Fix a terminal time 
and a probability space 
. Let ![{\displaystyle (B_{t})_{t\in [0,T]}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5df3acd3171fe6d3043cf659824788ec54a58c1b)
be a Brownian motion with natural filtration ![{\displaystyle ({\mathcal {F}}_{t})_{t\in [0,T]}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a82e12fc426e0596ce14ce461967be662ade6308)
. 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],}](//wikimedia.org/api/rest_v1/media/math/render/svg/17180d83e4f78ae7a196811558357512998fd870) |
|
(1) |
where ![{\displaystyle f:[0,T]\times \mathbb {R} \times \mathbb {R} \to \mathbb {R} }](//wikimedia.org/api/rest_v1/media/math/render/svg/cb5a07d360c3ab1e651a84d55d3ed4eac4e52574)
is called the generator of the BSDE, the terminal condition 
is an 
-measurable random variable, and the solution ![{\displaystyle (Y_{t},Z_{t})_{t\in [0,T]}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7657e9e989f222e34b2deb255a4865c7a70c8f59)
consists of stochastic processes ![{\displaystyle (Y_{t})_{t\in [0,T]}}](//wikimedia.org/api/rest_v1/media/math/render/svg/000976c421b7bbe36cc923abcd80451ba72e9573)
and ![{\displaystyle (Z_{t})_{t\in [0,T]}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8d1e34758f442803ae3d9a28f0be1441d12622b7)
which are adapted to the filtration .
Example
In the case 
, the BSDE (1) reduces to
-
![{\displaystyle Y_{t}=\xi -\int _{t}^{T}Z_{s}\mathrm {d} B_{s},\quad t\in [0,T].}](//wikimedia.org/api/rest_v1/media/math/render/svg/54ccbb034bf7321f8a155df3736e962d8ddd2c3c) |
|
(2) |
If 
, then it follows from the martingale representation theorem, that there exists a unique stochastic process such that ![{\displaystyle Y_{t}=\mathbb {E} [\xi |{\mathcal {F}}_{t}]}](//wikimedia.org/api/rest_v1/media/math/render/svg/22a566b1a0bce3de307338be1add02b9b37d5324)
and 
satisfy the BSDE (2).
![{\displaystyle (B_{t})_{t\in [0,T]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5df3acd3171fe6d3043cf659824788ec54a58c1b)
![{\displaystyle ({\mathcal {F}}_{t})_{t\in [0,T]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/a82e12fc426e0596ce14ce461967be662ade6308)
![{\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],}](http://wikimedia.org/api/rest_v1/media/math/render/svg/17180d83e4f78ae7a196811558357512998fd870)
![{\displaystyle f:[0,T]\times \mathbb {R} \times \mathbb {R} \to \mathbb {R} }](http://wikimedia.org/api/rest_v1/media/math/render/svg/cb5a07d360c3ab1e651a84d55d3ed4eac4e52574)
![{\displaystyle (Y_{t},Z_{t})_{t\in [0,T]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7657e9e989f222e34b2deb255a4865c7a70c8f59)
![{\displaystyle (Y_{t})_{t\in [0,T]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/000976c421b7bbe36cc923abcd80451ba72e9573)
![{\displaystyle (Z_{t})_{t\in [0,T]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/8d1e34758f442803ae3d9a28f0be1441d12622b7)
![{\displaystyle Y_{t}=\xi -\int _{t}^{T}Z_{s}\mathrm {d} B_{s},\quad t\in [0,T].}](http://wikimedia.org/api/rest_v1/media/math/render/svg/54ccbb034bf7321f8a155df3736e962d8ddd2c3c)
![{\displaystyle Y_{t}=\mathbb {E} [\xi |{\mathcal {F}}_{t}]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/22a566b1a0bce3de307338be1add02b9b37d5324)
