When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic PDE in the maturity and strike variables. By contrast, when the underlying asset is described by a discontinuous semimartingale, call prices solve a partial integro-differential equation (PIDE), containing a non-local integral term. We show that the two classes of equations sh...

Motivated by models of queues with server vacations, we consider a Le ́vy process modified to have random jumps at arbitrary stopping times. The extra jumps can counteract a drift in the Le ́vy process so that the overall Le ́vy process with secondary jump input, can have a proper limiting distribution. For example, the workload process in an M/G/1 queue with a server vacation each time the server...

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Lévy-type martingale subject to default. This class of models allows for local volatility, local default intensity and a locally dependent Lévy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci and Riga [SIAM J. Financial Math. 4 (2013) 265–296], we...

We first obtain exponential inequalities for martingales. Let (Xk)(1 ≤ k ≤ n) be a sequence of martingale differences relative to a filtration (Fk), and set Sn = X1 + ... + Xn. We prove that if for some δ > 0, Q ≥ 1, K > 0 and all k, a.s. E[ek Q |Fk−1] ≤ K, then for some constant c > 0 (depending only on δ,Q and K) and all x > 0, P [|Sn| > nx] ≤ 2e , where c(x) = cx if x ∈]0, 1], and c(x) = cx ...

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. D time units later the timer expires, and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the time...

The theory of noncommutative martingale inequalities has been rapidly developed since the establishment of the noncommutative Burkholder-Gundy inequalities in [12]. Many of the classical martingale inequalities has been transferred to the noncommutative setting. These include, in particular, the Doob maximal inequality in [3], the Burkholder/Rosenthal inequality in [5], [8], several weak type (...