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

Unified Systems of FB-SPDEs/FB-SDEs with Jumps/Skew Reflections and Stochastic Differential Games