On measure solutions of backward stochastic differential equations

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:119

Article

Ankirchner, Stefan²  Imkeller, Peter²  Popier, Alexandre¹ 

[1] Univ Maine, Lab Manceau Math, F-72085 Le Mans 9, France

[2] Humboldt Univ, Inst Math, D-10099 Berlin, Germany

关键词: Backward stochastic differential equation;    Stochastic control;    Hedging of contingent claim;    Martingale measure;    Martingale representation;    Girsanov's theorem;    Weak solution;    Measure solution;    Brownian motion;   

DOI  :  10.1016/j.spa.2009.02.003

来源: Elsevier

【 摘 要 】

We consider backward stochastic differential equations (BSDEs) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. For the case where the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. For the case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in the case of non-uniqueness, classical solutions that fail to be measure solutions can coexist with different measure solutions. (C) 2009 Elsevier B.V. All rights reserved.

【 授权许可】

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