# Linear Forward-Backward Stochastic Differential Equations

## Abstract

The problem of finding adapted solutions to systems of coupled linear forward-backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs.

- Authors:

- Department of Mathematics, Fudan University, Shanghai 200433 (China)

- Publication Date:

- OSTI Identifier:
- 21064289

- Resource Type:
- Journal Article

- Journal Name:
- Applied Mathematics and Optimization

- Additional Journal Information:
- Journal Volume: 39; Journal Issue: 1; Other Information: DOI: 10.1007/s002459900100; Copyright (c) 1999 Springer-Verlag New York Inc.; Article Copyright (c) Inc. 1999 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONTROL THEORY; DIFFERENTIAL EQUATIONS; FUNCTIONAL ANALYSIS; FUNCTIONS; MATHEMATICAL SOLUTIONS; MATRICES; STOCHASTIC PROCESSES

### Citation Formats

```
Yong Jiongmin.
```*Linear Forward-Backward Stochastic Differential Equations*. United States: N. p., 1999.
Web. doi:10.1007/S002459900100.

```
Yong Jiongmin.
```*Linear Forward-Backward Stochastic Differential Equations*. United States. doi:10.1007/S002459900100.

```
Yong Jiongmin. Fri .
"Linear Forward-Backward Stochastic Differential Equations". United States. doi:10.1007/S002459900100.
```

```
@article{osti_21064289,
```

title = {Linear Forward-Backward Stochastic Differential Equations},

author = {Yong Jiongmin},

abstractNote = {The problem of finding adapted solutions to systems of coupled linear forward-backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs.},

doi = {10.1007/S002459900100},

journal = {Applied Mathematics and Optimization},

issn = {0095-4616},

number = 1,

volume = 39,

place = {United States},

year = {1999},

month = {1}

}