On future drawdowns of Levy processes

For a given Levy process X = ( X t ) t 2 R + and for xed s 2 R + [f1g and t 2 R + we analyse the future drawdown extremes that are de ned as follows: The path-functionals D t;s and D t;s are of interest in various areas of application, including nancial mathematics and queueing theory. In the case that X has a strictly positive mean, we nd the exact asymptotic decay as x ! 1 of the tail probabilities P ( D t < x ) and P ( D t < x ) of D t = lim s !1 D t;s and D t = lim s !1 D t;s both when the jumps satisfy the Cram er assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the L evy 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