## On the Wellposedness of Forward-Backward SDEsâ€” A Unified Approach

- April 8, 2014
- 4:20 p.m.
- LeConte 312

## Abstract

The theory of Backward Stochastic Differential Equations (BSDEs) and Forward-Backward SDEs (FBSDEs) has been extensively studied for the past two decades, and its applications have been found in many branches of applied mathematics, especially the stochastic control theory and mathematical finance. The option pricing problem in the financial market and the celebrated principal-agent problem can be formulated as the forward-backward stochastic differential equations. It has been noted, however, that while in many situations the solvability of the original problems is essentially equivalent to the solvability of certain type of FBSDEs, these (mostly non-Markovian) FBSDEs are often beyond the scope of any existing frameworks.

In this talk, I will present our main result on the wellposedness of FBSDEs in a general non-Markovian framework. The main purpose is to build on all the existing methodology in the literature, and put them into a unified scheme. Our main device is a decoupling random field,and its uniform Lipschitz continuity in the spatial variable is crucial for the wellposedness of the original FBSDE. By analyzing a characteristic BSDE, which is a backward stochastic Riccati equation with quadratic growth in the Z component, we find various conditions under which such decoupling random field exists, which lead ultimately to the solvability of the original FBSDEs.