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Small-Time Solvability of a Flow of Forward–Backward Stochastic Differential Equations

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Abstract

Motivated from time-inconsistent stochastic control problems, we introduce a new type of coupled forward–backward stochastic systems, namely, flows of forward–backward stochastic differential equations. They are coupled systems consisting of a single forward SDE and a continuum of BSDEs, which are defined on different time-intervals. We formulate a notion of equilibrium solutions in a general framework and prove small-time well-posedness of the equations. We also consider discretized flows and show that their equilibrium solutions approximate the original one, together with an estimate of the convergence rate.

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Acknowledgements

I would like to thank Professor Masanori Hino, who is my supervisor, Professor Ichiro Shigekawa, and Professor Jun Sekine for helpful discussions. This work was supported by JSPS KAKENHI Grant Number JP18J20973.

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  1. Department of Mathematics, Kyoto University, Kyoto, 606–8502, Japan

    Yushi Hamaguchi

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  1. Yushi Hamaguchi
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Correspondence to Yushi Hamaguchi.

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Hamaguchi, Y. Small-Time Solvability of a Flow of Forward–Backward Stochastic Differential Equations. Appl Math Optim 84, 567–588 (2021). https://doi.org/10.1007/s00245-020-09654-7

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