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Shengjun Fan

Publications and source records attributed to Shengjun Fan.

At least 19 recordsLinked to original sources
arXiv2026

Weighted $L^p$ solutions of scalar BSDEs with general unbounded stochastic coefficients

This paper is devoted to solving one-dimensional backward stochastic differential equations (BSDEs in short) with a general random terminal time $\tau$ taking values in the extended nonnegative real numbers. The generator $g$ of BSDEs satisfies some stochastic growth/continuity conditions in the state variables $(y,z)$, featuring unbounded stochastic coefficients $\mu_\cdot\in\R$ and $\nu_\cdot\in\R_+$ satisfying $\int_0^\tau (|\mu_t|+\nu^2_t) {\rm d}t<+\infty$. For any given real $p>1$, let $\rho_\cdot\geq \mu_\cdot+\frac{\theta}{2(p-1)}\nu_\cdot^2$ (instead of $\rho_\cdot\geq\mu_\cdot+\frac{\theta}{2[1\wedge(p-1)]}\nu_\cdot^2$ used in Zhang, Li, Hu and Fan [2026, arXiv:2603.13873v1]) be a real-valued process for some constant $\theta>1$ such that $\int_0^\tau |\rho_t|{\rm d}t<+\infty$. We work within a weighted $L^p$ space with the weighting factor $e^{\int_0^t \rho_r{\rm d}r}$. Within this framework, we establish several innovative results on the weighted $L^p$ solutions of BSDEs: an existence result, an existence and uniqueness result, an existence and uniqueness result of the minimal (maximal) solution, and two comparison theorems. These findings unify and improve some existing results. Some novel ideas are employed to address the challenges posed by general unbounded stochastic coefficients and general weighted spaces.

math.PR↗
arXiv2026

Solvability of BSDEs with possibly unbounded stochastic coefficients on a general weighted $L^p$ space

This paper is devoted to solving a multidimensional backward stochastic differential equation (BSDE for short) with a general random terminal time $\tau$ taking values in $[0,+\infty]$. The generator $g$ of such BSDE satisfies a stochastic monotonicity condition in the state variable $y$ and a stochastic Lipschitz condition in the state variable $z$ with possibly unbounded stochastic coefficients $\mu_\cdot\in\R$ and $\nu_\cdot\in\R_+$ satisfying $\int_0^\tau (|\mu_t|+\nu^2_t) {\rm d}t<+\infty$, along with a very general growth in $y$ that is more easily verified and weaker than existing ones. Let $p>1$ be a given constant and $\rho_\cdot\geq \mu_\cdot+\frac{\theta}{2[1\wedge(p-1)]}\nu_\cdot^2$ be a given real-valued process for some constant $\theta>1$ such that $\int_0^\tau |\rho_t|{\rm d}t<+\infty$. In a general weighted $L^p$ space with a weighted factor $e^{\int_0^t \rho_r{\rm d}r}$, we establish an existence and uniqueness result for the adapted solution of previous BSDE when the terminal value satisfies an associated weighted integrability condition, broadening the scope of the process $\rho_\cdot$ in the weighted factor and thereby unifying and strengthening some corresponding existing results obtained in \citet{DarlingandPardoux1997}, \citet{Briand2003}, \citet{LiFan2024SD} and \citet{Li2025}. Some innovative ideas are presented in order to address the general weighted space and the very general growth condition. As applications, we prove the existence of viscosity solutions for parabolic and elliptic PDEs linked with previous BSDEs under some general assumptions on their nonlinear terms, and establish a dual representation of an unbounded dynamic concave utility defined on a general weighted $L^p$ space via the weighted $L^p$ solutions of previous BSDEs.

math.PR↗
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.

math.PR↗
arXiv2025

Existence and uniqueness on $L^1$ solutions of multidimensional BSDEs with generators of stochastic one-sided Osgood type

By imposing an additional integrability condition on the first component of the solution, this paper establishes an existence and uniqueness result for $L^1$ solutions of multidimensional backward stochastic differential equations (BSDEs) with a general terminal time when the generator $g$ satisfies a stochastic one-sided Osgood condition along with a general growth condition in the state variable $y$, and a stochastic Lipschitz condition in the state variable $z$, extending and strengthening Theorems 1 and 2 of Fan [J. Theor. Probab. 31(2018)]. Two general stochastic Gronwall-type and Bihari-type inequalities along with some innovative techniques dealing with stochastic coefficients and weaker integrability conditions play crucial roles in our proofs, and can be useful in further study on the adapted solution of BSDEs.

math.PR↗
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

Weighted $L^p~(p\geq1)$ solutions of random time horizon BSDEs with stochastic monotonicity generators

In this paper, we are concerned with a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time $\tau$, which may take values in $[0,+\infty]$. Firstly, we establish an existence and uniqueness result for a weighted $L^p~(p>1)$ solution of the preceding BSDE with generator $g$ satisfying a stochastic monotonicity condition with general growth in the first unknown variable $y$ and a stochastic Lipschitz continuity condition in the second unknown variable $z$. Then, we derive an existence and uniqueness result for a weighted $L^1$ solution of the preceding BSDE under an additional stochastic sub-linear growth condition in $z$. These results generalize the corresponding ones obtained in \cite{Li2024} to the $L^p~(p\geq 1)$ solution case. Finally, the corresponding comparison theorems for the weighted $L^p~(p\geq1)$ solutions are also put forward and verified in the one-dimensional setting. In particular, we develop new ideas and systematical techniques in order to establish the above results.

math.PR↗
arXiv2024

On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions: the certain exponential moment case

With the terminal value $|\xi|$ admitting some given exponential moments, we propose and prove several existence and uniqueness results for the unbounded solutions of quadratic backward stochastic differential equations whose generators may be represented as a uniformly continuous (not necessarily locally Lipschitz continuous) perturbation of some convex/concave function with quadratic growth. This perturbation satisfies various feasible conditions such as boundedness, sub-linear growth or linear growth. In particular, in some cases, the first component of the unique solution can be expressed as the value function of an optimal control problem. These results improves those posed in Delbaen, Hu and Richou [AIHP, 2011] and Fan, Hu and Tang [2020, CRM] to some extent. The critical case is also tackled, which strengthens the main result of Delbaen, Hu and Richou [DCDS, 2011].

math.PR↗
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

Weighted solutions of random time horizon BSDEs with stochastic monotonicity and general growth generators and related PDEs

This study focuses on a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time $\tau$ taking values in $[0,+\infty]$. The generator $g$ satisfies a stochastic monotonicity condition in the first unknown variable $y$ and a stochastic Lipschitz continuity condition in the second unknown variable $z$, and it can have a more general growth with respect to $y$ than the classical one stated in (H5) of \cite{Briand2003}. Without imposing any restriction of finite moment on the stochastic coefficients, we establish a general existence and uniqueness result for the weighted solution of such BSDE in a proper weighted $L^2$-space with a suitable weighted factor. This result is proved via some innovative ideas and delicate analytical techniques, and it unifies and strengthens some existing works on BSDEs with stochastic monotonicity generators, BSDEs with stochastic Lipschitz generators, and BSDEs with deterministic Lipschitz/monotonicity generators. Then, a continuous dependence property and a stability theorem for the weighted $L^2$-solutions are given. We also derive the nonlinear Feynman-Kac formulas for both parabolic and elliptic PDEs in our context.

math.PR↗
arXiv2024

On the existence and uniqueness of unbounded solutions to quadratic BSDEs with monotonic-convex generators

With the terminal value $\xi^-$ admitting a certain exponential moment and $\xi^+$ admitting every exponential moments or being bounded, we establish several existence and uniqueness results for unbounded solutions of backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a monotonicity condition with general growth in the first unknown variable $y$ and a convexity condition with quadratic growth in the second unknown variable $z$. In particular, the generator $g$ may be not locally-Lipschitz continuous in $y$. This generalizes some results reported in \cite{Delbaen 2011} by relaxing the continuity and growth of $g$ in $y$. We also give an explicit expression of the first process in the unique unbounded solution of a BSDE when the generator $g$ is jointly convex in $(y,z)$ and has a linear growth in $y$ and a quadratic growth in $z$. Finally, we put forward the corresponding comparison theorems for unbounded solutions of the preceding BSDEs. These results are proved by those existing ideas and some innovative ones.

math.PR↗
arXiv2024

Existence, uniqueness and comparison theorem on unbounded solutions of general time interval BSDEs with sub-quadratic generators

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of one-dimensional backward stochastic differential equations (BSDEs) with sub-quadratic generators, where the terminal time is allowed to be finite or infinite. We first establish existence of the unbounded solutions for this kind of BSDEs with generator $g$ satisfying a time-varying one-sided linear growth in the first unknown variable $y$ and a time-varying sub-quadratic growth in the second unknown variable $z$. Then, the uniqueness and comparison theorem of the unbounded solutions for this kind of BSDEs are proved under a time-varying extended convexity assumption. These results generalized those obtained in \cite{12} to the general time interval BSDEs. Finally, several sufficient conditions ensuring that the uniqueness holds are put forward and verified via some innovative ideas, which are explored at the first time even though for the case of finite time interval BSDEs.

math.PR↗
arXiv2023

1D nonlinear backward stochastic differential equations: a unified theory and applications

Since the celebrated paper by El Karoui, Peng and Quenez [Mathematical Finance, 7 (1997), 1--71], backward stochastic differential equations have found wide applications in stochastic control, financial technology and machine learning. In this paper, we present a comprehensive theory on the existence and uniqueness of adapted solutions to a one-dimensional nonlinear backward stochastic differential equation (1D BSDE for short), and assume that the generator $g$ has a unilateral linear or super-linear growth in the first unknown variable $y$, and has an at most quadratic growth in the second unknown variable $z$. We develop a unified methodology, featured by the test function method and the a priori estimate technique, to establish several existence theorems and comparison theorems, which immediately yield corresponding existence and uniqueness results. We also overview relevant known results and give some practical applications of our theoretical results. Finally, we list some open problems on the well-posedness of 1D BSDEs.

math.PR↗
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.

math.PR↗
arXiv2021

Invariant representation for generators of general time interval quadratic BSDEs under stochastic growth conditions

This paper is devoted to proving a general invariant representation theorem for generators of general time interval backward stochastic differential equations, where the generator $g$ has a quadratic growth in the unknown variable $z$ and satisfies some stochastic growth conditions in the unknown variable $y$. This unifies and strengthens some known results. And, a natural and innovative idea is used to prove the representation theorem.

math.PR↗
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 $|y||\ln |y||^{(\lambda+1/2)\wedge 1}+|z||\ln |z||^{\lambda}$ for some $\lambda\geq 0$. For the following four different ranges of the growth power parameter $\lambda$: $\lambda=0$, $\lambda\in (0,1/2)$, $\lambda=1/2$ and $\lambda>1/2$, 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 $\exp(L^\epsilon)$-integrability with $\epsilon\in (0,1)$. 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↗
arXiv2020

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: the general result

This paper is devoted to a general solvability of a multi-dimensional backward stochastic differential equation (BSDE) of a diagonally quadratic generator $g(t,y,z)$, by relaxing the assumptions of \citet{HuTang2016SPA} on the generator and terminal value. More precisely, the generator $g(t,y,z)$ can have more general growth and continuity in $y$ in the local solution; while in the global solution, the generator $g(t,y,z)$ can have a skew sub-quadratic but in addition "strictly and diagonally" quadratic growth in the second unknown variable $z$, or the terminal value can be unbounded but the generator $g(t,y,z)$ is "diagonally dependent" on the second unknown variable $z$ (i.e., the $i$-th component $g^i$ of the generator $g$ only depends on the $i$-th row $z^i$ of the variable $z$ for each $i=1,\cdots,n$ ). Three new results are established on the local and global solutions when the terminal value is bounded and the generator $g$ is subject to some general assumptions. When the terminal value is unbounded but is of exponential moments of arbitrary order, an existence and uniqueness result is given under the assumptions that the generator $g(t,y,z)$ is Lipschitz continuous in the first unknown variable $y$, and varies with the second unknown variable $z$ in a "diagonal" , "component-wisely convex or concave", and "quadratically growing" way, which seems to be the first general solvability of systems of quadratic BSDEs with unbounded terminal values. This generalizes and strengthens some existing results via some new ideas.

math.PR↗
arXiv2019

General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition

This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $g$ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $y$, and a stochastic-Lipschitz condition in the state variable $z$. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in \citet{XiaoFan2017Stochastics} and \citet{LiuLiFan2019CAM}. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other places. The martingale representation theorem, It\^{o}'s formula and the BMO martingale tool are used to prove these two inequalities.

math.PR↗