Continuous-Time Stochastic Processes: Discontinuous Paths
Thus far, the continuous-time stochastic processes that we have presented for stock price and interest rate modelling have all had continuous paths. Hence, if we monitor such a process closely, we cannot be caught by surprise by an exceptionally big move. However, there are many reasons such as catastrophes or surprising news (unexpected political changes, an economic scandal, and so forth) that can make a real-life process jump. More so, due to discreteness of measurement, it can also be argued that processes modelling the real world should allow for discontinuities in their paths, at least. Finally, in the insurance business the evolution of the absolute number of insurance cases is a fundamental issue that can only be modelled by a counting process. This might be the most natural example of a continuous-time stochastic process with piecewise constant paths that only increases by jumps.