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

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  • Nam, Kihun

Abstract

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=∫0tmsms∗dtr[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.

Suggested Citation

  • Nam, Kihun, 2021. "Locally Lipschitz BSDE driven by a continuous martingale a path-derivative approach," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 376-411.
    • Handle: RePEc:eee:spapps:v:141:y:2021:i:c:p:376-411
    DOI: 10.1016/j.spa.2021.09.009
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    References listed on IDEAS

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