  A backward stochastic differential equation (BSDE) is an SDE of the form
$-dY_t = f(t,Y_t,Z_t)dt - Z_t^*dW_t;\ Y_T = \xi$. The subject of BSDEs has seen
extensive attention since their introduction in the linear case by Bismut
(1973) and in the general case by Pardoux and Peng (1990). In contrast with
deterministic differential equations, it is not enough to simply reverse the
direction of time and treat the terminal condition as an initial condition, as
we would then run into problems with adaptedness. Intuitively, our "knowledge"
at time $t$ consists only of what has happened at all times $s \in [0,t]$, and
we cannot reverse the direction of time whilst keeping this true.
  The layout of this essay is as follows: In Section 1 we introduce BSDEs and
go over the basic results of BSDE theory, including two major theorems: the
existence and uniqueness of solutions and the comparison theorem. We also
introduce linear BSDEs and the notion of supersolutions of a BSDE. In Section 2
we set up the financial framework in which we will price European contingent
claims, and prove a result about the fair price of such claims in a dynamically
complete market. In Section 3 we extend the theory of linear BSDEs to include
concave BSDEs, and apply this to pricing claims in more complicated market
models. In Section 4 we take a look at utility maximisation problems, and see
how utilising BSDE theory allows for a relatively simple and neat solution in
certain cases.
