Fully coupled forward-backward stochastic differential equations and applications to optimal control. (English) Zbl 0931.60048

Peng, Shige; Wu, Zhen

SIAM J. Control Optimization 37, No. 3, 825-843 (1999).

Let \((\Omega,{\mathcal F},P)\) be a probability space, and let \(\{B_t\}_{t\geq 0}\) be a \(d\)-dimensional Brownian motion in this space; \({\mathcal F}_t\) denotes the natural filtration of this Brownian motion. The authors consider existence and uniqueness problems for the following fully coupled forward-backward stochastic differential equation (FBSDE): \[ x_t= a+ \int^t_0 b(s,x_s, y_s, z_s) ds+ \int^t_0 \sigma(s,x_s,y_s, z_s) dB_s, \]
\[ y_t= \Phi(x_T)+ \int^T_t f(s, x_s,y_s,z_s) ds- \int^T_t z_s dB_s,\quad t\in [0,T], \] where \((x,y,z)\) takes values in \(\mathbb{R}^n\times \mathbb{R}^m\times \mathbb{R}^{m+d}\), and \(b\), \(f\), \(\sigma\) are mappings with appropriate dimensions which are, for each fixed \((x,y,z)\), \({\mathcal F}_t\)-progressively measurable. They are also Lipschitz with respect to \((x,y,z)\); \(T>0\) is an arbitrarily prescribed number and the time interval is called the time duration. Finally, several examples of FBSDE related to stochastic optimal control and differential games problems are given.

Reviewer: N.G.Gamkrelidze (Moskva)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
93E03	Stochastic systems in control theory (general)

Keywords:

stochastic differential equations; stochastic optimal control; differential games; Hamiltonian system; stochastic analysis