  We study four systems and their interactions. First, we formulate a unified
system of coupled forward-backward stochastic partial differential equations
(FB-SPDEs) with Levy jumps, whose drift, diffusion, and jump coefficients may
involve partial differential operators. A solution to the FB-SPDEs is defined
by a 4-tuple general dimensional random vector-field process evolving in time
together with position parameters over a domain (e.g., a hyperbox or a
manifold). Under an infinite sequence of generalized local linear growth and
Lipschitz conditions, the well-posedness of an adapted 4-tuple strong solution
is proved over a suitably constructed topological space. Second, we consider a
unified system of FB-SDEs, a special form of the FB-SPDEs, however, with skew
boundary reflections. Under randomized linear growth and Lipschitz conditions
together with a general completely-S condition on reflections, we prove the
well-posedness of an adapted 6-tuple weak solution with boundary regulators to
the FB-SDEs by the Skorohod problem and an oscillation inequality.
Particularly, if the spectral radii in some sense for reflection matrices are
strictly less than the unity, an adapted 6-tuple strong solution is concerned.
Third, we formulate a stochastic differential game (SDG) with general number of
players based on the FB-SDEs. By a solution to the FB-SPDEs, we get a solution
to the FB-SDEs under a given control rule and then obtain a Pareto optimal Nash
equilibrium policy process to the SDG. Fourth, we study the applications of the
FB-SPDEs/FB-SDEs in queueing systems and quantum statistics while we use them
to motivate the SDG.
