A Class of Multi-dimensional Backward Stochastic Differential Equations with Singular Generators exhibiting Diagonally Quadratic Growth and Applications

Guangyan Jia; Wenbo Wang

REVIEW 4 major objections 4 minor 29 references

A Class of Multi-dimensional Backward Stochastic Differential Equations with Singular Generators exhibiting Diagonally Quadratic Growth and Applications

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that multi-dimensional BSDEs with singular diagonally quadratic generators admit unique positive solutions even for unbounded terminal conditions.

desk verdict Plausible multi-dimensional extension of singular quadratic BSDEs, but the central theorem rests on an external one-dimensional lemma whose coverage of the |z|^2/y case is not demonstrated. read the letter →

arxiv 2507.03348 v1 pith:SDYBOUKE submitted 2025-07-04 math.PR

Wenbo Wang , Guangyan Jia This is my paper

classification math.PR MSC 60H1060H30

keywords backwardstochasticdifferentialequationsdiagonallyquadraticgeneratorssingularunboundedterminalconditionspositivesolutionsoptimalinvestmentregime-switchingmarketLp

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Backward stochastic differential equations whose generator grows like $|z|^2/(2y)$ in each coordinate are singular: the quadratic term blows up as a component of the solution approaches zero, so positivity of the solution is part of the problem. This paper claims that such multi-dimensional BSDEs, with generator split as a Lipschitz linear-growth part plus a convex singular diagonal part bounded above by $\alpha_s+\beta_s y^i+\frac{\delta}{2y^i}|z^i|^2$, admit a unique positive solution $(Y,Z)$ in $S^{p(2+\delta)}(\mathbb{R}^n)\times M^p(\mathbb{R}^{n\times d})$ whenever the positive terminal condition satisfies the integrability condition in (H2). That matters because these singular diagonally quadratic systems were previously understood mainly in one dimension, and the multi-dimensional quadratic case with unbounded terminal lacked a general existence theory. The paper also shows the result yields the value function of an optimal investment problem in a regime-switching market with random coefficients.

What carries the argument

The load-bearing object is the one-dimensional weighted estimate of Lemma 2.1, imported from [12]: for a positive solution of a BSDE with $0\le g\le a_s+b_s y+\frac{\delta}{2y}|z|^2$, the lemma gives (1) an upper bound on $e^{(1+\delta)t+(2+\delta)\int_0^t b_u\,du}Y_t^{2+\delta}$ by a conditional expectation of the terminal and $a_s$, (2) an $L^p$ bound on $\int_0^T|Z_u|^2\,du$ in terms of norms of $Y$, $a$, and $b$, and (3) a unique-solution statement in $S^{p(2+\delta)}\times M^p$ under the stated integrability condition. The paper's proof inserts $H^i_1(s,V_s)$ as an additional bounded-linear-growth source, iterates the lemma coordinatewise, and uses a convexity-preserving change of variable $\Delta_\theta y=(y^{m+q}-\theta y^m)/(1-\theta)$ with Tanaka's formula and Doob's inequality to obtain the Cauchy estimate and the limit passage.

What would settle it

Run the one-dimensional BSDE with terminal $\zeta$ having a heavy tail just inside the (H2) integrability exponent, generator $\frac{\delta}{2y}|z|^2$ with $\delta$ close to 1, and $a_s$ chosen so $\int_0^T a_s^{2+\delta}\,ds$ is finite but large; check whether Lemma 2.1's bound on $\int_0^T|Z_u|^2\,du$ stays finite. An analytic or numerical example where the estimate fails at $\delta=1$ while the paper's hypotheses hold would isolate whether the $\delta\ne 1$ restriction is essential.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 2.1: under (H1)-(H2), the BSDE (1.1) with terminal vector $\xi>0$ and generator $H^i=H^i_1(s,y)+H^i_2(s,y^i,z^i)$, where $H^i_2$ is convex and satisfies $0\le H^i_2\le \alpha_s+\beta_s y^i+\frac{\delta}{2y^i}|z^i|^2$ with $\delta\ne 1$, has a unique solution $(Y,Z)$ with $Y\in S^{p(2+\delta)}(\mathbb{R}^n)$ and $Z\in M^p(\mathbb{R}^{n\times d})$, and $Y^i>0$ for every $i$. The existence proof runs a Picard-type iteration in which each coordinate solves a one-dimensional BSDE with the singular quadratic term, using weighted estimates to control the $L^{p(2+\delta)}$ norm of $Y$ and then the $M^p$ norm of $Z$, and then a convex-combination comparison argument gives the Cauchy property and uniqueness. The paper also proves a bounded-data corollary ($Y\in S^\infty$, $Z\in BMO$) and derives a portfolio application: the value function of a power-utility optimal investment problem in a regime-switching market is $x^\gamma Y^l_0/\gamma$, with the optimal strategy explicitly given by $\frac{1}{1-\gamma}(\lambda(s,l)'+\frac{1}{Y(s,l)}Z(s,l))$.

Load-bearing premise

Everything rests on Lemma 2.1, which the paper does not prove: if the one-dimensional weighted estimates or the existence statement it borrows from [12] fail under the stated integrability conditions, or if the constants depend on $p,\delta,T$ in a way the iteration cannot absorb, the coordinatewise construction has no starting point.

Editorial extensions

If this is right

If Theorem 2.1 is correct, singular diagonally quadratic multi-dimensional BSDEs are well-posed in $S^{p(2+\delta)}\times M^p$ for every $p$ and $\delta$ satisfying (H2), not just in the one-dimensional case.
Positivity of the solution is part of the conclusion, so the singular denominator $y^i$ never degenerates along the path; this makes the class usable in systems where positivity is economically meaningful.
With bounded $\alpha,\beta$ and bounded positive terminal data, the corollary gives bounded $Y$ and $Z\in BMO$, a convenient regularity package for applications.
In the regime-switching investment problem, the same BSDE gives both the value function and the optimal feedback strategy, so any numerical scheme for the BSDE directly yields the optimal portfolio.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implication left implicit is that the coordinatewise domination and convexity in the comparison argument are essential: a fully coupled quadratic system with cross terms $|z|^2/f(y)$ would not fall under this theorem and likely needs a genuinely different method.
The $\delta\ne 1$ restriction is likely a real threshold: at $\delta=1$ the quadratic weight $1/(2y)$ sits at an integrability boundary, so the boundary case may require different function spaces, but the paper does not explore it.
Because Lemma 2.1 is imported rather than proved, the theorem stands or falls with that one-dimensional estimate; a reader who wants to use the result in applications should check its constants and integrability conditions directly in [12].

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main theorem is plausible, and the proof strategy is recognizable from the quadratic-BSDE literature: per-component application of a one-dimensional comparison lemma, an iteration with time-slicing to get L^p bounds, and a theta-deformation argument for Cauchy and uniqueness. What is genuinely new is the combination of multi-dimensional diagonal quadratic growth, singular |z|^2/y generators, and unbounded positive terminal data, landing in S^{p(2+delta)} x M^p. The regime-switching investment application is also a real addition, with the verification step using the right completing-the-square and the delta = 2gamma/(1-gamma) choice correctly excluding gamma = 1/3.

The parts I could verify algebraically are consistent. The time-slicing constants, the Doob inequality steps, and the theta -> 1 limits all check out. The paper is not sloppy about the structure of the iteration.

The soft spot is exactly the one both the reader and the stress test flagged: Lemma 2.1. This lemma is the load-bearing input. The proof of Theorem 2.1 invokes it to solve the decoupled one-dimensional equations, to get the weighted y-estimates, and to control z. The paper states the lemma without proof and says more general cases can be found in [12], which is a preprint on L^p solutions of quadratic BSDEs. It is not shown that [12] covers the singular upper bound a + b y + (delta/(2y))|z|^2 with delta != 1 and convexity, nor that the exact integrability condition matches. If that lemma fails or its hypotheses are not satisfied by the constructed equations, the existence proof collapses. This is not a minor expository point; it is a missing link in the logical chain.

Secondary issues are less serious but real: Corollary 2.1's proof is omitted with a handwave, estimate (2.27) is asserted after a sketch, and the introduction contains a duplicated paragraph and many typos. None of these affect the central argument, but they lower clarity.

My overall take: the paper probably is correct, but as written it is conditional on an external result that may or may not cover the singular case. For a reader in the BSDE community, the result is worth knowing. For a referee, the main task is to check whether [12] actually implies Lemma 2.1, or to ask the authors to supply a full proof. The paper deserves serious peer review rather than desk rejection, but I would not cite it in my own work until the lemma is verified or proven in a revision.

Referee Report

4 major / 4 minor

Summary. The paper studies multi-dimensional backward stochastic differential equations of the form Y_t = ξ + ∫_t^T H(s,Y_s,Z_s)ds - ∫_t^T Z_s dW_s, where each component of the generator is the sum of a nonnegative Lipschitz linear-growth term H^i_1(s,y) and a convex singular term H^i_2(s,y_i,z_i) satisfying 0 ≤ H^i_2 ≤ α_s + β_s y_i + δ/(2y_i)|z_i|^2 with δ ≠ 1. The terminal condition ξ is positive and unbounded, with integrability conditions stated in (H2). Theorem 2.1 claims existence and uniqueness of a positive solution in S^{p(2+δ)}(R^n) × M^p(R^{n×d}). The proof proceeds by a fixed-point iteration driven by a one-dimensional existence/comparison lemma (Lemma 2.1), followed by a time-slicing argument, a θ-deformation Cauchy estimate, and a stability passage. The paper also contains a corollary for bounded terminals (Corollary 2.1) and an application to an optimal investment problem in a regime-switching market (Proposition 3.1), where the value function is represented through the solution of a multidimensional BSDE with diagonal quadratic growth.

Significance. If Theorem 2.1 is correct, the paper extends the one-dimensional singular quadratic BSDE theory of [11] to a multi-dimensional diagonally quadratic setting with unbounded positive terminal conditions, in the spirit of [16]. The proof architecture is standard and the algebraic estimates in the iteration are largely consistent, which is a genuine strength. The application to regime-switching optimal investment is also a natural and potentially useful contribution. However, the central result is conditional on Lemma 2.1, which is stated without proof and whose hypotheses are only asserted to be covered by an external preprint. Because that lemma is invoked at several load-bearing points of Theorem 2.1, the main theorem is not self-contained as it stands. The application section contains a further verification gap: the generator used there is defined as +∞ for non-positive y-component, which does not satisfy the standing hypotheses of Corollary 2.1 as formulated.

major comments (4)

[Section 2, Lemma 2.1 and proof of Theorem 2.1]
[Section 3, Proposition 3.1]
[Section 2, equations (2.27)-(2.32)]
[Section 2, inequality (2.32)]

minor comments (4)

[Throughout]
[Section 2, notation]
[Corollary 2.1]
[Section 3, hypotheses]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorem is built on external one-dimensional results, not on self-citation or definitional identities.

full rationale

The derivation chain in this paper is not circular. Theorem 2.1 is proved by an iteration scheme that reduces each step to Lemma 2.1, a one-dimensional BSDE estimate/existence result attributed to [12] (Yang, arXiv:1506.08146), an external work with no author overlap with Wang and Jia. The fact that Lemma 2.1 is stated without proof and that its hypotheses are tailored to make the iteration work is a completeness and reliability concern, not a circularity: the paper does not define Lemma 2.1's assumptions in terms of Theorem 2.1's conclusion, and the cited lemma is not a prior result of the present authors. Similarly, Corollary 2.1's omitted proof is an exposition gap, not a circular step. In Section 3, the application invokes Theorem 3.7 of [26] and Proposition 5.1 of [11]; both are external works by other authors, so this is ordinary external support rather than a self-citation chain or an imported uniqueness theorem. No equation in the paper reduces by construction to an earlier input, no fitted quantity is relabeled as a prediction, and no load-bearing argument depends on a self-citation. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper fits no data and introduces no new entities; it is a theorem paper whose ledger is the set of structural hypotheses and imported lemmas. The heavier items are Lemma 2.1 (unproved here, attributed to [12]) and the outsourced verification results [26] and [11]. All other inputs are standard stochastic calculus. The '+∞' convention for the generator when y^l ≤ 0 in Proposition 3.1 is a modeling convention, not an invented entity. No numbers are fitted; δ, p, q, A, α, β are hypothesis-level inputs describing the generator class.

assumptions (6)

ad hoc to paper Lemma 2.1 (parts 1-3): one-dimensional comparison and existence estimates for positive solutions of BSDE (2.1) with generator 0 ≤ g ≤ a_s + b_s y + δ/(2y)|z|², δ ≠ 1, under weighted Lp integrability.
Stated in Section 2 without proof; the only pointer given is 'more general cases can be found in [12]'. Theorem 2.1's iteration, z-estimates, and limit passage all invoke this lemma.
domain assumption Hypothesis (H1): H^i = H^i_1(s,y) + H^i_2(s,y^i,z^i), with H^i_1 nonnegative, linear growth, uniformly Lipschitz in y, H^i_2 continuous convex in (y^i,z^i), and 0 ≤ H^i_2 ≤ α+βy+δ/(2y)|z|².
Convexity makes the θ-deformation Cauchy and uniqueness arguments work; the upper bound defines the singular quadratic class. The condition δ ≠ 1 is required by Lemma 2.1.
domain assumption Hypothesis (H2): ξ^i > 0, E[(1/ξ^i)^q] < ∞ for some q > 1, and E[(1+|ξ|^{2+δ}+∫_0^T α^{2+δ} ds)^p e^{p(2+δ)∫_0^T β ds}] < ∞ for some p ≥ 2.
The exponential weights appear throughout (2.3)-(2.28); the (1/ξ^i)^q moment is used in estimate (2.32) to control the singular term near y = 0.
standard math Ito-Krylov and Ito-Tanaka formulas, Doob's maximal Lp inequality, the BDG inequality, and martingale convergence theorems.
Used in (2.3), (2.17)-(2.19), (2.24)-(2.28) and in the limit passage; these are background results from stochastic calculus.
ad hoc to paper Theorem 3.7 of [26] and Proposition 5.1 of [11] justify the regime-switching verification in Proposition 3.1.
The proof of Proposition 3.1 states 'We obtain remaining arguments by combining Theorem 3.7 in [26] and Proposition 5.1 in [11]' without restating either result, so the optimality argument is external to this manuscript.
standard math Kazamaki-type BMO martingale results ([28], Corollary 3.4) used to show the candidate optimal strategy is admissible (X ∈ class(D)).
Cited in the final step of Proposition 3.1's proof; standard results for exponential martingales and BMO.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Class of Multi-dimensional Backward Stochastic Differential Equations with Singular Generators exhibiting Diagonally Quadratic Growth and Applications." pith.science (2026). https://pith.science/paper/SDYBOUKE

@misc{pith2026250703348,
  author       = {Pith},
  title        = {Pith review of: A Class of Multi-dimensional Backward Stochastic Differential Equations with Singular Generators exhibiting Diagonally Quadratic Growth and Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDYBOUKE}},
  note         = {Machine review of arXiv:2507.03348}
}

read the original abstract

This paper investigate a class of multi-dimensional backward stochastic differential equations (BSDEs) with singualr generators exhibiting diagonally quadratic growth and unbounded terminal conditions, thereby extending results in the literature. We present an example of such equations in optimal investment decision.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

29 extracted references · 26 canonical work pages

[12]

Lp Solutions of Quadratic BSDEs

Yang, H. Lp solutions of quadratic BSDEs. Available online: https://arxiv.org/pdf/1506.08146v1.pdf (accessed on 1th June 2021)

work page Pith review arXiv 2021
[11]

BSDEs driven by $|z|^2/y$ and applications to PDEs and decision theory

Bahlali, K., Tangpi, L. BSDEs driven by |z|2/y and applications to PDEs and de- cision theory. Available online: https://arxiv.org/pdf/1810.05664.pdf. (accessed on 1th June 2021)

work page Pith review arXiv 2021
[16]

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

Fan, S., Hu, Y., Tang, S. Multi-dimensional backward stochastic differential equa- tions of diagonally quadratic generators: the general result. J. Differ. Equ. 2023, 368, 105–140

work page 2023
[1]

Adapted solution of a backward stochastic differential equation

Pardoux, E., Peng, S. Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 1990, 14, 55–61

work page 1990
[2]

BSDE with quadratic growth and unbounded terminal value

Briand, P., Hu, Y. BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Field 2006, 136, 604–618. DOI:10.1007/s00440-006-0497-0

doi:10.1007/s00440-006-0497-0 2006
[3]

Quadratic BSDEs with convex generators and unbounded terminal conditions

Briand, P., Hu, Y. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Relat. Field 2008, 141, 543–567. DOI: 10.1007/s00440- 007-0093-y

doi:10.1007/s00440- 2008
[4]

Solvability of backward stochastic differential equations with quadratic growth

Tevzadze, R. Solvability of backward stochastic differential equations with quadratic growth. Stoch. Process. Their Appl. 2008, 118, 503–515. DOI: 10.1016/j.spa.2007.05.009

work page doi:10.1016/j.spa.2007.05.009 2008
[5]

Barrieu, P., Karoui, N. E. Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. Ann. Probab. 2013, 41, 1831–

work page 2013

Show all 29 references

[6]

On the uniqueness of solutions to quadratic BSDEs with non- convex generators

Briand, P., Richou, A. On the uniqueness of solutions to quadratic BSDEs with non- convex generators. In: Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications; Cohen, S. N., Gy¨ ongy, I., dos Reis, G., Siska, D., Szpruch, L. Eds.; Springer: Cham, Switzerland, 2...

2019
[7]

Backward stochastic differential equations and partial differen- tial equations with quadratic growth

Kobylanski, M. Backward stochastic differential equations and partial differen- tial equations with quadratic growth. Ann. Probab. 2000, 28, 558–602. DOI: 10.1214/aop/1019160253
[8]

Stochastic differential utility

Duffie, D., Epstein, L.G. Stochastic differential utility. Econometrica 1992, 60, 353–

1992
[9]

General existence results for reflected BSDE and BSDE Bull

Essaky, E., Hassani, M. General existence results for reflected BSDE and BSDE Bull. Sci. Math. 2011, 135, 442–466. DOI: 10.1016/j.bulsci.2011.04.003

2011 doi
[10]

Quadratic BSDE with L2-terminal data: Krylov’s estimate, Itˆ o-Krylov’s formula and existence results

Bahlali, K., Eddahbi, M., Ouknine, Y. Quadratic BSDE with L2-terminal data: Krylov’s estimate, Itˆ o-Krylov’s formula and existence results. Ann. Probab. 2017, 45, 2377–2397. DOI: 10.1214/16-AOP1115

2017 doi
[13]

On the backward stochastic differential equation with generator f (y)|z|2

Zheng, S., Zhang, L., Feng, L. On the backward stochastic differential equation with generator f (y)|z|2. J. Math. Anal. Appl. 2021, 500

2021
[14]

Pricing principle via Tsallis relative entropy in incomplete markets

Tian, D. Pricing principle via Tsallis relative entropy in incomplete markets. SIAM J. Financ. Math. 2023, 14, 250–278

2023
[15]

Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth re- strictions

Zheng, S. Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth re- strictions. Available online: https://arxiv.org/abs/2412.21172 (accessed on 1th Jan. 2025)

2025 arXiv
[17]

Solving unbounded quadratic BSDEs by a domination method

Bahlali, K. Solving unbounded quadratic BSDEs by a domination method. Available online: https://arxiv.org/pdf/1903.11325 (accessed on 1th June 2021)

1903 arXiv
[18]

Solvability of some quadratic BS- DEs without exponential moments

Bahlali, K, Eddahbi, M, Ouknine, Y. Solvability of some quadratic BS- DEs without exponential moments. C. R. Math. 2013, 351, 229–233. DOI: 10.1016/j.crma.2013.04.003

2013 doi
[19]

Existence and uniqueness of multidimensional BSDEs and of systems of degenerate PDEs with superlinear growth generator

Bahlali, K., Essaky, E., Hassani, M. Existence and uniqueness of multidimensional BSDEs and of systems of degenerate PDEs with superlinear growth generator. SIAM J. Math. Anal. 2015, 47, 4251–4288

2015
[20]

Backward SDE and related g-expectation

Peng, S. Backward SDE and related g-expectation. In: Backward Stochastic Differ- ential Equations, Pitman Research Notes in Mathematics Series, No.364 ; Karoui, N. E., Mazliak, L. Eds.;Longman: Essex, England, 1997, pp. 141–159

1997
[21]

J., Jiang, L

Fan, S. J., Jiang, L. Lp(p > 1) solutions for one-dimensional BSDEs with linear- growth generators. J. Appl. Math. Comput. 2012, 38, 295–304

2012
[22]

Existence and uniqueness result for a backward stochastic differen- tial equation whose generator is Lipschitz continuous in y and uniformly continuous in z

Fan, S., Jiang, L. Existence and uniqueness result for a backward stochastic differen- tial equation whose generator is Lipschitz continuous in y and uniformly continuous in z. J. Appl. Math. Comput. 2011, 36, 1–10

2011
[23]

Portfolio optimization with Markov-modulated stock prices and interest rates

Bauerle, N., Rieder, U. Portfolio optimization with Markov-modulated stock prices and interest rates. IEEE Trans. Autom. Control 2004, 49, 442–447

2004
[24]

Markowitz’s mean-variance portfolio selection with regime switch- ing: from discrete-time models to their continuous-time limits

Yin, G., Zhou, X. Markowitz’s mean-variance portfolio selection with regime switch- ing: from discrete-time models to their continuous-time limits. IEEE Trans. Autom. Control 2004, 49, 349–360

2004
[25]

Hu, Y., Shi, X., Xu, Z.: Constrained stochastic LQ control with regime switching and application to portfolio selection. Ann. Appl. Probab. 2022, 32, 426–460

2022
[26]

Optimal Consumption–Investment with Constraints in a Regime Switching Market with Random Coefficients

Hu, Y., Shi, X., Xu, Z. Optimal Consumption–Investment with Constraints in a Regime Switching Market with Random Coefficients. Appl. Math. Optim. , 2025, 91, 1–44

2025
[27]

Systems of ergodic BSDEs arising in regime switching forward performance processes

Hu, Y., Liang, G., Tang, S. Systems of ergodic BSDEs arising in regime switching forward performance processes. SIAM J. Control Optim. , 2020, 58, 2503–2534

2020
[28]

Continuous exponential martingales and BMO

Kazamaki, N. Continuous exponential martingales and BMO. Springer, 2006

2006
[1863]

DOI: 10.1214/12-AOP743

doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [12]

Lp Solutions of Quadratic BSDEs

Yang, H. Lp solutions of quadratic BSDEs. Available online: https://arxiv.org/pdf/1506.08146v1.pdf (accessed on 1th June 2021)

work page Pith review arXiv 2021

[2] [11]

BSDEs driven by $|z|^2/y$ and applications to PDEs and decision theory

Bahlali, K., Tangpi, L. BSDEs driven by |z|2/y and applications to PDEs and de- cision theory. Available online: https://arxiv.org/pdf/1810.05664.pdf. (accessed on 1th June 2021)

work page Pith review arXiv 2021

[3] [16]

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

Fan, S., Hu, Y., Tang, S. Multi-dimensional backward stochastic differential equa- tions of diagonally quadratic generators: the general result. J. Differ. Equ. 2023, 368, 105–140

work page 2023

[4] [1]

Adapted solution of a backward stochastic differential equation

Pardoux, E., Peng, S. Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 1990, 14, 55–61

work page 1990

[5] [2]

BSDE with quadratic growth and unbounded terminal value

Briand, P., Hu, Y. BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Field 2006, 136, 604–618. DOI:10.1007/s00440-006-0497-0

doi:10.1007/s00440-006-0497-0 2006

[6] [3]

Quadratic BSDEs with convex generators and unbounded terminal conditions

Briand, P., Hu, Y. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Relat. Field 2008, 141, 543–567. DOI: 10.1007/s00440- 007-0093-y

doi:10.1007/s00440- 2008

[7] [4]

Solvability of backward stochastic differential equations with quadratic growth

Tevzadze, R. Solvability of backward stochastic differential equations with quadratic growth. Stoch. Process. Their Appl. 2008, 118, 503–515. DOI: 10.1016/j.spa.2007.05.009

work page doi:10.1016/j.spa.2007.05.009 2008

[8] [5]

Barrieu, P., Karoui, N. E. Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. Ann. Probab. 2013, 41, 1831–

work page 2013