On the expected discounted penalty function for a perturbed risk process driven by a subordinator

A note on scale functions and the time value of ruin for Lévy insurance risk processes

“…Many authors (including the aforementioned) also prefer to talk about perturbation of the classical Cramér-Lundberg process by another Lévy process, a typical and general case at hand being found in the work of Huzak et al (2004) and Morales and Olivares (2008). Since the sum of two independent Lévy processes is again a Lévy process, the concept of perturbation is contained in the above setting by decomposing Π = Π (1) +Π (2) , where Π (1) corresponds to the Lévy measure of the Cramér-Lundberg process and Π (2) is that of the perturbation process.…”

“…This expected discounted penalty, by now known as the Gerber-Shiu function, has been frequently and recursively studied in settings of increasing generality as well as being the named theme of two international workshops in 2006 and 2008. Although far from exhaustive on account of the sheer volume of relevant literature, a list of key papers which pertains to generalizations of the Cramér-Lundberg process includes for example Dickson (1992Dickson ( , 1993, Shiu (1997, 1998), Gerber and Landry (1998), Lin and Willmot (1999), Yang and Zhang (2001), Cai and Dickson (2002), Tsai and Willmot (2002), Cai (2004), Garrido and Morales (2006), Morales (2007) and Morales and Olivares (2008). The general setting which fits all of these papers is to model the risk process as having stationary and independent increments with no positive jumps.…”

A Note on Scale Functions and the Time Value of Ruin for Lévy Insurance Risk Processes

“…The case that Π(0, ∞) = ∞ but (0,1) xΠ(dx) < ∞ is the case of a bounded variation jump process and therefore X is necessarily the difference of a linear drift, also with rate a + (0,∞) xΠ(dx), and an infinite activity subordinator plus an independent Brownian motion with volatility σ. This model was the subject of the recent work of, for example, Morales (2007). The case that Π(0, ∞) = ∞ but (0,1) xΠ(dx) = ∞ is the case of an unbounded variation jump part.…”

“…This expected discounted penalty, by now known as the Gerber-Shiu function, has been frequently and recursively studied in settings of increasing generality as well as being the named theme of two international workshops in 2006 and 2008. Although far from exhaustive on account of the sheer volume of relevant literature, a list of key papers which pertains to generalizations of the Cramér-Lundberg process includes for example Dickson (1992Dickson ( , 1993; Shiu (1997, 1998); Gerber and Landry (1998); Lin and Willmot (1999); Yang and Zhang (2001); Cai and Dickson (2002); Tsai and Willmot (2002); Cai (2004); Garrido and Morales (2006); Morales (2007); Morales and Olivares (2008). The general setting which fits all of these papers is to model the risk process as having stationary and independent increments with no positive jumps.…”

A numerical method for the expected penalty–reward function in a Markov-modulated jump–diffusion process

“…Generalizations of the model are treated in Li and Garrido (2005), Sarkar and Sen (2005), and Morales (2007), whereas Ren (2005) gives explicit ∆ t and volatility κ(U t ), that possibly depends on the amount invested U t , driven by a Wiener process W I t independent of the risk process R t dU t = (∆ t dt + κ(U t )dW…”