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

TL;DR

This paper develops a unified framework for coupled forward-backward stochastic PDEs and SDEs with jumps and reflections, establishing well-posedness, and applies these to stochastic differential games and complex systems.

Contribution

It introduces a comprehensive system of FB-SPDEs and FB-SDEs with jumps and reflections, proving their well-posedness and linking them to stochastic differential games.

Findings

Proved well-posedness of FB-SPDEs with Levy jumps.

Established existence of solutions for FB-SDEs with skew reflections.

Applied the framework to queueing systems and quantum statistics.

Abstract

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.