In this note we consider a branching Brownian motion (BBM) on in which a particle at spatial position y splits into two at rate [beta]y2, where [beta]0 is a constant. This is a critical breeding rate for BBM in the sense that the expected population size blows up in finite time while the...

We study a class of self-similar jump type SDEs driven by Hölder continuous drift and noise coefficients. Using the Lamperti transformation for positive self-similar Markov processes we obtain a necessary and sufficient condition for almost sure extinction in finite time. We then show that in...

In Gapeev and Kühn (2005) [8], the Dynkin game corresponding to perpetual convertible bonds was considered, when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Lévy process. We...

We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type , where [nu] is the density of the stable Lévy measure and [gamma] is a positive parameter which depends on its characteristics. These processes were introduced in [M. E. Caballero,...

We consider the Russian option introduced by Shepp and Shiryayev (Ann. Appl. Probab. 3 (1993) 631, Theory Probab. Appl. 39 (1995) 103) but with finite expiry and show that its space-time value function characterizes the unique solution to a free boundary problem. Further, using a method of...

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of Lévy processes that is based on the Wiener–Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology....

In the spirit of Duquesne and Winkel (2007) and Berestycki et al. (2011), we show that supercritical continuous-state branching process with a general branching mechanism and general immigration mechanism is equivalent in law to a continuous-time Galton–Watson process with immigration (with...