Processes with jumps in Finance

We will addresses two particular investment problems that share, in particular, the following feature: the processes that model the uncertainty exhibit discontinuities in their sample paths. These discontinuities — or jumps — are driven by jump processes, hereby modelled by Poisson processes. Above all, the problems addressed are all problems that fall in the category of optimal stopping problems: choose a time to take a given action (in particular, the time to decide to invest, as here we consider investment problems) in order to maximize an expected payoff.

In the first problem, we assume that a firm is currently receiving a profit stream from an already operational project, and has the option to invest in a new project, with impact in its profitability. Moreover, we assume that there are two sources of uncertainty that influence the firm’s decision about when to invest: the random fluctuations of the revenue (depending on the random demand) and the changing investment cost. And, as already mentioned, both processes exhibit discontinuities in their sample paths.

The second problem is developed in the scope of technology adoption. The technology innovation is, by far, an example of a discontinuous process: the technological level does not increase in a steady pace, but instead from now and then some improvement or breakthrough happens. Thus it is natural to assume that technology innovations are driven by jump processes. As such, in this problem we consider a firm that is producing in a declining market, but with the option to undertake an innovation investment and thereby to replace the old product by a new one, paying a constant sunk cost. As the first product is a well established one, its price is deterministic. Upon investment in the second product, the price may fluctuate, according to a geometric Brownian motion. The decision is when to invest in a new product.