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Fan, Shengjun

doi:10.1016/j.spa.2015.11.012

Cited by 23 publications

(19 citation statements)

References 35 publications

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“…They are usually used when the generator g has a general growth in y and a linear growth in z. See [23,17,18] among others for more details. And, to study the uniqueness, Assumption (UN3) seems to be very natural for a non-linear growth function, see for example [13] and [15], where the convexity (concavity) condition of the generator g in z are required to ensure the uniqueness of the solution.…”

Section: Uniquenessmentioning

confidence: 99%

“…In this case, if the terminal condition (ξ, α • ) belongs to L p × L p for some p > 1, then BSDE(ξ, g) admits a solution in the space S p × M p , and the solution is unique in this space if g further satisfies the uniformly Lipschitz condition in (y, z). The reader is referred to [30,16,28,10,22,18] for details.…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, to have the existence of a solution to BSDE (ξ, g), the boundedness or at least some exponential integrability of the terminal condition (ξ, α • ) are required. The reader is referred to [27,12,13,15,11,9,25,18,21,20] among others. In addition, we mention that a class of quadratic BSDEs with L p -integrable (p > 1)…”

Section: Introductionmentioning

confidence: 99%

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

Fan¹,

Hu²,

Tang

2021

Preprint

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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)∧1 + |z|| ln |z|| λ for some λ ≥ 0. For the following four different ranges of the growth power parameter λ: λ = 0, λ ∈ (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 ∈ (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.

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Section: Uniquenessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Fan¹,

Hu²,

Tang

2021

Preprint

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“…Finally, in view of the assumptions of g 1 and g 2 together with ξ 1 ≤ ξ 2 , the rest proof runs as the proof of Theorem 2.4 and Theorem 2.1 in Fan [13] with u(t) = v(t) ≡ 1 and λ(t) ≡ γ, which is omitted.…”

Section: Comparison Theoremmentioning

confidence: 99%

$L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions

Fan¹

2019

Electron. J. Probab.

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We establish several existence, uniqueness and comparison results for L 1 solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumption that the generator g satisfies a one-sided Osgood condition together with a very general growth condition in y, a uniform continuity condition and/or a sub-linear growth condition in z, and a generalized Mokobodzki condition which relates the growth of g and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of L 1 solutions of the reflected BSDEs. We also prove that the L 1 solutions of reflected BSDEs can be approximated by the penalization method and by some sequences of L 1 solutions of reflected BSDEs. These results strengthen some existing work on the L 1 solutions of non-reflected BSDEs and reflected BSDEs.And, we say dK 1 ≤ dK 2 means that for each (F t )-progressively measurable set D ⊂ Ω × [0, T ],

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“…for BSDE(ξ, g) to have a minimal (maximal) adapted solution and a unique solution when the generator g satisfies (H1) and (H2) respectively. See for example [1,2,6] for more details.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of solution to scalar BSDEs with $L\exp (\mu \sqrt{2\log (1+L)} )$-integrable terminal values: the critical case

Fan¹,

Hu²

2019

Electron. Commun. Probab.

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In [8], the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) when the terminal value is L exp µ 2 log(1 + L)integrable for a positive parameter µ > µ0 with a critical value µ0, and a counterexample is provided to show that the preceding integrability for µ < µ0 is not sufficient to guarantee the existence of the solution. Afterwards, the uniqueness result (with µ > µ0) is also given in [3] for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: µ = µ0.Keywords: Backward stochastic differential equation ; L exp µ 2 log(1 + L) -integrability ;Existence and uniqueness ; Critical case. AMS MSC 2010: 60H10. Submitted to ECP on FIXME!, final version accepted on FIXME!.

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Bounded solutions, Lp(p>1) solutions and

Cited by 23 publications

References 35 publications

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

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

$L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions

Existence and uniqueness of solution to scalar BSDEs with $L\exp (\mu \sqrt{2\log (1+L)} )$-integrable terminal values: the critical case

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