# Small-time solvability of a flow of forward-backward stochastic differential equations

## TL;DR

This paper introduces a new class of coupled forward-backward stochastic differential equations called flows, proves their small-time well-posedness, and shows discretized solutions approximate the continuous flow with quantifiable convergence rates.

## Contribution

It defines a novel flow of forward-backward SDEs motivated by time-inconsistent control, establishes their well-posedness, and analyzes discretized approximations.

## Key findings

- Proved small-time well-posedness of the flow equations.
- Established convergence of discretized flows to the continuous solution.
- Provided estimates for the convergence rate of approximations.

## 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 systems consisting of a single forward SDE and a continuum of BSDEs, which are defined on different time-intervals and connected via an equilibrium condition. 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.