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

Abstract

This paper is devoted to solving a multidimensional backward stochastic differential equation (BSDE for short) with a general random terminal time τ taking values in [0,+∞]. 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 μ. ∈ ℝ and ν.∈ ℝ₊ satisfying ∫₀^τ(|μ_t|+ν²_t) dt<+∞, 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 ρ_·≥ μ_·+θ/(2[1∧(p−1)])ν_·² be a given real-valued process for some constant θ>1 such that ∫₀^τ|ρ_t|dt<+∞. In a general weighted L^p space with a weighted factor e^{∫₀^t ρ_rdr}, 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 ρ_· 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.

Well-posedness of Filtering Equations in Weighted Sobolev Spaces with Unbounded System Coefficients
Zeju Sun, Songlin Zhou, Stephen S. -T. Yau
arXiv3 Sep 2026PreprintarXiv licenceFull textOptimisation & controlarXiv
Backward stochastic partial differential equations with unbounded random coefficients
Jie Xiong, Wen Xu, Zuo Quan Xu +1
arXiv26 Oct 2026PreprintCC BY-SA 4.0Full textProbabilityarXiv
Global solvability of the heat equations in two unbounded domains and the interface
Hajime Koba
arXiv23 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
L^p-regularity for fourth-order Schrödinger operators with unbounded or degenerate coefficients
Federica Gregorio, Chiara Spina, Cristian Tacelli
arXiv16 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
The L¹-realization of an elliptic operator with unbounded coefficients
Sallah Eddine Boutiah
arXiv14 Oct 2026PreprintCC BY-NC-ND 4.0Full textAnalysis & PDEs Mathematical physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{zhang2026solvability,
  title = {{Solvability of BSDEs with possibly unbounded stochastic coefficients on a general weighted $L^p$ space}},
  author = {Yaqi Zhang and Xinying Li and Ying Hu and Shengjun Fan},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  eprint = {2603.13873},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05751215v1},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Solvability of BSDEs with possibly unbounded stochastic coefficients on a general weighted L^p space
AU  - Yaqi Zhang
AU  - Xinying Li
AU  - Ying Hu
AU  - Shengjun Fan
PY  - 2026
DA  - 2026/09/15
PB  - HAL
UR  - https://hal.science/hal-05751215v1
AB  - This paper is devoted to solving a multidimensional backward stochastic differential equation (BSDE for short) with a general random terminal time τ taking values in [0,+∞]. 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 μ. ∈ ℝ and ν.∈ ℝ₊ satisfying ∫_0^τ(|μ_t|+ν^2_t) dt<+∞, 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 ρ_·≥ μ_·+θ/(2[1∧(p−1)])ν_·^2 be a given real-valued process for some constant θ>1 such that ∫_0^τ|ρ_t|dt<+∞. In a general weighted L^p space with a weighted factor e^∫_0^t ρ_rdr, 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 ρ_· 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.
ER  -

Abstract

Related

Cite