Backward stochastic differential equation

A backward stochastic differential equation (BSDE) is a stochastic differential equation with a terminal condition in which the solution is required to be adapted with respect to an underlying filtration. BSDEs naturally arise in various applications such as stochastic control, mathematical finance, and nonlinear Feynman-Kac formulae. ^[1]

Background

Backward stochastic differential equations were introduced by Jean-Michel Bismut in 1973 in the linear case ^[2] and by Étienne Pardoux and Shige Peng in 1990 in the nonlinear case. ^[3]

Mathematical framework

Fix a terminal time ${\displaystyle T>0}$ and a probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$. Let ${\displaystyle (B_{t})_{t\in [0,T]}}$ be a Brownian motion with natural filtration ${\displaystyle ({\mathcal {F}}_{t})_{t\in [0,T]}}$. A backward stochastic differential equation is an integral equation of the type

${\displaystyle Y_{t}=\xi +\int _{t}^{T}f(s,Y_{s},Z_{s})\mathrm {d} s-\int _{t}^{T}Z_{s}\mathrm {d} B_{s},\quad t\in [0,T],}$

(1)

where ${\displaystyle f:[0,T]\times \mathbb {R} \times \mathbb {R} \to \mathbb {R} }$ is called the generator of the BSDE, the terminal condition ${\displaystyle \xi }$ is an ${\displaystyle {\mathcal {F}}_{T}}$-measurable random variable, and the solution ${\displaystyle (Y_{t},Z_{t})_{t\in [0,T]}}$ consists of stochastic processes ${\displaystyle (Y_{t})_{t\in [0,T]}}$ and ${\displaystyle (Z_{t})_{t\in [0,T]}}$ which are adapted to the filtration ${\displaystyle ({\mathcal {F}}_{t})_{t\in [0,T]}}$.

Example

In the case ${\displaystyle f\equiv 0}$, the BSDE ( 1 ) reduces to

${\displaystyle Y_{t}=\xi -\int _{t}^{T}Z_{s}\mathrm {d} B_{s},\quad t\in [0,T].}$

(2)

If ${\displaystyle \xi \in L^{2}(\Omega ,\mathbb {P} )}$, then it follows from the martingale representation theorem, that there exists a unique stochastic process ${\displaystyle (Z_{t})_{t\in [0,T]}}$ such that ${\displaystyle Y_{t}=\mathbb {E} [\xi |{\mathcal {F}}_{t}]}$ and ${\displaystyle Z_{t}}$ satisfy the BSDE ( 2 ).

See also

• Martingale representation theorem
• Stochastic control
• Stochastic differential equation

References

1. Ma, Jin; Yong, Jiongmin (2007). Forward-Backward Stochastic Differential Equations and their Applications. Lecture Notes in Mathematics. Vol. 1702. Springer Berlin, Heidelberg. doi:10.1007/978-3-540-48831-6. ISBN   978-3-540-65960-0.
2. Bismut, Jean-Michel (1973). "Conjugate convex functions in optimal stochastic control". Journal of Mathematical Analysis and Applications. 44 (2): 384–404. doi:10.1016/0022-247X(73)90066-8.
3. Pardoux, Etienne; Peng, Shi Ge (1990). "Adapted solution of a backward stochastic differential equation". Systems & Control Letters. 14: 55–61. doi:10.1016/0167-6911(90)90082-6.

Further reading

• Pardoux, Etienne; Rӑşcanu, Aurel (2014). Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic modeling and applied probability. Springer International Publishing Switzerland.
• Zhang, Jianfeng (2017). Backward stochastic differential equations. Probability theory and stochastic modeling. Springer New York, NY.