Jump processes

Although historically models in mathematical finance were based on Brownian motion and thus are models with continuous price paths, jump processes play now a key role across all areas of finance (see e.g. [5]). One reason for this move into a new class of processes is that because of their distributional properties diffusions in many cases cannot provide a realistic picture of empirically observed facts. Another reason is the enormous progress which has been made in understanding and handling jump processes due to the development of semimartingale theory on one side and of computational power on the other side. The simplest jump process is a process with just one jump. Let T be a random time – actually a stopping time with respect to an information structure given by a filtration (Ft)t≥0 – then Xt = 1l{T≤t} (t ≥ 0) (1)