Classical and varitional differentiability of BSDEs with quadratic growth

Stefan Ankirchner, Peter Imkeller, Goncalo Dos Reis

Research output: Contribution to journalArticlepeer-review

Abstract

We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter x. We give sufficient conditions for the solution pair of the BSDE to be differentiable in x. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational sense (Malliavin differentiable).

Original languageEnglish
Article number53
Pages (from-to)1418-1453
Number of pages36
JournalElectronic journal of probability
Volume12
DOIs
Publication statusPublished - 9 Nov 2007

Keywords

  • BSDE
  • foward-backward SDE
  • quadratic growth
  • differentiability
  • stochastic calculas of variations
  • Malliavin calculas
  • Feyman-Kac formula
  • BMO martingale
  • reverse Holder inequality
  • STOCHASTIC DIFFERENTIAL-EQUATIONS

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