Exponential growth BSDE driven by a marked point process

Abstract

In this study, we investigate the well-posedness of exponential growth backward stochastic differential equations (BSDEs) driven by a marked point process (MPP) under unbounded terminal conditions. Our analysis utilizes a fixed point argument, the theta-method, and an approximation procedure. Additionally, we establish the solvability of mean-reflected exponential growth BSDEs driven by the MPP using the theta-method.

Summary

This paper establishes full well-posedness (existence + uniqueness) for exponential growth BSDEs driven by a general marked point process with unbounded terminal conditions possessing exponential moments of all orders. The exponential (Qexp) structure is central: the driver is bounded between barriers involving $j_{\lambda }(t,u)={\int }_E(e^{\lambda u(e)}-1-\lambda u(e)){\varphi }_t(de)$, which captures exponential growth in the jump integrand $u$. Remark 3.14 extends to the full quadratic-exponential case (quadratic in $z$, exponential in $u$) when Brownian motion is also present.

This is the most directly relevant well-posedness reference for the Qexp BSDE in BUETGOLFOUSE2026, which has a driver with quadratic growth in $Z$, exponential growth in $u$ (from the default jump), and linear growth from the book-funding discount.

Key Contributions

• Existence + uniqueness for exponential growth BSDEs driven by MPP (Theorem 3.12) — prior works (El Karoui, Matoussi & Ngoupeyou 2016) only had existence without uniqueness
• Unbounded terminal conditions with exponential moments (not just bounded terminals as in Kazi-Tani et al. 2015)
• General MPP compensator (no absolute continuity w.r.t. Lebesgue measure required)
• Mean-reflected exponential growth BSDEs driven by MPP (Theorem 4.2) — first result of its kind

Methodology

The proof proceeds via: (1) Comparison theorem (Theorem 3.6) under convexity/concavity of the driver in $u$; (2) Uniqueness (Corollary 3.8) from comparison; (3) Existence for bounded terminal (Theorem 3.11) via Lipschitz approximation + theta-method; (4) Extension to unbounded terminal (Theorem 3.12) via approximation with truncated terminals.

Key Findings

• The growth condition (H3)(c): $g_{-}(t,y,u)\le f(t,y,u)\le \bar{g}(t,y,u)$ where the barriers involve $j_{\lambda }$, is the precise Qexp structure
• Convexity or concavity of $f$ in $u$ (H3)(e) is essential for uniqueness — this replaces the absolute continuity assumption on the compensator used in earlier works
• Solution space: $\mathcal{E}\times H_{\nu }^{2,p}$ for all $p\ge 1$

Critical Notes

The XVA HJB paper should cite this, not Quenez & Sulem (2013)

Quenez & Sulem (2013) only covers Lipschitz drivers — their Theorem 2.3 requires uniform Lipschitz in $(y,z)$. The Qexp driver in BUETGOLFOUSE2026 (quadratic in $Z$, exponential in $u$) is NOT Lipschitz. Gu, Lin & Xu (2023) is the correct reference for Qexp well-posedness with jumps. The extension to include Brownian quadratic growth (Remark 3.14) covers the full structure of the XVA BSDE.

Important References

1. El Karoui, Matoussi & Ngoupeyou 2016 — Extended quadratic-exponential structure for BSDEJs, but only existence without uniqueness
2. Kazi-Tani, Possamai & Zhou 2015 — Quadratic BSDEs with jumps, but required bounded terminal and Frechet differentiability
3. Briand & Hu 2006 — Quadratic BSDEs in the Brownian case; Gu et al. extend their solution space results

Atomic Notes

• exponential utility BSDE
• comparison theorem for BSDEs

---

paper