On path-dependent multidimensional forward-backward SDEs

Kaitong Hu; Zhenjie Ren; Nizar Touzi

doi:10.3934/naco.2022010

Article Contents

2023, Volume 13, Issue 3&4: 413-430. Doi: 10.3934/naco.2022010

This issue Previous Article Mean-field type quadratic BSDEs Next Article Mean-field doubly reflected backward stochastic differential equations

On path-dependent multidimensional forward-backward SDEs

1.
CMAP, Ecole Polytechnique, France
2.
CEREMADE, Université Paris Dauphine-PSL, France

^*Corresponding author: Nizar Touzi
^*Corresponding author: Nizar Touzi

Received: January 2022

Revised: April 2022

Early access: May 2022

Published: September 2023

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Abstract

This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time $ t $ can depend on the whole path of the forward process up to time $ t $. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling field on small time interval. Then by introducing the characteristic BSDE, we show that a global decoupling field can be constructed by patching local solutions together as long as the solution of the characteristic BSDE remains bounded. Finally, we provide a stability result for path-dependent forward-backward SDEs.

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Mathematics Subject Classification: 60H07, 60H30, 35R60, 34F05.

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References

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