A Unified System of FB-SDEs with Levy Jumps and Double Completely-S Skew Reflections

Wanyang Dai Nanjing University

Dynamical Systems Information Theory Probability mathscidoc:2103.11004

Communications in Mathematical Sciences, 16, (3), 659-704, 2018.4
We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs) with Levy jumps and double completely-S skew reflections. Owing to the reflections, the solution to an embedded Skorohod problem may be not unique, i.e., bifurcations may occur at reflection boundaries, the well-known contraction mapping approach can not be extended directly to solve our problem. Thus, we develop a weak convergence method to prove the well-posedness of an adapted 6-tuple weak solution in the sense of distribution to the unified system. The proof heavily depends on newly established Malliavin calculus for vector-valued Levy processes together with a generalized linear growth and Lipschitz condition that guarantees the well-posedness of the unified system even under a random environment. Nevertheless, if a more strict boundary condition is imposed, i.e., the spectral radii in certain sense for the reflections are strictly less than the unity, a unique adapted 6-tuple strong solution in the sense of sample pathwise is concerned. In addition, as applications and economical studies of our unified system, we also develop new techniques including deriving a generalized mutual information formula for signal processing over possible non-Gaussian channels with multi-input multi-output (MIMO) antennas and dynamics driven by Levy processes.
Stochastic differential equation, Levy jump, completely-S skew reflection, Skorohod problem, weak convergence, Malliavin calculus, mutual information, queueing network, Big Data, economical modeling.
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  title={A Unified System of FB-SDEs with Levy Jumps and Double Completely-S Skew Reflections},
  author={Wanyang Dai},
  booktitle={Communications in Mathematical Sciences},
Wanyang Dai. A Unified System of FB-SDEs with Levy Jumps and Double Completely-S Skew Reflections. 2018. Vol. 16. In Communications in Mathematical Sciences. pp.659-704. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210331102149372033772.
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