# A Stackelberg Game of Backward Stochastic Differential Equations with Applications

## TL;DR

This paper studies a Stackelberg game involving backward stochastic differential equations, providing optimality conditions, Riccati equation solutions, and applications to financial market consumption strategies.

## Contribution

It introduces a novel Stackelberg game framework for BSDEs, deriving explicit Riccati-based feedback controls and solvability conditions, with practical financial applications.

## Key findings

- Derived necessary and sufficient optimality conditions for both players.
- Presented Riccati equations for optimal controls in linear-quadratic cases.
- Applied the theoretical results to a financial market consumption problem.

## Abstract

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs), where the coefficients of the backward system and the cost functionals are deterministic, and the control domain is convex. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs is investigated under standard assumptions. The state feedback representation for the optimal control of the follower is first given via two Riccati equations. Then the leader's problem is formulated as an optimal control problem of FBSDE with the control-independent diffusion term. Two high-dimensional Riccati equations are introduced to represent the state feedback for the optimal control of the leader. The solvability of the four Riccati equations are discussed. Theoretic results are applied to an optimal consumption rate problem of two players in the financial market.