  In this paper, we consider a general time-inconsistent optimal control
problem for a non homogeneous linear system, in which its state evolves
according to a stochastic differential equation with deterministic
coefficients, when the noise is driven by a Brownian motion and an independent
Poisson point process. The running and the terminal costs in the objective
functional, are explicitly dependent on some general discounting coefficients
which cover the non-exponential and the hyperbolic discounting situations.
Furthermore, the presence of some quadratic terms of the conditional
expectation of the state process as well as a state-dependent term in the
objective functional makes the problem time-inconsistent. Open-loop Nash
equilibrium controls are constructed instead of optimal controls, by using a
version of the stochastic maximum principle approach. This approach involves a
stochastic system that consists of a flow of forward-backward stochastic
differential equations and an equilibrium condition. As an application, we
study some concrete examples.
