The purpose of the master thesis is to look at the classical article «Backward Stochastic Differential Equations in Finance» by Karoui, Peng and Quenez. Here, theory about Backward Stochastic Differential Equations (BSDEs) is developed with Brownian motion as noise. In the master thesis, some of the results are generalized to include jumps. Mainly, this is done by adapting the techniques in Karoui et. al.
Topics that is investigated are existence and uniqueness, linear BSDEs, the comparison theorem, dependence on a parameter and Malliavin differentiability. The last chapter also looks at the combined derivative in a parameter and in the Malliavin sense.
The difficulty of jumps is that one usually get more terms to handle, that the Lévy measure is not in general finite and that we may need to differentiate in an operaor.