This article proposes a model for stock prices which incorporates shot-noise effects. This means, that sudden jumps in the stock price are allowed, but their effect may decline as time passes by. Our model is general enough to capture arbitrary effects of this type. Generalizing previous approaches to shot-noise we in particular allow the decay to be stochastic. This model describes an incomple...

We study Fourier multipliers which result from modulating jumps of Lévy processes. Using the theory of martingale transforms we prove that these operators are bounded in Lp(Rd) for 1 < p < ∞ and we obtain the same explicit bound for their norm as the one known for the second order Riesz transforms.

We prove existence of weak martingale solutions satisfying an almost sure version of the energy inequality and which constitute a (almost sure) Markov process.

Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f , the backward Kolmogorov equation gives a condition for f(t,X) to be a local martingale. We generalize the backward equation in two main ways. First, it is extended to non-differentiable functions. Second, the process X is not required to...