Locally Lipschitz BSDE driven by a continuous martingale a path-derivative approach

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Using a new notion of path-derivative, we study the well-posedness of backward stochastic differential equation driven by a continuous martingale M when f(s,γ,y,z) is locally Lipschitz in (y,z): Yt=ξ(M[0,T])+∫tTf(s,M[0,s],Ys−,Zsms)dtr[M,M]s−∫tTZsdMs−NT+Nt.Here, M[0,t] is the path of M from 0 to t and m is defined by [M,M]t=∫0tmsmsdtr[M,M]s. When the BSDE is one-dimensional, we show the existence and uniqueness of the solution. On the contrary, when the BSDE is multidimensional, we show the existence and uniqueness only when [M,M]T is small enough: otherwise, we provide a counterexample. Then, we investigate the applications to optimal control of diffusion and optimal portfolio selection under various restrictions.

Original languageEnglish
Pages (from-to)376-411
Number of pages36
JournalStochastic Processes and their Applications
Publication statusPublished - Nov 2021


  • Backward stochastic differential equation
  • Coefficients of superlinear growth
  • Functional derivative
  • Path differentiability
  • Utility maximization

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