quasilinear Forward-Backward Parabolic Equations Arising from Mean Field Games

In this paper, we study fully coupled nonlocal second order quasilinear forward-backward partial differential equations (FBPDEs), which arise from solution of the mean field game (MFG) suggested by Lasry and Lions [Japan. J. Math. 2 (2007), p. 237 (Remark iv)]. We show the existence of solutions $(u,m)\in C^{1+\frac{1}{4},2+\frac{1}{2}}([0,T]\times\mathbb{R}^n)\times C^{\frac{1}{2}}([0,T],\mathcal{P}_1(\mathbb{R}^n))$, and also the uniqueness under an additional monotonicity condition. Then, we improve the regularity of our weak solution $m$ to get a classical solution under appropriate regularity assumptions on coefficients. The FBPDEs can be used to investigate a system of mean field equations (MFEs), where the backward one is a Hamilton-Jacobi-Bellman equation and the forward one is a Fokker-Planck equation. Moreover, we prove a verification theorem and give an optimal strategy of the associated MFG via the solution of MFEs. Finally, we address the linear-quadratic problems.

math.PR↗
arXiv2022

State-Density Flows of Non-Degenerate Density-Dependent Mean Field SDEs and Associated PDEs

In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,μ}$ with initial $(t,μ)\in[0,T]\times L^2(\mathbb{R}^d)$, and a stochastic differential equation for $X^{t,x,μ}$ with initial $(t,x)\in[0,T]\times \mathbb{R}^d$, whose coefficients depend on the solution of FP equation. We develop a combined probabilistic and analytical method to explore the regularity of the functional $V(t,x,μ)=\mathbb{E}[Φ(X^{t,x,μ}_T,m^{t,μ}(T,\cdot))]$. Our main result states that, under a non-degenerate condition and appropriate regularity assumptions on the coefficients, the function $V$ is the unique classical solution of a nonlocal partial differential equation of mean-field type. The proof depends heavily on the differential properties of the flow $μ\mapsto (m^{t,μ}, X^{t,x,μ})$ over $μ\in L^2(\mathbb{R}^d)$. We also give an example to illustrate the role of our main result. Finally, we give a discussion on the $L^1$ choice case in the initial $μ$.

math.PR↗
arXiv2022

Optimal control of SDEs with expected path constraints and related constrained FBSDEs

In this paper, we consider optimal control of stochastic differential equations subject to an expected path constraint. The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs. In particular, the compensated process in our adjoint equation is deterministic, which seems to be new in the literature. For the typical case of linear stochastic systems and quadratic cost functionals (i.e., the so-called LQ optimal stochastic control), a verification theorem is established, and the existence and uniqueness of the constrained reflected FBSDEs are also given.

math.OC↗
arXiv2022

Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection

We study the quantitative stability of the solutions to Markovian quadratic reflected BSDEs with bounded terminal data. By virtue of BMO martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs, and obtain the explicit rate of convergence by applying the quantitative stability result.

math.PR↗
arXiv2021

Convergence of Gradient Algorithms for Nonconvex C^{1+alpha} Cost Functions

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball, and Nesterov's accelerated gradient, is analyzed in a general framework under mild assumptions. Based on the convergence result of expected gradients, we prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of upcrossings. It is worth noting that there are not additional restrictions imposed on the objective function and stepsize. Another improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of Holder continuity. As a byproduct, we apply a localization procedure to extend our results to stochastic stepsizes.

math.OC↗
arXiv2021

Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables $y$ and $z$) in a super-linear way like $|y||\ln |y||^{(λ+1/2)\wedge 1}+|z||\ln |z||^λ$ for some $λ\geq 0$. For the following four different ranges of the growth power parameter $λ$: $λ=0$, $λ\in (0,1/2)$, $λ=1/2$ and $λ>1/2$, we give reasonably weakest possible different integrability conditions of the terminal value for the existence of an unbounded solution to the BSDE. In the first two cases, they are stronger than the $L\ln L$-integrability and weaker than any $L^p$-integrability with $p>1$; in the third case, the integrability condition is just some $L^p$-integrability for $p>1$; and in the last case, the integrability condition is stronger than any $L^p$-integrability with $p>1$ and weaker than any $\exp(L^ε)$-integrability with $ε\in (0,1)$. We also establish the comparison theorem, which yields naturally the uniqueness, when either generator of both BSDEs is convex (concave) in both unknown variables $(y,z)$, or satisfies a one-sided Osgood condition in the first unknown variable $y$ and a uniform continuity condition in the second unknown variable $z$.

math.PR↗
arXiv2021

Mean Field Games with Common Noises and Conditional Distribution Dependent FBSDEs

In this paper, we consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. We assume that the cost function satisfies a convexity and a weak monotonicity property. We use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE). We prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by Hu and Peng \cite{YH2}. We apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.

math.OC↗
arXiv2021

Quadratic $G$-BSDEs with convex generators and unbounded terminal conditions

In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by $G$-Brownian motions ($G$-BSDEs) with unbounded terminal values. With the help of a $θ$-method of Briand and Hu [4] and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.

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