We consider interstage dependent stochastic linear programs where both the random right-hand side and the model of the underlying stochastic process have a special structure. Namely, for stage $t$, the right-hand side of the equality constraints (resp. the inequality constraints) is an affine function (resp. a given function $b_t$) of the process value for this stage. As for $m$-th component of the process at stage $t$, it depends on previous values of the process through a function $h_{t m}$. For this type of problem, to obtain an approximate policy under some assumptions for functions $b_t$ and $h_{t m}$, we detail a stochastic dual dynamic programming algorithm. Our analysis includes some enhancements of this algorithm such as the definition of a state vector of minimal size, the computation of feasibility cuts without the assumption of relatively complete recourse, as well as efficient formulas for sharing cuts between nodes of the same stage. The algorithm is given for both a non-risk averse and a risk averse model. We finally provide preliminary results comparing the performances of the recourse functions corresponding to these two models for a real-life application.