Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs

Bahlali, Khaled; Essaky, El Hassan; Hassani, M.

doi:10.1016/j.crma.2010.03.006

Cited by 23 publications

(18 citation statements)

References 8 publications

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“…Since there exists p > 1 such that inequality (3.14) holds, then equation (3.19) has a unique solution (Y 5 , Z 5 ) such that (details can be found in [3,4]): Assertion (i) is proved. We shall prove assertion (ii).…”

Section: 22mentioning

confidence: 99%

Solving Unbounded Quadratic BSDEs by a Domination Method

Bahlali¹

2019

Preprint

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We introduce a domination argument which asserts that: if we can dominate the parameters of a quadratic backward stochastic differential equation (QBSDE) with continuous generator from above and from below by those of two BSDEs having ordered solutions, then also the original QBSDE admits at least one solution. This result is presented in a general framework: we do not impose any integrability condition on none of the terminal data of the three involved BSDEs, we do not require any constraint on the growth nor continuity of the two dominating generators. As a consequence, we establish the existence of a maximal and a minimal solution to BSDEs whose coefficient H is continuous and satisfies, where α t , β t , θ t are positive processes and the function f is positive, continuous and increasing (or even only positive and locally bounded) on R. This is done with unbounded terminal value. We cover the classical QBSDEs where the function f is constant ([10], [12], [23], [25]) and when f (y) = y p ([21]) and also the cases where the generator has super linear growth such as y|z|, e |y| k |z| p , e e |y| |z| 2 , (k ≥ 0, 0 ≤ p < 2) and so on. In contrast to the works [10,12,21,23,25], we get the existence of a a maximal and a minimal solution and we cover the BSDEs with at most linear growth (take f = 0). In particular, we cover and extend the results of [22] and [24]. Furthermore, we establish the existence and uniqueness of solutions to BSDEs driven by f (y)|z| 2 when f is merely locally integrable on R.

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Section: 22mentioning

confidence: 99%

Solving Unbounded Quadratic BSDEs by a Domination Method

Bahlali¹

2019

Preprint

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“…Since ψ is of linear growth, we can get the existence of a constant k such that ψ(x) ≤ k(1 + x) for all x ∈ R + . Then, in view of b n ≤ b 1 , β(·) ∈ L 1 ([0, T ]; R + ) and the Fubinin Theorem, by (2) we can obtain that for each t ∈ [0, T ], dP − a.s.,…”

Section: Comparison Theorems Of L P (P > 1) Solutions and L 1 Solutionsmentioning

confidence: 99%

Bounded solutions, Lp(p>1) solutions and
Fan
¹

2016
Stochastic Processes and their Applications
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This paper aims at solving one-dimensional backward stochastic differential equations (BSDEs) under weaker assumptions. We establish general existence, uniqueness, and comparison results for bounded solutions, L p (p > 1) solutions and L 1 solutions of the BSDEs. The time horizon is allowed to be finite or infinite, and the generator g is allowed to have a general growth in y and a quadratic growth in z. As compensation, the generator g needs to satisfy a kind of one-sided linear or super-linear growth condition in y, instead of the monotonicity condition in y as is usually done. Many of our results improve virtually some known results, even though for the case of the finite time horizon and the case of the L 2 solution.

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“…From the beginning, many authors attempted to improve the result of [36] by weakening the Lipschitz hypothesis on g, see [1,2,4,[6][7][8][13][14][15][16][17][18][19][20][22][23][24][26][27][28][29][31][32][33][34][35]37,40,43,45,46], or the L 2 integrability assumptions on ξ, see [5][6][7]11,21,23,38,44], or relaxing the finite terminal time T to a stopping time or infinity, see [12,25,29,34,35,45]. From these results we can see that the case of one-dimensional BSDEs is easier to handle due to the presence of the comparison theorem of solutions (see [6][7][8]…”

Section: Introductionmentioning

confidence: 99%

Lp solutions of multidimensional BSDEs with weak monotonicity and general growth generators

Fan

2015

Journal of Mathematical Analysis and Applications

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In this paper, we first establish the existence and uniqueness of L p (p > 1) solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition together with a general growth condition in y for the generator g. Then, we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way. Finally, we put forward and prove a stability theorem and a comparison theorem of L p (p > 1) solutions for this kind of BSDEs.

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Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs

Cited by 23 publications

References 8 publications

Solving Unbounded Quadratic BSDEs by a Domination Method

Solving Unbounded Quadratic BSDEs by a Domination Method

Bounded solutions, Lp(p>1) solutions and
Fan
¹

2016
Stochastic Processes and their Applications
23
0
19
0
View full text Add to dashboard Cite

Lp solutions of multidimensional BSDEs with weak monotonicity and general growth generators

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Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs

Cited by 23 publications

References 8 publications

Solving Unbounded Quadratic BSDEs by a Domination Method

Solving Unbounded Quadratic BSDEs by a Domination Method

Bounded solutions, Lp(p>1) solutions and Fan1 2016Stochastic Processes and their Applications230190View full textAdd to dashboardCite

Lp solutions of multidimensional BSDEs with weak monotonicity and general growth generators

Contact Info

Product

Resources

About

Bounded solutions, Lp(p>1) solutions and
Fan
¹

2016
Stochastic Processes and their Applications
23
0
19
0
View full text Add to dashboard Cite