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

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): Y_t=ξ(M_[0,T])+∫_t^Tf(s,M_[0,s],Y_s−,Z_sm_s)dtr[M,M]_s−∫_t^TZ_sdM_s−N_T+N_t.Here, M_[0,t] is the path of M from 0 to t and m is defined by [M,M]_t=∫₀^tm_sm_s^∗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.