  Bi-Directional Grid Constrained (BGC) stochastic processes (BGCSPs) constrain
the random movement toward the origin steadily more and more, the further they
deviate from the origin, rather than all at once imposing reflective barriers,
as does the well-established theory of It^o diffusions with such reflective
barriers. We identify that BGCSPs are a variant rather than a special case of
the multi-skew Brownian motion (M-SBM). This is because they have their own
complexities, such as the barriers being hidden (not known in advance) and not
necessarily constant over time. We provide an M-SBM theoretical framework and
also a simulation framework to elaborate deeper properties of BGCSPs. The
simulation framework is then applied by generating numerous simulations of the
constrained paths and the results are analysed. BGCSPs have applications in
finance and indeed many other fields requiring graduated constraining, from
both above and below the initial position.
