A unified system of FB-SDEs with Lévy jumps and double completely-$\mathcal{S}$ skew reflections

We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs) with Lévy jumps and double completely-$\mathcal{S}$ 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 and 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 Lévy 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 stricter boundary condition is imposed, i.e., the spectral radii of each square submatrix at a corner of the reflections are strictly less than unity, a unique adapted 6-tuple strong solution (in the sense of sample paths) is considered. In addition, as applications and economic 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 Lévy processes.

Keywords

stochastic differential equation, Lévy jump, completely-$\mathcal{S}$ skew reflection, Skorohod problem, weak convergence, Malliavin calculus, mutual information, queueing network, Big Data, economic modeling

2010 Mathematics Subject Classification

60H10, 60J75, 60K25, 60K37, 94A17

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This research is supported by National Natural Science Foundation of China with Grant No. 11771006 and Grant No. 11371010.

Received 23 November 2016

Received revised 20 October 2017

Accepted 20 October 2017

Published 30 August 2018