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

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