  Using a new notion of path-derivative, we study well-posedness of backward
stochastic differential equation driven by a continuous martingale $M$ when
$f(s,\gamma,y,z)$ is locally Lipschitz in $(y,z)$:
\[Y_{t}=\xi(M_{[0,T]})+\int_{t}^{T}f(s,M_{[0,s]},Y_{s-},Z_{s}m_{s})d{\rm
tr}[M,M]_{s}-\int_{t}^{T}Z_{s}dM_{s}-N_{T}+N_{t}\] Here, $M_{[0,t]}$ is the
path of $M$ from $0$ to $t$ and $m$ is defined by
$[M,M]_{t}=\int_{0}^{t}m_{s}m_{s}^{*}d{\rm tr}[M,M]_{s}$. When the BSDE is
one-dimensional, we could show the existence and uniqueness of solution. On the
contrary, when the BSDE is multidimensional, we show existence and uniqueness
only when $[M,M]_{T}$ is small enough: otherwise, we provide a counterexample
that has blowing-up solution. Then, we investigate the applications to utility
maximization problems.
