  In this paper, we study one-dimensional backward stochastic differential
equation with jump under logarithmic growth assumption in the z-variable
(|z|\sqrt{|\ln|z|}|) and an L^p terminal value (for a suitable p>2). We show
the existence and the uniqueness of the solution when the noise is driven by a
Brownian motion and an independent Poisson random measure. In addition, we
highlight the connection of such BSDEs with stochastic optimal control problem,
where we show the existence of an optimal strategy for the stochastic control
problem.
