L1 solution to scalar BSDEs with logarithmic sub-linear growth generators

With the test function method and a localization technique, a scalar backward stochastic differential equation (BSDE for short) subject to an L1 terminal condition is shown to have an L1 solution when the generator g(t,y,z) has a one-sided linear growth in y and a logarithmic sub-linear growth in z,...

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Bibliographic Details
Published inSystems & control letters Vol. 177
Main Authors Fan, Shengjun, Hu, Ying, Tang, Shanjian
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.07.2023
Elsevier
Subjects
Backward stochastic differential equation
Comparison theorem
Existence and uniqueness
L1 solution
Logarithmic sub-linear growth generator
Mathematics
Probability
Logarithmic sub-linear growth generator
L1 solution
Backward stochastic differential equation
Comparison theorem
Existence and uniqueness
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Summary:With the test function method and a localization technique, a scalar backward stochastic differential equation (BSDE for short) subject to an L1 terminal condition is shown to have an L1 solution when the generator g(t,y,z) has a one-sided linear growth in y and a logarithmic sub-linear growth in z, which improves some existing results. A new idea to study the existence of an adapted solution to a BSDE is given. When the generator g(t,y,z) additionally has an extended monotonicity in y and a logarithmic uniform continuity in z, we further establish a comparison theorem for the L1 solutions to the above BSDEs, which yields immediately the uniqueness of the solution.
ISSN:0167-6911
1872-7956
DOI:10.1016/j.sysconle.2023.105553