Multi-dimensional backward stochastic differential equations of diagonally quadratic generators

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• 作者：Ying Hu^a ; ^{Ying.Hu@univ-rennes1.fr" class="auth_mail" title="E-mail the corresponding author} ; Shanjian Tang^b ; ^c ; ^{sjtang@fudan.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60H15 ; 35R60 ; 93E20
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：126
• 期：4
• 页码：1066-1086
• 全文大小：273 K

文摘

In this paper, we study a multi-dimensional BSDE with a “diagonally” quadratic generator, the quadratic part of whose class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002598&_mathId=si1.gif&_user=111111111&_pii=S0304414915002598&_rdoc=1&_issn=03044149&md5=76a1f311823633f00c31ff82949c6998" title="Click to view the MathML source">iclass="mathContainer hidden">class="mathCode">$i$th component depends only on the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002598&_mathId=si1.gif&_user=111111111&_pii=S0304414915002598&_rdoc=1&_issn=03044149&md5=76a1f311823633f00c31ff82949c6998" title="Click to view the MathML source">iclass="mathContainer hidden">class="mathCode">$i$th row of the second unknown variable. Local and global solutions are given, which seem to be the first systematic (positive) results on the general solvability of multi-dimensional quadratic BSDEs. In our proofs, it is natural and crucial to apply both John–Nirenberg and reverse Hölder inequalities for BMO martingales. Our results are finally illustrated to solve the system of “diagonally” quadratic BSDEs arising from a nonzero-sum risk-sensitive stochastic differential game, which answers the open problem posed in El Karoui and Hamadène [Stochastic Process. Appl. 107 (2003), page 164].