On future drawdowns of Lévy processes

Abstract

For a given Lévy process X=(Xt)t∈R+ and for fixed s∈R+∪{∞} and t∈R+ we analyse the future drawdown extremes that are defined as follows: D¯t,s∗=sup0≤u≤tinfu≤w<t+s(Xw−Xu),D¯t,s∗=inf0≤u≤tinfu≤w<t+s(Xw−Xu). The path-functionals D¯t,s∗ and D¯t,s∗ are of interest in various areas of application, including financial mathematics and queueing theory. In the case that X has a strictly positive mean, we find the exact asymptotic decay as x→∞ of the tail probabilities P(D¯t∗<x) and P(D¯t∗<x) of D¯t∗=lims→∞D¯t,s∗ and D¯t∗=lims→∞D¯t,s∗ both when the jumps satisfy the Cramér assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the Lévy process X are of single sign and X is not subordinator, we identify the one-dimensional distributions in terms of the scale function of X. By way of example, we derive explicit results for the Black–Scholes–Samuelson model.