Backward stochastic differential equations with central value reflection

Abstract

In this study, we investigate the well-posedness of a backward stochastic differential equation with jumps and a central value reflection constraint. The reflection condition is imposed on the real-valued function obtained by solving the equation $\mathbb{E}(\mathrm{arctan}({Y}_{t}-x))=0$ at each time $t\in [0,T]$. The driver depends on the distribution of the solution process $Y$ and follows a general quadratic-exponential structure. The terminal value is assumed to be bounded. Using a fixed-point argument and Bounded Mean Oscillation (BMO in short) martingale theory, we establish the existence and uniqueness of local solutions, which are then extended to construct a global solution over the entire time interval $[0,T]$.

Keywords

Central value reflection / Bounded mean oscillation martingales / Jumps

Acknowledgements

The authors would like to thank the Editor-in-Chief for his attention to the article and appreciate the reviewers for their constructive feedback and valuable suggestions, which have greatly improved the paper.

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