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.
The main result of this study is to obtain, using the localization method in Briand et al. [3] . Levi, Fatou and Lebesgue type theorems for the solutions of certain one-dimensional backward stochastic differential equation (BSDEs) with integrable parameters with respect to the terminal condition.
PreliminariesLet (Ω, F , P ) be a probability space carrying a standard d-dimensional Brownian motionFor t ∈ [0, T ] and real p > 0, let L p (Ω, F t , P ) denote the set of all F t -measurable random variable ξ such that E|ξ| p < +∞. Let S p denote the set of R R R-valued, F t -adapted and càdlàg process {X t , t ∈ [0, T ]} such thatand let S ∞ denote the set of predictable bouned processes. Let M p denote the set of F t -In this paper, we are concerned with the following one-dimensional BSDE:where the random function g (ω, t, y, z) :termed the generator of BSDE(1), and ξ is an F T -measurable random variable termed the terminal condition. (ξ, g) is called the parameters of BSDE(1). A solution Manuscript
Endophytic fungi play an important role in plant growth. The composition and structure of endophytes vary in different plant tissues, which are specific habitats for endophyte colonization. To analyze the diversity and structural composition of endophytic fungi from toothed clubmoss (Huperzia serrata) that was artificially cultivated for 3 years, we investigated endophytic fungi from the roots, stems and leaves using comparative sequence analysis of the ITS2 region of the fungal rRNA genes sequenced with high-throughput sequencing technology. Seven fungal phyla were identified, and fungal diversity and structure varied across different tissues, with the most distinctive community features found in the roots. A total of 555 operational taxonomic units (OTUs) were detected, and 198 were common to all samples, and 43, 16, 16 OTUs were unique to the root, stem, leaf samples, respectively. Taxonomic classification showed that Ascomycota and Basidiomycota were dominant phyla, and Cladosporium, Oidiodendron, Phyllosticta, Sebacina and Ilyonectria were dominant genera. The relative abundance heat map at the genus level suggested that H. serrata had characteristic endophytic fungal microbiomes. Line discriminant analysis effect size analysis and principal coordinate analysis demonstrated that fungal communities were tissue-type and tissue-site specific. Overall, our study provides new insights into the complex composition of endophytic fungi in H. serrata.
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